# Photoluminescence of _Cis_-Polyacetylene Semiconductor Material
## 3 Results and Discussion  
Figure 2 indicates the energy patterns of the frontier orbital. We used two labels for unoccupied and occupied orbitals: (a) each orbital was counted based on its occupation as \(LU+(n-1)\) or \(HO-(n^{\prime}-1)\), and (b) through an analogy with the particle in the box, we labelled the unoccupied orbitals with the index \(n=1,2,3\) and the occupied orbitals with the index \(n^{\prime}=1,2,3\). IN Figure 2a, the LU orbital shows that the energy is \(E_{LU}=-3\) eV. The energies of the unoccupied orbitals increase with an increase in the number of indexes n. When \(n=2\), the energy of the LU + 1 orbital is \(E_{LU+1}=-2.5\) eV and the sub-gap between the LU and LU + 1 orbitals is around 0.4 eV. However, the HO orbital shows lower energy (\(E_{HO}=-4\) eV). It was observed that the energy gap between the HO and LU was \(\Delta E_{gap}=1.2\) eV, which was observed in the single-oligomer model. Because of the single oligomer model, there was less chance of hybridization between the electronic state of the orbital compared with a molecular crystal of multiple oligomers. Figure 2a shows that the gap between the HO and LU orbitals was more than the sub-gap between the HO and HO + 1 orbitals, and between the LU and LU-1 orbitals. Figure 2b illustrates the spatial isocontours of the frontier orbital \(\left|\varphi(r)\right|^{2}\) as computed by Equations (12) and (13). The maximum amount of the charge density of the LU orbital is shown in the middle portion of \(\left|\varphi(r)\right|^{2}\), whereas a lower amount of the charge density appears at the edge of the orbital. The same trend is observed for the HO orbital: the maximum amount of the charge density appears in the middle portion of the particle in the box and a lower amount of charge density is shown at the edge of the particle in the box. When \(n=2\) (the LU + 1 orbital and HO-1), the maximum amount of charge density changes from its original position. Because of the increase in the positive index of the eigenstate, the orbital was divided into two parts,  
Figure 1: Geometry-optimized structure (ground state) of the polymer model aligned in the y\(-\)z plane. Cyan and grey colors represent C and H atoms, respectively. and the maximum amount of the charge density was observed in the two middle portions of the orbital, and at the end of the edge and mid-point of the orbital, the charge density was almost zero. Figure 3a shows the absorption spectrum as a function of the transition energies computed by Equations (8)-(10), with intense peaks appearing from 1 eV to 4 eV. In Figure 3a,b, the peaks are labeled in the order of ascending transition energy. Here, Peak A indicates the lowest transition energy but the highest intensity of absorption, which means that bright transition occurred, whereas Peak B, Peak D, and Peak F show the highest transition energies and lower intensities of absorption, which means that they show darker transitions compared with Peaks A', C' and G'. To validate this using the PBE GGA function, we simulated the absorption spectra by time-dependent DFT (TDDFT) (Equation (9)) theory with an HSE06 hybrid function using Equation (5). A comparison of the TDDFT simulated and experimental absorption spectra (Table S1) indicates that the PBE GGA function underestimated the KS orbital energies, which demonstrates good qualitative agreement with the spectral features. [The comparison between the TDDFT simulated and experimental absorption spectra (Table S1) indicates that the GGA functional PBE underestimated the KS orbital energies, which shows a good qualitative agreement with the spectral features] The absorption spectra can be calculated using the PBE function shifted by a ratio (R) of the bandgap in the HSE06 and PBE function calculations, R = E\({}_{\text{gap(HSE06)}}\)/E\({}_{\text{gap(PBE)}}\).