SOME ASPECTS OF THE SYMMETRY AND TOPOLOGY OF POSSIBLE CARBON ALLOTROPE STRUCTURES

Elemental carbon has recently been shown to form molecular polyhedral allotropes known as fullerenes in addition to the familiar graphite an d diamond known since antiquity. Such fullerenes contain polyhedral ca rbon cages in which all vertices have degree 3 and all faces are eithe r pentagons or hexagons. All known fullerenes are found to satisfy the isolated pentagon rule (IPR) in which all pentagonal faces are comple tely surrounded by hexagons so that no two pentagonal faces share an e dge. The smallest fullerene structures satisfying the IPR are the know n truncated icosahedral C-60 Of I-h symmetry and ellipsoidal C-70 of D -5h symmetry. The multiple IPR isomers of families of larger fullerene s such as C-76, C-78, C-82 and C-84 can be classified into families re lated by the so-called pyracylene transformation based on the motion o f two carbon atoms in a pyracylene unit containing two linked pentagon s separated by two hexagons. Larger fullerenes with 3 upsilon vertices can be generated from smaller fullerenes with upsilon vertices throug h a so-called leapfrog transformation consisting of omnicapping follow ed by dualization. The energy levels of the bonding molecular orbitals of fullerenes having icosahedral symmetry and 60n(2) carbon atoms can be approximated by spherical harmonics. If fullerenes are regarded as constructed from carbon networks of positive curvature, the correspon ding carbon allotropes constructed from carbon networks of negative cu rvature are the polymeric schwarzites. The negative curvature in schwa rzites is introduced through heptagons or octagons of carbon atoms and the schwarzites are constructed by placing such carbon networks on mi nimal surfaces with negative Gaussian curvature, particularly the so-c alled P and D surfaces with local cubic symmetry. The smallest unit ce ll of a viable schwarzite structure having only hexagons and heptagons contains 168 carbon atoms and is constructed by applying a leapfrog t ransformation to a genus 3 figure containing 24 heptagons and 56 verti ces described by the German mathematician Klein in the 19th century an alogous to the construction of the C60 fullerene truncated icosahedron by applying a leapfrog transformation to the regular dodecahedron Alt hough this C-168 schwarzite unit cell has local O-h point group symmet ry based on the cubic lattice of the D or P surface, its larger permut ational symmetry group is the PSL(2,7) group of order 168 analogous to the icosahedral pure rotation group, I, of order 60 of the C-60 fulle rene considered as the isomorphous PSL(2,5) group. The schwarzites, wh ich are still unknown experimentally, are predicted to be unusually lo w density forms of elemental carbon because of the pores generated by the infinite periodicity in three dimensions of the underlying minimal surfaces.