# Calculations of Cross-Sections for Positron Scattering on Benzene
## 2 Theory and Computational Details
### Model for Direct Ionization of Molecules by Positron Impact  
The binary-encounter Bethe (BEB) model [35; 36] is one of the most frequently used and successful models for the calculation of cross-sections for ionization by electron impact. [PERSON] and [PERSON] [28] derived binary-encounter Bethe models for positrons BEB-0 and BEB-W. The BEB-W model takes into account the Wannier threshold law [37]. [PERSON] et al. [29] present the models BEB-A and BEB-B, which follow the threshold law of [PERSON] et al. [38]. These models allow accurate results to be obtained over a large range of energies and give ionization cross-sections which are close to experimental data [14; 39] and ab initio calculations [40]. Here we give a short summary of the BEB-0, BEB-W, BEB-A and BEB-B models. More details can be found in the original publications [28; 29]. The total cross-section (\(\sigma_{\rm ion}\)) for the direct ionization of a molecule by positron impact can be written as the sum of the partial ionization cross-section (\(\sigma_{i}^{\rm ion}\)) for ionization out of the occupied molecular orbitals. \[\sigma_{\rm ion}(E)=\sum_{i}^{n_{\rm occ}}\sigma_{i}^{\rm ion}(E). \tag{1}\]  
Here, the sum runs over all \(n_{\rm occ}\) occupied orbitals. In the following, \(E\) is the kinetic energy of the incoming positron, \(B_{i}\) is the binding energy of the electron in molecular orbital \(i\), and \(U_{i}\) is the expectation value of the kinetic energy of the electron in orbital \(i\). The formulas can be expressed in reduced quantities:  
\[t_{i}=\frac{E}{B_{i}}\ \ and\ \ u_{i}=\frac{U_{i}}{B_{i}}. \tag{2}\]  
The general formula for the partial ionization cross-section is given by  
\[\sigma_{i}^{\rm ion}=\frac{S_{i}}{t_{i}+u_{i}+1+f_{i}}\left[\frac{\ln{(t_{i}) }}{2}(1-\frac{1}{t_{i}^{2}})+h_{i}(1-\frac{1}{t_{i}})+G_{i}\right]\, \tag{3}\]  
where the prefactor is given by  
\[S_{i}=4\pi a_{0}^{2}N_{i}(\frac{R}{B_{i}})^{2}\, \tag{4}\]  
where \(a_{0}=0.529\times 10^{-10}\) m is the Bohr radius, \(R=13.6\) eV is the Rydberg constant, and \(N_{i}\) is the number of electrons in orbital number \(i\). The various models can be defined as follows:  
* BEB-0 model: \(f_{i}=0\), \(h_{i}=1\), \(G_{i}=0\). * BEB-W model: \(f_{i}=f_{i}^{\rm W}\), \(h_{i}=1\), \(G_{i}=0\) with the term \[f_{i}^{\rm W}=\frac{1}{(t_{i}-1)^{1.65}}\.\] (5)* BEB-A model: \(f_{i}=f_{i}^{\rm A}\), \(h_{i}=1\), \(G_{i}=0\) with the term \[f_{i}^{\rm A}=\frac{1}{(t_{i}-1)^{\alpha-1}e^{-\beta_{i}\sqrt{t_{i}-1}}}\,\] (6) where \(\alpha=2.64\) and \(\beta_{i}=0.489\sqrt{\frac{B_{i}}{2K}}\) to fulfill the threshold law of [PERSON] et al. (2018). * BEB-B model: \(f_{i}=0\), \(h_{i}=1-g_{i}(t_{i})\), \(G_{i}=g_{i}(t_{i})\Big{(}1-\frac{1}{t_{i}}\Big{)}^{\alpha}\) and \(g_{i}(t_{i})=e^{\beta_{i}\sqrt{t_{i}-1}}\) with the same values for \(\alpha\) and \(\beta_{i}\) as in the BEB-A model. For the calculation of the ionization cross-section, the binding energies and the expectation values of the kinetic energy of the electron before its removal are presented in Table 1. The binding energies for the lowest orbitals are calculated using [PERSON] theorem ([PERSON], 1966). For the outer valence orbitals, [PERSON] theorem is not accurate enough, and therefore, we employ the Outer Valence Green Function (OVGF) method ([PERSON], 1966) for the highest seven orbitals.. The program package Gaussian v16 ([PERSON], 1966) is used for all calculations with the 6-311++G(2 df,2p) basis set ([PERSON], 1966). In our computation, we obtain 9.26 eV for the lowest ionization energy, which is close to the experimental value of 9.24 eV for the vertical ionization energy obtained by [PERSON] et al. (2013). The total cross-section for electronic excitation is given by the sum of cross-sections for electronic excitation from the electronic ground state to excited states \(Y\).