# Electron Impact Cross Sections and Transport Studies of C\({}_{3}\)F\({}_{6}\)[PERSON]
## 2 Calculation Methods  
### Electron Impact Cross Section Calculation  
In this section, we outline the methods utilized for the cross section calculation. The R-matrix is a powerful method to study low-energy electron-molecule scattering. Here, we provide a brief overview of the approach. The method is fundamentally based on the division of the configuration space into an inner and outer region. The inner region accommodates the entire N target electron wave function. Due to the indistinguishability of the target and scattering electron, the \(e^{-}-e^{-}\) correlation and exchange interaction becomes important and makes the inner region problem complicated. The complex problem in the inner region is solved and propagated to the outer region for the calculation of cross sections. In the outer region, the exchange and correlation effects are negligible, and only a long-range multipolar interaction between the scattering electron and the target states is included, which makes the problem simpler. The R-matrix serves as a connection between the two areas. Once the R-matrix at the boundary is established, the wave function in the outer area can be determined by extending the R-matrix from the boundary of the two areas to infinity. In the asymptotic limit, the solution consists of various distinct functions that are directly influenced by the coefficients \(K_{ij}\) related to the \(K\)-matrix. The \(K\)-matrix is diagonalized to calculate the sum of eigenphases, which is utilized to determine the location and size of the resonances. Finally, the scattering matrix \(S\) is obtained from the \(K\)-matrix by means of the following transformation:  
\[S=\frac{1+iK}{1-iK} \tag{1}\]  
The \(T\)-matrix is then obtained from this \(S\)-matrix by the relation  
\[T=S-1 \tag{2}\]  
Thus, the K-matrix is used to obtain \(T\)-matrices using the definition  
\[T=\frac{2 iK}{1-iK} \tag{3}\]  
This \(T\) matrix is then used to obtain the integral cross sections. A detailed overview of the method can be found in the reference papers [12; 13]. The fixed nuclei (FN) T-matrices are used to determine rotational state-to-state cross sections for low partial waves up to \(1=4\). For dipole-forbidden transitions with J \(>1\), the cross sections quickly converge. However, for dipole-allowed transitions (J = 1), the FN approximation is not suitable owing to the long-range electron-dipole interaction. To tackle this situation, Born approximation is used to determine the cross sections for J \(>4\). These cross sections then estimate the contribution of partial waves up to l = 4 through a top-up procedure to achieve reliable outcomes [15]. Considering the nonpolar nature of the present target, contributions from lower partial waves were sufficient to obtain reliable cross sections. QEC implemented three scattering models for the calculations: static exchange (SE), static exchange polarization (SEP), and close coupling (CC). The QEC manual recommends the usage of the SEP method for elastic, momentum transfer, rotational excitation, and dissociative electron attachment (DEA) cross sections. The CC method is used for electronic excitation cross sections. The dissociative electron attachment (DEA) is investigated through a semi-quantitative estimate, which uses the resonance information from the R-matrix results. The DEA cross section directly depends on the resonance cross section. This resonance cross section, in turn, requires the Breit-Wigner cross section, which depends on the resonance width and potential. In QEC, the Morse potential is used as the interatomic interaction potential. Further details can be found in [16]. Considering the bigger size of the present molecule, convergence of the cross sections was a concern. In this context, a DEA calculation was performed with an R-matrix sphere of 10 \(a_{0}\). Further, SEP and CC calculations were performed with an R-matrix radius of 12 \(a_{0}\) and 14 \(a_{0}\), respectively. The QEC software also computes the electronic excitation and ionization cross section.