# Photoluminescence of _Cis_-Polyacetylene Semiconductor Material
## 2 Methods
### Nonadiabatic Calculations  
In this current study, nonadiabatic calculations were used to understand the charge transfer of the system. Currents and electronic densities can be obtained using RDM (reduced density matrix) elements \(\rho_{jk}\), where \(j\) and \(k\) are the electronic orbitals. All the calculations formed here are based on Kohn-Sham (KS) orbitals, which were used for DFT calculations. We used ab initio molecular dynamics (MD) to compute the nonadiabatic coupling of the system, which provided dissipative transitions. Electronic dissipative transitions \(\left(\frac{d\rho_{jk}}{dt}\right)_{diss}\) were computed along a molecular dynamic trajectory for the positions of ions \(\{\overset{\rightarrow}{R}_{I}(t)\}\) with the initial conditions for the positions \(\{\overset{\rightarrow}{R}_{I}(t=0)\}\) and velocities \(\left\{\frac{d}{dt}\overset{\rightarrow}{R}_{I}(t=0)\right\}\) representing ambient temperature. Along with the nuclear trajectory, the nonadiabatic couplings of the system can be computed as follows [24; 25]:  
\[V_{ij}(t)=\frac{-i\hbar}{\Delta t}\int d\overset{\rightarrow}{r}\ \varphi_{i}^{KS *}\bigg{(}\left\{\overset{\rightarrow}{R}_{I}(t)\right\}\overset{ \rightarrow}{r}\bigg{)}\varphi_{j}^{KS}\bigg{(}\left\{\overset{\rightarrow} {R}_{I}(t+\Delta t)\right\}\overset{\rightarrow}{r}\bigg{)}+h.c. \tag{14}\]  
The autocorrelation function of the coupling is maximal at \(t=0\) and decays rapidly thereafter, and it also provides components of the Redfield tensor:  
\[M_{ijkl}(\tau)=\frac{1}{T}\int_{0}^{T}dtV_{ij}(t+\tau)V_{kl}(t) \tag{15}\]A Fourier transform of the coupling autocorrelation function [19] provides the partial component \(\Gamma^{\pm}\) as follows:  
\[\Gamma^{+}_{ljik}=\int d\tau\ M_{ljik}(\tau)exp\ (-i\omega_{ik}\tau) \tag{16}\]  
\[\Gamma^{-}_{ljik}=\int d\tau\ M_{ljik}(\tau)exp\ (-i\omega_{lj}\tau) \tag{17}\]  
Equations (15) and (16) express the partial components of the autocorrelation function. Combining them together will give us the Redfield tensor which controls the dynamics of the density matrix:  
\[R_{ijkl}=\Gamma^{+}_{ljik}+\Gamma^{-}_{ljik}-\delta_{jl}\sum_{m}\Gamma^{+}_{ immk}-\delta_{ik}\sum_{m}\Gamma^{-}_{jmml} \tag{18}\]  
\[\left(\frac{d\rho_{jk}}{dt}\right)_{diss}=\sum_{lm}R_{jklm}\rho_{lm} \tag{19}\]  
The initial excitations by a photon, \(\hbar\Omega_{AB}=E^{tot}_{A\to B}\ -E^{tot}_{ground}\approx\varepsilon_{B}- \varepsilon_{A}\), occur between Orbitals \(A\) and \(B\). At time \(t=0\), the excitation energy can be defined by a density matrix as follows [26]:  
\[\rho_{ij}(0)=\delta_{ij}(f_{i}-\delta_{iA}+\delta_{jB}) \tag{20}\]  
where \(f_{i}\) is the Fermi-Dirac thermal population of the \(i\)th orbital. By solving the equation of motion, the time evolution of the electronic state can be calculated as follows:  
\[\dot{\rho}_{ij}=-\frac{i}{\hbar}\sum_{k}\Bigl{(}F_{ik}\rho_{kj}-\rho_{jk}F_{ ki}\Bigr{)}+\left(\frac{d\rho_{ij}}{dt}\right)_{diss} \tag{21}\]  
The thermal fluctuations of the ions of the system help us to understand the electronic transitions between the orbitals. The numerical solution of Equation (15) provides the time-dependent elements of the density matrix \(\rho_{kj}(t)\), and \(F_{ik}\) is the matrix element of the KS Hamiltonian term; this term is produced due to the dissipation of energy from electronic to nuclear degrees of freedom. The dissipative rates can be calculated using Equation (19). Some of the most important parameters are the diagonal elements of the system \(\rho_{jj}(t)\), which can be described by the time-dependent occupations of the Kohn-Sham orbitals. Using the above information, we can easily calculate the charge density distribution, the rate of energy dissipation and the rate of charge transfer of the system. Therefore, the distribution of charge of the system can be explained as a function of energy as follow:  
\[n^{*}(\varepsilon,t)=\sum_{i}\rho_{ii}(t)\delta(\varepsilon-\varepsilon_{i}) \tag{22}\]  
The difference above is an equilibrium distribution.