Fullerenes -- Further fullerenes

Further fullerenes

Further fullerenes

In [6] and [5] huge tables with numbers of isomers and of chiral isomers of substituted fullerenes from C₂₀ to C₇₀ are computed, but the corresponding cycle indices are not given. In [9] an even greater list of fullerene structures can be found. The cycle indices of the symmetry groups of these fullerenes can be computed by determining the permutation representation of the symmetry groups of these fullerenes acting on their sets of vertices, edges or faces. Here we want to present a list of the 3-dimensional cycle indices for these fullerenes.

The point group of C₂₄ is D_6d which leads to the following cycle indices.

Z₃(R(C₂₄))= ^._. 1

12
( 2 e₆⁶ f₁² f₆² v₆⁴+ 2 e₃¹² f₁² f₃⁴ v₃⁸+ e₂¹⁸ f₁² f₂⁶ v₂¹²+ 6 e₁² e₂¹⁷ f₂⁷ v₂¹²+ e₁³⁶ f₁¹⁴ v₁²⁴)^._.

Z₃(S(C₂₄))= ^._. 1

2
Z₃(R(C₂₄)) + ^._. 1

24
( 4 e₁₂³ f₂ f₁₂ v₁₂²+ 2 e₄⁹ f₂ f₄³ v₄⁶ + 6 e₁⁴ e₂¹⁶ f₁⁴ f₂⁵ v₁⁴ v₂¹⁰ ).^._.

Since the C₂₆ has a symmetry group of the form D_3h we have

Z₃(R(C₂₆))= ^._. 1

6
( 3 e₁ e₂¹⁹ f₁ f₂⁷ v₂¹³+ 2 e₃¹³ f₃⁵ v₁² v₃⁸ + e₁³⁹ f₁¹⁵ v₁²⁶ ) ^._.

Z₃(S(C₂₆))= ^._. 1

2
Z₃(R(C₂₆)) + ^._. 1

12
( 2 e₃ e₆⁶ f₃ f₆² v₂ v₃² v₆³ + 3 e₁⁵ e₂¹⁷ f₁⁵ f₂⁵ v₁⁴ v₂¹¹ + e₁³ e₂¹⁸ f₁³ f₂⁶ v₁⁶ v₂¹⁰).^._.

The fullerene C₂₈ is of tetrahedral symmetry T_d so the cycle indices can be computed as

Z₃(R(C₂₈))= ^._. 1

12
( 8 e₃¹⁴ f₁ f₃⁵ v₁ v₃⁹ + 3 e₁² e₂²⁰ f₂⁸ v₂¹⁴ + e₁⁴² f₁¹⁶ v₁²⁸ )^._.

Z₃(S(C₂₈))= ^._. 1

2
Z₃(R(C₂₈)) + ^._. 1

24
( 6 e₂ e₄¹⁰ f₄⁴ v₄⁷ + 6 e₁⁴ e₂¹⁹ f₁⁴ f₂⁶ v₁⁶ v₂¹¹ ).^._.

The cycle index of R(C₂₈) acting on its set of vertices can be found in [4] as well.

The C₃₀ has D_5h symmetry and the cycle indices for the action on the sets of edges, faces and vertices are

Z₃(R(C₃₀))= ^._. 1

10
( 4 e₅⁹ f₁² f₅³ v₅⁶ + 5 e₁ e₂²² f₁ f₂⁸ v₂¹⁵ + e₁⁴⁵ f₁¹⁷ v₁³⁰)^._.

Z₃(S(C₃₀))= ^._. 1

2
Z₃(R(C₃₀)) + ^._. 1

20
( 4 e₅ e₁₀⁴ f₂ f₅ f₁₀ v₁₀³ + 5 e₁⁵ e₂²⁰ f₁⁵ f₂⁶ v₁⁶ v₂¹² + e₁⁵ e₂²⁰ f₁⁵ f₂⁶ v₂¹⁵ ). ^._.

The symmetry group of C₃₂ is D₃, so it consists only of rotations and we have

Z₃(R(C₃₂))= ^._. 1

6
( 2 e₃¹⁶ f₃⁶ v₁² v₃¹⁰ + 3 e₁² e₂²³ f₂⁹ v₂¹⁶ + e₁⁴⁸ f₁¹⁸ v₁³² ). ^._.

Since C₃₄ has C_3v symmetry the cycle indices are

Z₃(R(C₃₄))= ^._. 1

3
( 2 e₃¹⁷ f₁ f₃⁶ v₁ v₃¹¹ + e₁⁵¹ f₁¹⁹ v₁³⁴)^._.

Z₃(S(C₃₄))= ^._. 1

2
(Z₃(R(C₃₄)) + e₁⁵ e₂²³ f₁⁵ f₂⁷ v₁⁴ v₂¹⁵ ). ^._.

The point group of C₃₆ is D_6h, from which the following cycle indices can be computed.

