**1. Introduction**

Progress towards the theoretical understanding of the fundamental mechanism underlying stable negative-ion formation in low-energy electron collisions with complex heavy multi-electron atoms and fullerene molecules has been very slow. This physical mechanism is of fundamental importance in physics and chemistry. More specifically, it has important implications for a wide range of applications, from catalysis to drug delivery and water purification. Unfortunately, the complexity of the interactions among electrons in heavy multi-electron atoms and fullerene molecules has, for a long time, made it virtually impossible to reliably predict the energetics of the electron binding and the properties of the resulting negative ions. A theoretical breakthrough was achieved in low-energy electron scattering from complex heavy multi-electron systems through our rigorous Regge pole method, wherein is embedded the electron–electron correlation effects and the corepolarization interaction, identified as the two crucial physical effects responsible for electron attachment resulting in stable negative-ion formation.

Consequently, the robust Regge pole method has allowed us to reliably explore, for the first time ever, negative-ion formation in complex heavy multi-electron systems such as the lanthanide and actinide atoms, as well as the fullerene molecules through the electron elastic total cross sections (TCSs) calculation. Importantly, these directly yield the anionic binding energies (BEs), the shape resonances (SRs) and the Ramsauer–Townsend (R-T) minima. From the TCSs unambiguous and reliable ground, metastable and excited state negativeion BEs of the formed anions during the collisions are extracted and compared with the measured and/or calculated electron affinities (EAs) of the atoms and fullerene molecules. The novelty and generality of the Regge pole approach is in the extraction of rigorous negative-ion BEs from the TCSs, without any assistance whatsoever from either experiments or any other theories. Whether the measured EAs are identified with the ground state anionic BEs or with the excited states anionic BEs of the formed negative ions during the collisions, the rigorous Regge pole-calculated BEs are available to guide measurements.

**Citation:** Msezane, A.Z.; Felfli, Z. Recent Progress in Low-Energy Electron Elastic-Collisions with Multi-Electron Atoms and Fullerene Molecules. *Atoms* **2022**, *10*, 79. https://doi.org/10.3390/ atoms10030079

Academic Editor: Anatoli Kheifets

Received: 19 May 2022 Accepted: 12 July 2022 Published: 29 July 2022

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Essential to the understanding of chemical reactions involving negative ions are accurate and reliable atomic and molecular affinities [1]. Moreover, low-energy electron collisions, resulting in negative-ion formation, provide a special insight into quantum dynamics [2]. Consequently, the careful determination of the EAs is needed. The Ramsauer– Townsend (R–T) effect is an important *inter alia* for understanding sympathetic cooling and the production of cold molecules using natural fermions and SRs. Additionally, the EAs provides a stringent test of theoretical calculations when their results are compared with those from reliable measurements. For ground state collisions, the Regge pole-calculated negative-ion BEs correspond to the challenging to calculate theoretically EAs, yielding outstanding agreemen<sup>t</sup> with the standard measured EAs for Au, Pt and the highly radioactive At atoms as well as for the C60 and the C70 fullerene molecules. In general, for the fullerenes C20 through C92, our Regge pole-calculated ground-state anionic BEs have been found to excellently match with the measured EAs. These results give grea<sup>t</sup> credence to the power and ability of the Regge pole method to produce unambiguous and reliable ground state anionic BEs of complex heavy systems through the TCSs calculation. Significantly, the Regge pole method achieves the remarkable feat without the assistance from experiments or any other theories.

