# Electron Impact Cross Sections and Transport Studies of C\({}_{3}\)F\({}_{6}\)[PERSON]
## 2 Calculation Methods
### Electron Impact Cross Section Calculation
Here, we outline the Bef method [17], which is used for the electronic excitation process. The plane wave Born (PWB) approximation is used as the starting point in the Bef scaling. The first-order PWB cross section for inelastic collision is given as  
\[\sigma_{PWB}=\frac{4\pi a_{0}^{2}R}{T}F_{PWB}(T) \tag{4}\]  
where \(a_{0}\) is the Bohr radius, \(R\) is the Rydberg energy, and \(T\) is the incident energy. \(F_{PWB}(T)\) is the collision strength. Nevertheless, this approach does not account for electron exchange effects, the distortion of plane waves near the target, or the target polarization in the presence of the incident electron. To overcome these shortcomings, BE scaling was performed. \[\sigma_{BE}=\frac{T}{T+B+E}\sigma_{PWB} \tag{5}\]  
where \(B\) and \(E\) are the binding energy of the excited electron and the excitation energy, respectively. This scaling reduces the cross section in the low-energy region and also improves the peak value when compared to the experiments. In this way, the effective incident energy as seen by the target electron improves the drawbacks of the PWB method. Further, the accuracy of the PWB cross sections mainly depends on the correct wave function description. To address a poor wave function in the calculation, dipole oscillator strengths, which are commonly known as optical oscillator strengths (OOS), can be used. We refer to the OOS values as the \(f\) values here. When accurate OOSs are available, the PWB cross sections derived from poor wave function can be improved. \[\sigma_{f}=\frac{f_{accu}}{f_{PWB}}\sigma_{PWB} \tag{6}\]  
The two scalings, when combined, give us the Bef scaled cross sections. \[\sigma_{Bef}=\frac{f_{accu}}{f_{PWB}}\sigma_{BE} \tag{7}\]  
[PERSON] and [PERSON] introduced the binary-encounter-Bethe (BEB) method to predict the total ionization cross section for ejecting one electron from each orbital [14]. This model combines the dipole part of the Bethe cross section for high energy with the Mott cross section for low energy. The dipole interaction between the incident and ionized electrons is described by the former, while the latter represents close collisions with small impact parameters. The BEB method proved to be highly successful in accurately determining the total ionization cross section and exhibited good agreement with the experimental results. The formula for the total ionization cross section for an orbital according to BEB is provided below. \[\sigma_{BEB}=\frac{S}{t+u+1}[\frac{Int}{2}(1-\frac{1}{t^{2}})+1-\frac{1}{t}- \frac{Int}{t+1}] \tag{8}\]  
Here, \(S=4\pi a_{0}^{2}N(R/B)^{2}\); \(t=\frac{T}{B}\), \(u=\frac{U}{B}\). The variables in the equation are defined as follows: \(T\) represents the incident energy, \(U\) represents the kinetic energy of the electron on a molecular orbital, and \(B\) represents the binding energy of the electron. The occupational number is denoted by \(N\), the Rydberg energy by \(R\), and the Bohr radius by \(a_{0}\). These parameters can be determined easily using quantum chemistry tools, such as the GAUSSIAN 09 software (Revision D.01) [18], with density functional theory (DFT) and the \(\omega\)B97X-D functional. DFT is a reliable and powerful approach for obtaining accurate energetics in molecular systems. Once the cross section for each orbital was obtained using Equation (8), they were summed to calculate the total ionization cross section for the given molecular target.