Kekulé structures of different carbon species have been determined. On the basis of Kekulé structure and C-C bond counts as well as the surface curvature, stability of diverse carbon species, driving force for curling of graphite fragments and formation of fullerenes and nanotubes, have been discussed. Curling of graphite flat fragments, end-capping of nanotubes, and closure of curved structures are driven by a tremendous increase in Kekulé structures as terminal carbon atoms couple their dangling bonds into C-C σ bonds. The increasing tendency becomes particularly striking for large cages and nanotubes. Resonance among numerous Kekulé structures will stabilize the curved structure and dominate formation of closed carbon species. For similar carbon cages with comparable Kekulé structure counts in magnitude, the surface curvature of carbon cages, as a measure for the strain energy, also plays an important role in determining their most stable forms.
Using density functional theory, geometries and vibrational frequencies of linear chains NC2nN and HC2n+1N (n = 1～10) have been investigated. Time-dependent density functional theory (TD-DFT) has been used to calculate the vertical transition energies and oscillator strengths for the X 1 Σ +g →11 Σu+ transition in NC2nN (n = 1～10) and X 1Σ + →11Σ+ transition in HC2n+1N (n = 1～7). On the basis of present calculations, the explicit expressions for the size dependence of the excitation energy and the first adiabatic ionization energy in both carbon chains have been suggested.
The geometries, bondings, and vibrational frequencies of C 2n H ( n =3-9) and C 2n -1 N( n =3-9) were investigated by means of density functional theory(DFT). The vertical excitation energies for the X 2Π→ 2Π transitions of C 2n H( n =3-9) and for the X 2Σ→ 2Π and the X 2Π→ 2Π transitions of C 2n -1 N( n =3-9) have been calculated by the time-dependent density functional theory(TD-DFT) approach. On the basis of present calculations, the explicit expression for the wavelengths of the excitation energies in linear carbon chains is suggested, namely, λ 0=[1240 6A/(2+[KF(]3n+6-3n+3)](1-B e -Cn ), where A=3 24463, B=0 90742 , and C = 0 07862 for C 2n H, and A=2 94714, B=0 83929 , and C =0 08539 for C 2n -1 N. In consideration of a comparison of the theory with the experiment, both the expressions are modified as λ 1=0 92( λ 0+100) and λ 1= 0 95( λ 0+90) for C 2n H and C 2n -1 N, respectively. (1-B e -Cn ), where A=3 24463, B=0 90742 , and C = 0 07862 for C 2n H, and A=2 94714, B=0 83929 , and C =0 08539 for C 2n -1 N. In consideration of a comparison of the theory with the experiment, both the expressions are modified as λ 1=0 92( λ 0+100) and λ 1= 0 95( λ 0+90) for C 2n H and C 2n -1 N, respectively.