Unfortunately, for most of the lanthanide atoms, producing sufficient anions that can be used in photodetachment experiments is challenging [3]. Due to their radioactive nature the actinide atoms are difficult to handle experimentally. Thus, there is a grea<sup>t</sup> need for reliable theoretical EAs to guide measurements. The EAs of atomic Au, Pt, and At have been measured [4–9], including those of the C60 and C70 fullerene molecules [10–14]. For the highly radioactive At atom, various sophisticated theoretical calculations, including the Multiconfiguration Dirac Hartree–Fock (MCDHF) value [15] agree excellently with the measured EAs [9]. Reference [9] employed the Coupled-Cluster method, while [15] used the MCDHF method. Furthermore, in [15] an extensive comparison among various sophisticated theoretical EAs has been carried out. For all these atoms, the measured EAs matched excellently the Regge pole-calculated BEs of the anionic ground states of the formed negative ions during the collisions, see Table 1 for comparisons. Moreover, the measured EAs of the fullerenes C20 through to C92 agree excellently with the Regge polecalculated anionic ground states BEs [16,17]. This gives grea<sup>t</sup> credence to our interpretation of the EAs of these complex systems, viz. as corresponding to the ground state BEs of the formed negative ions during the collisions.

Recently, the EAs of the highly radioactive actinide atoms Th [18] and U [19,20] were measured as well. The experimentalists concluded that the EAs of both Th and U corresponded to the BEs of the weakly bound electron to the neutral atoms. For the Ti atom, two measurements obtained the EAs as 0.377 eV [21] and 0.075 eV [22]. The former value is close to various theoretical calculations [23,24], including the Regge pole-calculated BE of the second excited state, namely 0.281 eV [25]. However, the value of 0.075 eV [22] is close to the Regge pole BE of the highest excited state of the formed Ti; anion, 0.0664 eV; its ground state BE is 2.42 eV [25]. The measured EA of Hf is 0.178 eV [26]. It is close to the Regge pole SR at 0.232 eV, the RCI EA of 0.114 eV [27] and the Regge pole second excited state anionic BE of 0.113 eV [28]. The Hf highest excited state BE is at 0.017 eV [29]. Indeed, here we are faced with the problem of interpretation of what is meant by the EA.

For the lanthanide atoms problems regarding what is meant by the EA have already been discussed [29,30]. Briefly, for the Nd atom, there are two measured EA values, viz. 1.916 eV [31] and 0.0975 eV [32]. The value of [31] is close to the Regge pole ground state anionic BE value of 1.88 eV [33], while the EA of [32] is close to the RCI EA [34] and the Regge pole anionic BE of the highest excited state [33]. Similarly, the measured EAs for the Eu atom are 0.116 eV [3] and 1.053 eV [35]. The former value agrees excellently with the Regge pole BE of the highest excited state, viz. 0.116 eV [33], and with the RCI EA of 0.117 eV [36]. The EA of [35] agrees very well with the Regge pole-metastable anionic BE value of 1.08 eV [33]. For the large Tm atom, the measured EA [37] is close to the Regge pole-metastable BE (1.02 eV) [33]. Clearly, the results here demonstrate the need for an

unambiguous meaning of the EA. The crucial question also considered here is: does the EAs of heavy multi-electron systems (atoms and fullerene molecules) correspond to the BE of the attached electron in the ground, metastable or excited state of the formed negative ion during the collision? Indeed, the meaning of the measured EAs of multi-electron atoms and fullerene molecules is also discussed here within the context of two prevailing viewpoints:


We conclude the paper with a demonstration of the first ever use of the Regge polecalculated TCSs as probes of the electronic structures of the actinide atoms Bk and Cf to identify the transitional element in the actinide series [38].

#### **2. Method of Calculation**

In this paper we have used the rigorous Regge pole method to calculate the electron elastic TCSs. Regge poles, singularities of the S-matrix, rigorously define resonances [39,40] and in the physical sheets of the complex plane they correspond to bound states [41]. In [42], it was confirmed that the Regge poles formed during low-energy electron elastic scattering become stable bound states. In the Regge pole method, also known as complex angular momentum (CAM), the important and revealing energy-dependent Regge Trajectories are also calculated. Their effective use in low-energy electron scattering has been demonstrated in [33,43], for example. The near-threshold electron–atom/fullerene collision TCS resulting in negative-ion formation as resonances is calculated using the Mulholland formula [44]. In the form below, the TCS fully embeds the essential electron–electron correlation effects [45,46] (atomic units are used throughout):

$$\sigma\_{\rm bot}(E) = 4\pi k^{-2} \int\_0^\infty \text{Re}[1 - S(\lambda)] \lambda d\lambda - 8\pi^2 k^{-2} \sum\_n \text{Im} \frac{\lambda\_n \rho\_n}{1 + \exp(-2\pi i \lambda\_n)} + I(E) \tag{1}$$

