Reprinted with permission from Kuklin, A. The inset shows the frontier states behavior near the Fermi level in polymer III. The high-symmetry k-points (C) in the Brillouin zone and corresponding electronic band structures calculated by GGA-PBE ( black lines) and HSE06 hybrid functional ( red lines) for polymers I (D) III (E) and II (F). Carbon, oxygen, and hydrogen atoms are presented in black, red, and beige, respectively. Unit cells are marked by dashed gray lines. Illustration of 2D tetraoxacirculenes formation (A) via attaching the same monomers in different ways (from left to right: polymer I, polymer III, and polymer II) (B). The 1D ribbon-like tetraoxacirculenes are also characterized by significantly low HLG values compared with other conjugated 1D polymers. This suggests that the studied models of tetraoxacirculene-based polymers of type I demonstrate relatively low HLG values (~ 1.66 eV under the boundary condition n → ∞) compared to most of the known 2D conjugated polymers, 138 i.e., they belong to organic semiconductors. Thus, the principal difference between the 1D and 2D tetraoxacirculene-based conjugated polymers is absolutely clear: at the increasing n number, the HLG values decrease rapidly in the 1D polymers, whereas in the 2D polymers the HLG values show a much more rapid contraction ( Fig. It was found that the HLG values decrease by ~ 1.2 eV in the 1D polymers (when going from tetraoxacirculene monomer n = 1 to an infinite linear structure n → ∞), whereas in the case of the 2D polymers the HLG decrease is abrupt at large n and goes to ≈ 2.0 eV in the limit n → ∞ ( Fig. 138 A strong electronic conjugation in the 1D and 2D polymeric tetraoxacirculenes leads to relatively small HLG values. The concept of HOMO–LUMO gap (HLG) engineering was used in order to interpret differences between 1D and 2D materials based on tetraoxacirculenes. The absorption spectra obtained by Gaussian broadening (width 0.05 eV) of the transition energies and by multiplying with the oscillator strengths for different clusters are shown in Fig. QMC calculation on this structure gave the value of 3.5 eV in good agreement with the calculations within B3PW91. Another allowed weak transition was obtained at ˜3.5 eV, but the oscillator strength was very low. The optical gap for Si 29H 24 was calculated to be 3.72 eV. Therefore similar behaviour is expected from the metal encapsulated silicon clusters. This cluster is supposed to represent the structure of a magic cluster that exhibits bright photoluminescence. Similar oscillator strength of 0.003 was obtained for Si 29H 24 with a tetrahedral structure. These values are comparable to those of 0.005–0.15 obtained for small elemental silicon clusters terminated with hydrogen. The oscillator strengths for these transitions were calculated to be 0.003 and 0.012, respectively. For the fullerene isomer of 16 the optical gap is 1.96 eV (HOMO–LUMO gap of 2.44 eV within B3PW91) which lies in the red region. This lies in the deep blue region of the optical spectrum. Accordingly the optical gap for the FK isomer of 16 was calculated to be 2.85 eV compared with the value of 3.44 eV for the HOMO–LUMO gap within B3PW91 hybrid functional for the exchange-correlation energy. The excitation of an electron creates an electron-hole pair and the exciton binding energy lowers the excitation energy. calculated the absorption spectra for Ti and Zr doped fullerene and FK isomers. The large HOMO–LUMO gaps of metal encapsulated silicon clusters and weak reactivity could be used for optical absorption and photoluminescence in the visible region.