Z₃(R(C₃₆))= ^._. 1

12
( 2 e₆⁹ f₂ f₆³ v₆⁶ + 2 e₃¹⁸ f₁² f₃⁶ v₃¹² + 4 e₂²⁷ f₁² f₂⁹ v₂¹⁸ + 3 e₁² e₂²⁶ f₂¹⁰ v₂¹⁸ + e₁⁵⁴ f₁²⁰ v₁³⁶ ) ^._.

Z₃(S(C₃₆))= ^._. 1

2
Z₃(R(C₃₆)) + ^._. 1

24
( 2 e₃² e₆⁸ f₂ f₃² f₆² v₆⁶ + e₂²⁷ f₂¹⁰ v₂¹⁸ + 2 e₆⁹ f₁² f₆³ v₆⁶ + 3 e₁⁴ e₂²⁵ f₁⁴ f₂⁸ v₁⁸ v₂¹⁴ + 3 e₁⁶ e₂²⁴ f₁⁶ f₂⁷ v₁⁴ v₂¹⁶ + e₁⁶ e₂²⁴ f₁⁶ f₂⁷ v₂¹⁸ ). ^._.

C₃₈ is of C_3v symmetry so we have

Z₃(R(C₃₈))= ^._. 1

3
( 2 e₃¹⁹ f₃⁷ v₁² v₃¹² + e₁⁵⁷ f₁²¹ v₁³⁸) ^._.

Z₃(S(C₃₈))= ^._. 1

6
( 2 e₃¹⁹ f₃⁷ v₁² v₃¹² + 3 e₁⁵ e₂²⁶ f₁⁵ f₂⁸ v₁⁶ v₂¹⁶ + e₁⁵⁷ f₁²¹ v₁³⁸). ^._.

For the C₄₀ two possible symmetry groups are given, namely T_d and D_5d. In the first case the cycle indices are

Z₃(R(C₄₀))= ^._. 1

12
( 3 e₂³⁰ f₁² f₂¹⁰ v₂²⁰ + 8 e₃²⁰ f₁ f₃⁷ v₁ v₃¹³ + e₁⁶⁰ f₁²² v₁⁴⁰ )^._.

Z₃(S(C₄₀))= ^._. 1

2
Z₃(R(C₄₀)) + ^._. 1

24
( 6 e₄¹⁵ f₂ f₄⁵ v₄¹⁰ + 6 e₁⁶ e₂²⁷ f₁⁶ f₂⁸ v₁⁴ v₂¹⁸ ).^._.

For the symmetry group D_5d we compute

Z₃(R(C₄₀))= ^._. 1

10
( 4 e₅¹² f₁² f₅⁴ v₅⁸ + 5 e₁² e₂²⁹ f₂¹¹ v₂²⁰ + e₁⁶⁰ f₁²² v₁⁴⁰ )^._.

Z₃(S(C₄₀))= ^._. 1

2
Z₃(R(C₄₀)) + ^._. 1

20
( 4 e₁₀⁶ f₂ f₁₀² v₁₀⁴ + e₂³⁰ f₂¹¹ v₂²⁰ + 5 e₁⁶ e₂²⁷ f₁⁶ f₂⁸ v₁⁴ v₂¹⁸ ).^._.

Since the point group of C₄₂ is D₃ we have

Z₃(R(C₄₂))= ^._. 1

6
( 2 e₃²¹ f₁² f₃⁷ v₃¹⁴ + 3 e₁ e₂³¹ f₁ f₂¹¹ v₂²¹ + e₁⁶³ f₁²³ v₁⁴² ). ^._.

For the D_3h symmetry group of C₄₄ we can compute

Z₃(R(C₄₄))= ^._. 1

6
( 2 e₃²² f₃⁸ v₁² v₃¹⁴ + 3 e₁² e₂³² f₂¹² v₂²² + e₁⁶⁶ f₁²⁴ v₁⁴⁴) ^._.

Z₃(S(C₄₄))= ^._. 1

2
Z₃(R(C₄₄)) + ^._. 1

12
( 2 e₃² e₆¹⁰ f₃² f₆³ v₂ v₃² v₆⁶ + 4 e₁⁶ e₂³⁰ f₁⁶ f₂⁹ v₁⁶ v₂¹⁹ ). ^._.

A second form of C₄₄ is chiral and has T as its symmetry group which gives the following cycle index:

Z₃(R(C₄₄))= ^._. 1

12
( 3 e₁²e₂³² f₂¹² v₂²² + 8 e₃²² f₃⁸ v₁²v₃¹⁴ + e₁⁶⁶ f₁²⁴ v₁⁴⁴ ). ^._.

The symmetry group of C₄₆ is C₃ so we compute

Z₃(R(C₄₆))= ^._. 1

3
( 2 e₃²³ f₁ f₃⁸ v₁ v₃¹⁵ + e₁⁶⁹ f₁²⁵ v₁⁴⁶ ). ^._.