In Equation (1), *S( λ)* is the S-matrix, *k* = √2*mE*, *m* being the mass and *E* the impact energy, *ρn* is the residue of the S-matrix at the *n*th pole, λ*n*and *I(E)* contains the contributions from the integrals along the imaginary λ-axis (λ is the complex angular momentum); its contribution has been demonstrated to be negligible [33].

As in [47], here we consider the incident electron to interact with the complex heavy system without consideration of the complicated details of the electronic structure of the system itself. Therefore, within the Thomas–Fermi theory, Felfli et al. [48] generated the robust Avdonina-Belov-Felfli (ABF) potential which embeds the vital core-polarization interaction.

$$\mathcal{U}(r) = -\frac{Z}{r(1 + aZ^{1/3}r)(1 + \beta Z^{2/3}r^2)}\tag{2}$$

In Equation (2), *Z* is the nuclear charge, *α* and *β* are variation parameters. For small *r*, the potential describes Coulomb attraction between an electron and a nucleus, *U*(*r*) ~ <sup>−</sup>Z/*<sup>r</sup>*, while at large distances it has the appropriate asymptotic behavior, *viz.* ~ −1/(αβr4) and accounts properly for the polarization interaction at low energies. For an electron, the source of the bound states giving rise to Regge Trajectories is the attractive Coulomb well it experiences near the nucleus. The addition of the centrifugal term to the well 'squeezes' these states into the continuum [49]. For larger complex angular momentum (CAM) *λ,* the effective potential develops a barrier. Consequently, a bound state crossing the threshold energy E = 0 in this region may become a long-lived metastable state or an excited state. As a result, the highest "bound state" formed during the collision is identified with the highest excited state, here labeled as EXT-1. As E increases from zero, the second excited state may form with the anionic BE labeled, EXT-2. For the metastable states, similar labeling is used

as MS-1, MS-2, etc. However, it should be noted here that the metastable states are labeled relative to the anionic ground state. Regge poles are generalized bound-states, namely solutions of the Schrödinger equation where the energy (E) is real and positive, and *λ* is complex. The CAM methods have the advantage in that the calculations are based on a rigorous definition of resonances, viz. as singularities of the S-matrix, see [49,50].

The strength of this extensively studied potential [51,52] lies in that it has five turning points and four poles connected by four cuts in the complex plane. The presence of the powers of Z as coefficients of *r* and *r*2 in Equation (2) ensures that spherical and nonspherical atoms and fullerenes are correctly treated. Moreover, small and large systems are appropriately treated. The effective potential *<sup>V</sup>*(*r*) = *<sup>U</sup>*(*r*) + *λ*(*λ* + <sup>1</sup>)/2*r*<sup>2</sup> is considered here as a continuous function of the variables *r* and complex *λ*. The details of the numerical evaluations of the TCSs have been described in [46] and references therein; see also [53]. In the solution of the Schrödinger equation as described in [46], the parameters "*α*" and "*β*" of the potential, Equation (2) are varied. With the optimal value of *α* = 0.2 the *β*-parameter is then varied carefully and when the dramatically sharp resonance appears in the TCS, it is indicative of negative ion formation; this energy position matches the measured EAs of the atom/fullerene, for example Au or C60 fullerene. This has been found to be the case in all the atoms and fullerenes we have investigated thus far.
