Fullerenes -- The Leapfrog principle

The Leapfrog principle

The Leapfrog principle

Finally let me describe the so called leapfrog principle, which was invented by Fowler [6][9]. It can be used for the construction of a fullerene C_3v from a fullerene C_v having the same or even a bigger symmetry group than C_v. In general the leapfrog transformation can be defined for any polyhedron P as capping all the faces of P and switching to the dual of the result. The leapfrog L(P) is always a trivalent polyhedron having 2e_P vertices, v_P+f_P faces and 3e_P edges, where v_P, f_P and e_P are the numbers of vertices, faces and edges of the parent P. When starting from a trivalent parent, the leapfrog has always 3v_P vertices. If we are starting with a C_v cluster with icosahedral symmetry all the new clusters will be of the same symmetry, since this is the biggest symmetry group in 3-dimensional space.

Knowing the 3-dimensional cycle indices of both S and R acting on the sets of vertices, edges and faces of a fullerene C_v and using some facts from representation theory it is possible to determine the 3-dimensional cycle index for the action on C_3v. More general for trivalent polyhedra or for deltahedra the 3-dimensional cycle indices of S and the rotational subgroup R of the leapfrog can be computed from the 3-dimensional cycle indices of S and R on the parent.

From [7][8] we know that the permutation characters for the actions on the sets of faces, vertices and edges of the leapfrog L can be computed from the permutation characters of the parent P by

χ_f,L(g)=χ_v,P(g)+χ_f,P(g) ^._.

χ_v,L(g)=χ_e,P(g)+χ_f,P(g)+χ_v,P(g) χ_ε(g)-(1+ χ_ε(g)) ^._.

χ_e,L(g)=χ_f,L(g)χ_T(g)-(χ_T(g)+χ_R(g)) ^._.

So we have expressed the permutation characters for the action on the components of the leapfrog in the permutation characters of the components of the parent and in χ₀, χ_ε and χ_T, where χ₀ is the totally symmetric character given by χ₀(g)=1 for all g∈ G, χ_ε is the antisymmetric character which takes the values +1 for all proper rotations and -1 for all improper rotations. Furthermore χ_T is the translational character, and χ_R is given by χ_R=χ_Tχ_ε. Since we usually know the cycle indices both for the group of all symmetries and the subgroup of all rotational symmetries we can assume that the antisymmetric character is known. Only for computing χ_e,L we furthermore have to compute the translational character. In some cases however we already get all the necessary information from the 3-dimensional cycle index for the action on the parent P. For instance in the case that P is a trivalent polyhedron (see [5]), we have

χ_e,P(g)=χ_f,P(g)χ_T(g)-(χ_T(g)+χ_R(g))^._.

such that χ_e,L can be written as

χ_e,L(g)=χ_v,P(g)χ_T(g)+χ_e,P(g)^._.

Furthermore there is a formula that

χ_v,P(g)χ_T(g) = (χ_f,P(g)-χ₀(g))χ_ε(g)+ (χ_v,P(g)-χ₀(g)) +χ_e,P(g). ^._.

So finally we get

χ_e,L(g)=2χ_e,P(g)+(χ_f,P(g)-χ₀(g)) χ_ε(g)+ (χ_v,P(g)-χ₀(g)). ^._.

Similar formulae hold when P is a deltahedron.

Using some standard methods we can compute the cycle type of g∈ G from the permutation character and vice versa by

a_k(g)=∑_{d |
k}μ(k/d)a₁(g^d) a₁(g^k)=∑_{d | k} a_d(g). ^._.

In order to give an example we realize that C₆₀ is the leapfrog of C₂₀ which is of . of icosahedral symmetry I_h as well. In [10] you can find the following 3-dimensional cycle indices for the actions on the components of C₂₀ (which corresponds to the pentagonal dodecahedron).

Z₃(I,C₂₀)=(1)/(60)( v₁²⁰e₁³⁰f₁¹² + 20v₁²v₃⁶e₃¹⁰f₃⁴ + 15v₂¹⁰e₁²e₂¹⁴f₂⁶ + 24v₅⁴e₅⁶f₁²f₅²) ^._.

Z₃(I_h,C₂₀)= (1)/(2) Z₃(I,C₂₀) + (1)/(120) ( v₂¹⁰e₂¹⁵f₂⁶ + 20v₂v₆³e₆⁵f₆² + 15v₁⁴v₂⁸e₁⁴e₂¹³f₁⁴f₂⁴ + 24v₁₀²e₁₀³f₂f₁₀).^._.

Applying the formulae above we get the 3-dimensional cycle indices of section *. Iterating the leapfrog method once more we derive the 3-dimensional cycle index of C₁₈₀ as

Z₃(I,C₁₈₀)=(1)/(60)( v₁¹⁸⁰e₁²⁷⁰f₁⁹²+ 20 v₃⁶⁰e₃⁹⁰f₁²f₃³⁰+ 15 v₂⁹⁰e₁²e₂¹³⁴f₂⁴⁶+ 24 v₅³⁶e₅⁵⁴f₁²f₅¹⁸ ) ^._.

and

Z₃(I_h,C₁₈₀)= (1)/(2) Z₃(I,C₁₈₀) + (1)/(120) ( v₂⁹⁰e₂¹³⁵f₂⁴⁶+ 20 v₆³⁰e₆⁴⁵f₂f₆¹⁵+ 15 v₁¹²v₂⁸⁴e₁¹²e₂¹²⁹ f₁¹²f₂⁴⁰+ 24 v₁₀¹⁸e₁₀²⁷f₂f₁₀⁹ ). ^._.

In order to compute the number of essentially different colourings of C_3v it is necessary to compute the 3-dimensional cycle index for C_3v. Only for the determination of the number of different colourings of the faces of C_3v with k colours the 3-dimensional cycle index of C_v will do the job in the following way. Replace all the indeterminates in this cycle index corresponding to the action on the sets of vertices and faces of C_v by k and all the indeterminates corresponding to the action on the edges by 1, then the expansion of this cycle index gives the number of different colourings of the faces of C_3v. For example the number of essentially different simultaneous colourings of C₂₀ with 2 colours for the vertices, 1 colour for the edges and 2 colours for the faces is computed as

Z₃(C₂₀,I_h, v_i=2, e_i=1, f_i=2)=35 931 952,^._.

which is the number of different colourings of the faces of C₆₀ with 2 colours (cf. table). Please take care that this number is not the product of the numbers of different colourings of the vertices and faces of C₂₀ with 2 colours. (These two numbers are given as 9 436 and 82 respectively.)

harald.fripertinger@kfunigraz.ac.at

The Leapfrog principle