The C₄₈ has D₃ as its symmetry group, so we have

Z₃(R(C₄₈))= ^._. 1

6
( 2 e₃²⁴ f₁²f₃⁸ v₃¹⁶ + 3 e₁² e₂³⁵ f₂¹³ v₂²⁴ + e₁⁷² f₁²⁶ v₁⁴⁸ ). ^._.

For the D_5h symmetry of C₅₀ we derive

Z₃(R(C₅₀))= ^._. 1

10
( 4 e₅¹⁵ f₁² f₅⁵ v₅¹⁰ + 5 e₁ e₂³⁷ f₁ f₂¹³ v₂²⁵ + e₁⁷⁵ f₁²⁷ v₁⁵⁰ )^._.

Z₃(S(C₅₀))= ^._. 1

2
Z₃(R(C₅₀)) + ^._. 1

20
( 4 e₅ e₁₀⁷ f₂ f₅ f₁₀² v₅² v₁₀⁴ + 5 e₁⁷ e₂³⁴ f₁⁷ f₂¹⁰ v₁⁴ v₂²³ + e₁⁵ e₂³⁵ f₁⁵ f₂¹¹ v₁¹⁰ v₂²⁰ ) .^._.

The symmetry group of C₅₂ is T, which consists only of rotational symmetries.

Z₃(R(C₅₂))= ^._. 1

12
( 3 e₁²e₂³⁸ f₂¹⁴ v₂²⁶ + 8 e₃²⁶ f₁ f₃⁹ v₁ v₃¹⁷ + e₁⁷⁸ f₁²⁸ v₁⁵² ). ^._.

C₅₄ has point group D₃, so we compute

Z₃(R(C₅₄))= ^._. 1

6
( 3 e₁ e₂⁴⁰ f₁ f₂¹⁴ v₂²⁷ + 2 e₃²⁷ f₁² f₃⁹ v₃¹⁸ + e₁⁸¹ f₁²⁹ v₁⁵⁴ ). ^._.

For the D_3d symmetry group of C₅₆ the cycle indices are

Z₃(R(C₅₆))= ^._. 1

6
( 2 e₃²⁸ f₃¹⁰ v₁² v₃¹⁸ + 3 e₁² e₂⁴¹ f₂¹⁵ v₂²⁸ + e₁⁸⁴ f₁³⁰ v₁⁵⁶ )^._.

Z₃(S(C₅₆))= ^._. 1

2
Z₃(R(C₅₆)) + ^._. 1

12
( 2 e₆¹⁴ f₆⁵ v₂ v₆⁹ + 3 e₁⁶ e₂³⁹ f₁⁶ f₂¹² v₁⁸ v₂²⁴ + e₂⁴² f₂¹⁵ v₂²⁸ ) .^._.

The symmetry group of C₅₈ is C_3v so we have

Z₃(R(C₅₈))= ^._. 1

3
( 2 e₃²⁹ f₁ f₃¹⁰ v₁ v₃¹⁹ + e₁⁸⁷ f₁³¹ v₁⁵⁸)^._.

Z₃(S(C₅₈))= ^._. 1

2
(Z₃(R(C₅₈)) + e₁⁷ e₂⁴⁰ f₁⁷ f₂¹² v₁⁶ v₂²⁶ ). ^._.

C₈₀ is the next fullerene with I_h symmetry. Its cycle index is

Z₃(R(C₈₀))=^._. 1

60
( 24 e₅²⁴f₁²f₅⁸v₅¹⁶+ 20 e₃⁴⁰f₃¹⁴v₁² v₃²⁶+ 15 e₂⁶⁰f₁² f₂²⁰v₂⁴⁰+ e₁¹²⁰f₁⁴²v₁⁸⁰) ^._.

Z₃(S(C₈₀))=^._. 1

2
Z₃(R(C₈₀)) + ^._. 1

120
( e₂⁶⁰f₂²¹v₂⁴⁰ + 20e₆²⁰f₆⁷v₂v₆¹³ + e₁⁸e₂⁵⁶f₁⁸f₂¹⁷15v₁⁸v₂³⁶ + 24e₁₀¹²f₂f₁₀⁴v₁₀⁸).^._.

The first fullerene with symmetry group I is the C₁₄₀. Its cycle index is

Z₃(R(C₁₄₀))=^._. 1

60
( 24 e₅⁴²f₁²f₅¹⁴v₅²⁸+ 20 e₃⁷⁰f₃²⁴v₁²v₃⁴⁶+ 15 e₁²e₂¹⁰⁴f₂³⁶v₂⁷⁰+ e₁²¹⁰f₁⁷²v₁¹⁴⁰) ^._.

We could go on listing the cycle indices of many more fullerenes. In many cases it is possible to arrange the v vertices of C_v in several different ways leading to different symmetry groups and to different cycle indices. For instance for the fullerene C₇₈ there are 4 possible isomers given in [9]. So from chemical properties we first have to determine the actual shape of the molecule. In [11] it is shown that C₇₆ is of D₂ symmetry and not of T_d symmetry which would be possible as well.

harald.fripertinger "at" uni-graz.at, May 10, 2016

Further fullerenes