Calculation of electron-impact and photoionization cross sections of methane using analytical Gaussian integrals

Abdallah Ammar, Arnaud Leclerc, and Lorenzo Ugo Ancarani

Phys. Rev. A 109, 052810 – Published 9 May 2024

Abstract

The ionization by photon or electron impact of the inner ( $2 a_{1}$ ) and outer ( $1 t_{2}$ ) valence orbitals of the ${CH}_{4}$ molecule is investigated theoretically. In spite of a number of approximations, including a monocentric approach and a rather simple distorting molecular potential, the calculated cross sections are overall similar to those of other theoretical methods and in reasonable agreement with experimental data. The originality of the present approach stands in the way we evaluate the transition matrix elements. The key ingredient of the calculation scheme is that the continuum radial wave function of the ejected electron is represented by a finite sum of complex Gaussian-type orbitals. This numerically expensive optimization task is then largely compensated by rather simple and rapid analytical calculations of the necessary integrals, and thus all ionization observables, including cross section angular distributions. The proposed and implemented Gaussian approach is proved to be numerically very reliable in all considered kinematical situations with ejected electron energy up to 2.7 a.u. The analytical formulation of the scheme is provided here for bound molecular states described by monocentric Slater-type orbitals; alternatively, one may also use monocentric Gaussian-type orbitals for which the formulation is even simpler. In combination with complex Gaussian functions for the continuum states, an all-Gaussian approach with multicentric bound states can be envisaged.

Received 15 December 2023
Accepted 9 April 2024

DOI:https://doi.org/10.1103/PhysRevA.109.052810

Physics Subject Headings (PhySH)

Electronic excitation & ionization Photoemission Scattering theory

Molecules

Electronic structure

Atomic, Molecular & Optical

Authors & Affiliations

Abdallah Ammar ^*, Arnaud Leclerc ^†, and Lorenzo Ugo Ancarani

Université de Lorraine, CNRS, Laboratoire de Physique et Chimie Théoriques, F-57000 Metz, France

^*Present address: Université de Toulouse, CNRS, Laboratoire de Chimie et Physique Quantiques, 31062 Toulouse, France.
^†arnaud.leclerc@univ-lorraine.fr

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Vol. 109, Iss. 5 — May 2024

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Images

Figure 1
Partial photoionization cross section $σ^{(G)} (k_{e})$ (top panels) and asymmetry parameter $β^{(G)} (k_{e})$ (bottom panels) as a function of the photon energy $E_{γ}$ [in eV, see Eq. (3)], for orbitals $2 a_{1}$ (left panels) and $1 t_{2}$ (right panels) of ${CH}_{4}$ . Present results using cGTOs in both length (L) and velocity (V) gauges are compared with results from other theoretical methods: TD-DFT by Stener et al. [48] (dashed line) and single-center method of Novikovskiy et al. [50] (dark red dotted line), and experimental points: Backx and Van der Wiel [44] (green diagonal crosses), Van der Wiel et al. [45] (straight violet crosses), and Marr et al. [46] (red spots).
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Figure 2
TDCS for ionization by electronic impact of the inner ( $2 a_{1}$ ) valence orbital of ${CH}_{4}$ , as a function of ejection angle $θ_{e}$ , at fixed scattering angle of $θ_{s} = - 6^{\circ}, E_{s} = 500 eV, E_{e} = 37 eV$ (as in the experiments of Lahmam-Bennani et al. [52]). For a Coulomb wave for the ejected electron (red line) or a distorted wave (black thick line), the results obtained with the cGTO approach are compared to the purely numerical ones (blue and orange dashed lines). The vertical lines indicate the momentum transfer direction and its opposite.
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Figure 3
TDCS for ionization by electronic impact of the inner ( $2 a_{1}$ , left panels) and outer ( $1 t_{2}$ , right panels) valence orbitals of ${CH}_{4}$ , as a function of ejection angle $θ_{e}$ , at fixed scattering angle of $θ_{s} = - 6^{\circ}$ , with kinematical parameters of Lahmam-Bennani et al. [52], $E_{s} = 500 eV, E_{e} = 12 eV$ (upper panels), $37 eV$ (middle panels), and $74 eV$ (lower panels). The present cGTO calculations with the ejected electron described by a Coulomb wave (red line) and a distorted wave (black thick line) are compared with the complex Kohn method (CKM) results (blue dashed line) [57] and the experimental points [52]. The vertical lines indicate the momentum transfer direction and its opposite. The relative experimental data have been normalized at the binary peak maximum to the cGTO theoretical curve obtained with the distorted potential.
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Figure 4
TDCS for ionization by electronic impact of the outer ( $1 t_{2}$ ) valence orbital of ${CH}_{4}$ , as a function of the ejection angle $θ_{e}$ , for different values of the scattering angle (from top to bottom) $θ_{s} = - 20^{\circ}, - 22 . 5^{\circ}, - 25^{\circ}, - 27 . 5^{\circ}$ , and $- 30^{\circ}$ with kinematical parameters of Ali et al. [66], $E_{i} = 250 eV, E_{e} = 50 eV$ . The present cGTO calculations with the ejected electron described by a Coulomb wave (red line) and a distorted wave (black thick line) are compared with the theoretical (M3DW, dashed blue, and GSF, dashed orange) curves and with the experimental points published in [66]. For $θ_{s} = - 20^{\circ}$ , the experimental points of [59] are also shown. The vertical dotted lines indicate the momentum transfer direction. The experimental data, as well as the M3DW and the GSF results, have been normalized to the cGTO curve obtained with distorted potential at the binary peak maximum for $θ_{s} = - 20^{\circ}$ .
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Figure 5
Same as Fig. 4 but for $E_{e} = 30 eV$ . The experimental data, as well as the M3DW and the GSF results, have been normalized to the cGTO curve obtained with distorted potential at the binary peak maximum for $θ_{s} = - 20^{\circ}$ .
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