Exotic Allotropes of Carbon - Exotic Properties of Carbon Nanomatter - page 191

Table 8.7 Comparative data for some small structures involved in hyper-graphenes at Hartree-Fock
HF, DFT (B3LYP) and DFTB levels of theory
Structure

C atoms Theory E tot (au) E tot /C (au) Gap (eV)

C 20 _Cy5J5_75 75 HF − 2838.062 − 37.841 4.158

C 20 _Cy6J5_90 90 − 3405.751 − 37.842 5.990

C 28 _Cy5J6_110 110 − 4163.361 − 37.849 5.533

C 28 _Cy6J6_132 132 − 4996.056 − 37.849 5.421

C 60 _HF 60 − 2271.830 − 37.864 7.418

C 60 P2J5_115 115 − 4354.333 − 37.864 7.597

C 60 P2J6_114 114 − 4316.491 − 37.864 6.270

C 20 _Cy5J5_75 75 DFT − 2856.161 − 38.082 0.600

− 3427.462 − 38.083

C 20 _Cy6J5_90 90 0.900

C 28 _Cy5J6_110 110 − 4189.837 − 38.089 1.072

C 28 _Cy6J6_132 132 − 5027.845 − 38.090 1.059

C 60 _DFT 60 − 2286.174 − 38.103 2.760

C 60 P2J5_115 115 − 4381.797 − 38.103 2.907

C 60 P2J6_114 114 − 4343.730 − 38.103 1.908

C 20 _Cy5J5_75 75 DFTB − 126.324 − 1.684 0.113

C 20 _Cy6J5_90 90 − 151.694 − 1.684 0.195

C 28 _Cy5J6_110 110 − 185.928 − 1.690 0.006

C 28 _Cy6J6_132 132 − 223.166 − 1.691 0.035

C 60 _DFTB 60 − 102.185 − 1.703 1.930

C 60 P2J5_115 115 − 195.708 − 1.702 2.044

C 60 P2J6_114 114 − 194.183 − 1.703 1.444

For some edges of a connected graph G there are the following relations satisfied
(John et al. 2007 ):

ecoe
ecof

⇔

fcoe
ecof & fcoh ⇒ ecoh

though the relation (11) is not always valid.
Let C ( e ): ={ f ∈ E ( G ); fcoe } denote the set of edges in G , codistant to the

edge e E ( G ). If relation co is an equivalence relation ( i.e ., C ( e ) obeys (4) to (6)),

then G is called a co-graph . Consequently, C ( e ) is called an orthogonal cut oc of G
and E ( G ) is the union of disjoint orthogonal cuts: C 1 ∪

∈

C 2 ∪

∪ ∩ =

... C k and Ci Cj Ø

for i = j , i , j = 1,2, .. , k . Observe co is a relation, (Djokovic 1973 ; Klavžar 2008 ;

Winkler 1984 ).