Towards semi-empirical partial cross sections for ionization of molecules by electron impact

Several successful approaches are known for total cross sections for the ionization of molecules by electron impact but the pattern of partitioning into fragment ions is still far from being resolved. Binary-Encounter-Bethe (BEB) is quite useful in predicting total ionization cross sections, for molecules, like CH₄, N₂, N₂O in agreement with best measurements within 5%, i.e. the experimental uncertainties. Input data for BEB are – bound and kinetic energies of electrons from particular orbitals (the relation between the two values is governed by the virial theorem) and the number of electrons on these orbitals.

By combining available experimental ion‐yield data with Bayesian inference, we assess the feasibility of estimating the probability that ionization of a given molecular orbital produces a specific ionic fragment. Unlike classical fitting, Bayesian inference does not yield a single point estimate in parameter space; instead, it produces posterior probability density functions for the model parameters shaped by the experimental data. Because Bayesian parameter estimation requires evaluating multidimensional integrals, we employed a Markov chain Monte Carlo (MCMC) method implemented in the MATLAB toolbox by Laine (https://mjlaine.github.io/mcmcstat/). We adopted uninformative Gaussian priors for all fitting parameters, with effectively infinite variance and means set to the initial values obtained from a nonlinear regression fit to the experimental data (using the Levenberg–Marquardt algorithm for nonlinear least squares). This choice of priors assigns equal probability to all possible values. After discarding the burn-in period, we generated all posterior distributions from 100,000 MCMC samples.

As an example owe show the MCMC-based BEB fit to the cross section for CO⁺ production from CO. The results show that ionization of the outermost (5σ) orbital is exclusively responsible for CO⁺ production, whereas ionization of the 1π orbital yields CO⁺ with approximately 80% probability. Contributions from other molecular orbitals are far less significant, as evidenced by their near-zero mean values and/or large standard deviations. We will apply the same analysis to specific ion production via electron-impact ionization of other molecules, including CO₂, O₂, CH₄, and C₂H₂.