Effective one-electron approach to proton collisions with molecular hydrogen

THE EUROPEAN PHYSICAL JOURNAL D Eur. Phys. J. D (2022)76:31 https://doi.org/10.1140/epjd/s10053-022-00359-w Regular Article – Atomic and Molecular Collisions Effective one-electron approach to proton collisions with molecular hydrogen Corey T. Plowman1,a , Ilkhom B. Abdurakhmanov2 , Igor Bray1 , and Alisher S. Kadyrov1,b 1 Department of Physics and Astronomy and Curtin Institute for Computation, Curtin University, GPO Box U1987, Perth, WA 6845, Australia 2 Pawsey Supercomputing Centre, 1 Bryce Ave, Kensington, WA 6151, Australia Received 17 December 2021 / Accepted 31 January 2022 © The Author(s) 2022 Abstract. The two-centre wave-packet convergent close-coupling approach to ion–atom collisions is extended to study proton collisions with molecular hydrogen including electron-capture channels. We use a model potential to represent the molecular target as an effective one-electron spherically symmetric system. This greatly simplifies the target structure, allowing us to use already existing code developed for ion collisions with single-electron targets. Calculated total cross sections for electron capture, single ionisation, and excitation processes generally agree well with experimental data and other theoretical calculations where available. However, the total electron capture cross section is found to overestimate the experimental data at low energies, while the total ionisation cross section is slightly underestimated. Additionally, we present state-resolved cross sections for capture into the 1s, 2, and 3 states of the projectile where deviation between various previous calculations is substantial. Our results lead to overall improvement over previous theoretical studies although discrepancies with experiment are observed for 3p and 3d capture. We conclude that treating molecular hydrogen as an effective one-electron system within the two-centre coupled-channel approach to one-electron targets can give reasonably accurate total cross sections at intermediate and high energies, without the need for a complex and computationally demanding two-electron target representation. 1 Introduction The simplest homonuclear diatomic molecule is twoelectron molecular hydrogen. The multicentre nature of H2 makes it difficult to accurately represent its structure, requiring complex theoretical descriptions and computationally demanding codes. However, as the most abundant molecule in nature and the simplest molecular target it represents a useful first step towards scattering on more complex targets. Molecular hydrogen has attracted significant attention with a number of recent works published analysing collisions with ions, see e.g. Ref. [1] and references therein. This is partly due to the emergence of hadron therapy for treatment of cancer and the consequential requirements [2,3] for accurate scattering calculations of ion collisions with complex molecules. Proton scattering on molecular hydrogen has been extensively investigated experimentally. Stier and Barnett [4] performed a comprehensive experiment to determine both total electron-loss and electron-capture cross sections in p + H2 collisions at low incident energies. The measurements of the total electron-loss cross a e-mail: (corresponding author) b e-mail: 0123456789().: V,-vol section by Hooper et al. [5] provided data for ionisation at high energies where the electron-capture contribution to total electron loss is negligible in comparison with ionisation. Electron capture was measured across a wide energy range by Barnett and Reynolds [6], McClure [7] and Toburen et al. [8]. No distinction was made between capture that left the residual molecular ion intact or in a dissociative state. However, measurements by Shah et al. [9] and Shah and Gilbody [10] of the separate dissociative and non-dissociative capture channels showed that the contribution from capture events leading to dissociation are approximately an order of magnitude smaller than non-dissociative capture processes. Additionally, Rudd et al. [11] made empirical calculations and estimated uncertainties by fitting an analytical formula to the range of available experimental data. Total ionisation cross sections were measured by Toburen and Wilson [12] at high impact energies where dissociative ionisation is negligible. Edwards et al. [13] and Shah et al. [9] explicitly measured nondissociative ionisation and showed that dissociative ionisation becomes negligible in comparison with nondissociative ionisation at impact energies higher than 20 keV. Electron-capture cross sections into the 2s state were measured by Andreev et al. [14], Bayfield [15], Birely and McNeal [16], Hughes et al. [17], and Shah 123 31 Page 2 of 13 et al. [18], although only the apparatus of Andreev et al. [14], Shah et al. [18] were calibrated to give absolute cross sections. Capture into the 2p state was experimentally measured by Birely and McNeal [16] and Hughes et al. [19]. Hughes et al. [20] measured cross sections for 3s, 3p, and 3d capture, while Williams et al. [21] obtained data only for 3s and Dawson and Loyd [22] for 3p and 3d. Furthermore, the total ionisation cross section for antiproton collisions with molecular hydrogen was measured by Knudsen et al. [23], Hvelplund et al. [24], and Andersen et al. [25]. Thus far, the majority of theoretical works on ion collisions with molecular hydrogen is limited to negatively charged projectiles such as antiprotons [1,26–30]. This removes the possibility of charge-exchange processes, significantly simplifying the collisional problem. Modelling scattering of positively charged projectiles brings additional challenges due to electron capture into bound and continuum states of the projectile having significant contributions to the total electron-loss cross section. Separating these processes from direct ionisation requires more elaborate theories such as two-centre expansion approaches. However, this greatly increases computational complexity [31]. An alternative approach that projects a bound state of the projectile atom onto the total scattering wave function has been recently developed to calculate electron capture using only a one-centre expansion [32]. This significantly simplifies the theory compared to a two-centre approach and provided very good agreement with both experiment and two-centre calculations for p+H collisions. This approach was used for multielectron atom targets and is currently being extended to molecular hydrogen. The boundary-corrected first Born (B1B) approximation developed by Belkić et al. [33] was extended by Corchs et al. [34] to calculate capture cross sections into the ground state of the projectile in p+H2 collisions for collision energies from 100 to 1000 keV. This perturbative approach works well in the high-energy region, but its assumptions break down at lower incident energies. Another type of perturbative methods is based on the continuum-distorted-wave (CDW) approach. The CDW approach was used to cal (...truncated)