**1. Introduction**

Collisions between particles on the atomic scale are ubiquitous throughout the universe. Our interest is in the collisions of fundamental particles such as electrons, positrons, photons, and protons with atoms and molecules. The field relies on strong interactions between experimental and theoretical approaches to the collision problems. The expectation is that experiments provide benchmark measurements for theorists to test their models, and when deemed sufficiently accurate the models provide extensive data for use in applications. The latter include astrophysics, fusion, lighting, nanolithography, and medical imaging and therapy. Accordingly, it is of grea<sup>t</sup> concern whenever there are substantial discrepancies between theory and experiment that are not understood. One such case was the discrepancy for the angular correlation parameters in the fundamental Coulomb three-body collision problem of e-H excitation of the 2p state [1,2].

From the theoretical side, the difficulties associated with calculating e-H scattering relate to the fact that the target has a countably infinite number of discrete states, an uncountably infinite continuum, and that the Coulomb interaction extends to infinite distances for charged particles. The convergen<sup>t</sup> close-coupling (CCC) method [3,4] was developed

**Citation:** Bray, I.; Weber, X.; Fursa, D.V.; Kadyrov, A.S.; Schneider, B.I.; Pamidighantam, S.; Cytowski, M.; Kheifets, A.S. Taking the Convergent Close-Coupling Method beyond Helium: The Utility of the Hartree-Fock Theory. *Atoms* **2022**, *10*, 22. https://doi.org/10.3390/ atoms10010022

Academic Editor: Grzegorz Piotr Karwasz

Received: 19 January 2022 Accepted: 5 February 2022 Published: 11 February 2022

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in response to the abovementioned discrepancy by providing a systematic and mathematically rigorous approach to the underlying computational difficulties. By expanding the target wave-functions in a truncated complete Laguerre basis the target is represented with a finite number of target states *N*. Furthermore, due to the exponential falloff of the basis, the interactions do not extend to infinite distances. The problem is then reduced to obtaining convergence in the scattering amplitudes of interest with increasing *N*. Despite demonstrating such convergence, the CCC method was unable to resolve the discrepancy with the experiment, and yielded results similar to other sophisticated theoretical approaches to the problem [4]. However, it was able to yield excellent agreemen<sup>t</sup> with the measurements of the total ionization cross section and its spin asymmetry [5], which is a particularly stringent test of any theory.

While atomic hydrogen is the ideal starting point for testing any electron–atom scattering theory, it is not ideal from the experimental point of view. Alkali atoms such as Li and Na are easier to prepare in the laboratory, and ye<sup>t</sup> their chemistry is very similar to that of H, having just one valence electron. This allowed for some very accurate measurements of e-Li [6–8], e-Na [9–12], and e-K [13–15] scattering systems. So the next stage in trying to understand the discrepancy for the e-H system was to extend the CCC method to the alkali atoms. The group of Miron Amusia generously provided the Hartree-Fock computational code [16,17] for a self-consistent treatment of the core electrons. This allowed the reduction of the electron–alkali atom collision problem to also be a Coulomb three-body problem, albeit with some more complicated nonlocal potentials [18]. The Hartree-Fock code also enabled the extension of the two-electron CCC code [19] to quasi two-electron targets such as alkaline earth metals [20,21]. Reviews of the development of the CCC method, including application to differential ionization, are available [22–25].

Unlike the case of the e-H system, the agreemen<sup>t</sup> between the CCC calculations and experiment for the electron–alkali collisions was spectacular, so much so, that the e-H collision system was remeasured by other groups, who found excellent agreemen<sup>t</sup> with the CCC calculations [26,27]. Subsequently, errors in the original measurements were identified [28].

A similar development track was taken by DPI on He, where the final state is determined from the e-He<sup>+</sup> collision system. Here too. theoretical predictions did not always agree with the initial experiment. Mergel et al. [29] measured the DPI of He using circularly polarized light. However, the corresponding CCC calculations did not yield agreemen<sup>t</sup> with the measurements [30]. Subsequently, Achler et al. [31] revisited the experiment and found excellent agreemen<sup>t</sup> with the CCC calculations. The CCC method was then extended to describe double ionization of He by electron impact (the so-called (e,2e) reaction [32,33]). Then an extension was made to the process of two-photon double ionization of He [34]. The utility of the HF theory was instrumental to describe the valence-shell DPI of alkaline– earth metal atoms [35]. These calculations were later found in good agreemen<sup>t</sup> with experiments [36]. Most recently, the CCC technique aided by the HF theory was applied to time-resolved atomic photoemission. The time delay in photoemission, expressed via the phase of the complex ionization amplitude, became experimentally accessible [37,38]. While the measurements are restricted at present to single active electron targets, theoretical predictions for two-electron targets have been made [39,40]. Most recently, photodetachment time delay was analyzed to reveal the implications of the fundamental threshold laws [41].

Here we shall demonstrate the agreemen<sup>t</sup> between theory and experiment by focusing on just the simplest electron–alkali collision system, that of e-Li scattering. This collision system is also a key component in calculating the DPI of Li<sup>−</sup>, as upon single or double photoionization the e-Li wave-function corresponds to the final state of Li<sup>−</sup>. The CCC computer codes utilize modern computational infrastructure including massive parallelism and GPU acceleration and are readily accessible for execution via the Atomic, Molecular and Optical Science Gateway, https://amosgateway.org (accessed on 4 February 2022) [42].
