3.6.2. Results

The electron elastic TCSs for the typical large fullerenes C100, C120 and C140 demonstrate negative-ion formation [92–94] with significant differences among their EAs, namely 3.67 eV, 3.74 eV and 3.94 eV, respectively, see also Table 2. It is now clear that the ground state anionic BEs located at the absolute R–T minima of the ground state TCSs yield the challenging to calculate theoretically EAs. Indeed, the R–T minimum provides an excellent environment that is conducive to negative-ion catalysis and the creation of new molecules. The underlying physics in the fullerene TCSs has already been explained in [93,94]. The obtained results are consistent with the observation that low-energy electron-fullerene interactions are characterized by rich resonance structures [95,96] and that the experimentally detected fullerene isomers correspond to the metastable TCSs [84]. They also support the conclusion that the EAs of fullerene molecules are relatively large. The results of [83] including those presented in Table 2 should satisfy part of the requirement to increase fullerene acceptor resistance to degradation by the photo-oxidation mechanism through the use in organic solar cells of fullerenes with high EAs [82]. The extracted EAs from the TCSs could also be used to construct the widely used simple model potentials for the fullerene shells, including endohedral fullerenes [97–105], as well as in the study of the stability of An@C40(An = Th, —–, Md) [106]. Notably, the EAs are at the hearts of many of the model potentials. Indeed, the rich resonance structures in the fullerenes TCSs and their large Eas explain the tendency of fullerenes to form compounds with electron-donor anions and their vast applications as well.

The utility of the generated fullerene anions has been demonstrated in the catalysis of water oxidation to peroxide and water synthesis from H2 and O2 using the anionic fullerene catalysts C20− to C136− [92]. Figure 5 taken from [92], demonstrates the Density Functional Theory (DFT) calculated TS energy barriers for both processes. DFT and dispersion corrected DFT approaches have been employed for the TS evaluations. Geometry optimization of the structural molecular conformation utilized the gradient-corrected Perdew-Burke-Ernzerhof parameterizations [107] of exchange-correlation as implemented in DMol3 [108]. DFT calculated energy barriers reduction in the oxidation of H2O to H2O2 catalyzed using the anionic fullerene catalysts C20− to C136− are shown in the Figure 5 (left panel). The results in Figure 5 (right panel), also from [92] are for the water synthesis from H2 and O2 catalyzed using the anionic fullerene catalysts C20− to C136− as well. For both water oxidation and water synthesis DFT TS calculations found the C52; and C60− anions to be numerically stable and the C36− and C100− anions to increase the energy barriers the most in the water oxidation to H2O2 and water synthesis using H2 and O2, respectively. The C136− anion has proved to be the most effective in reducing the energy barrier significantly when catalyzing both water oxidation to peroxide and synthesis from H2 and O2. Importantly, a single large fullerene such as the C136, or even the C74 could replace the Au, Pd and Sn atoms in the catalysis of H2O2 from H2O in the experiments [87–89] acting as a multiple-functionalized catalyst. These fullerenes have their metastable Bes

close to the EAs of the atoms used in the experiments. Thus, an inexpensive dynamic water purification system for the developing world could be realized [89].

**Figure 5.** Transition state energy barriers of anionic fullerenes sizes C20− to C136− for catalyzing water oxidation to peroxide (**left panel**) and catalyzing hydrogen and oxygen synthesis to water (**right panel**).

Indeed, the utility of the fullerene molecular anions has been demonstrated in the catalysis of water oxidation to peroxide and water synthesis from H2 and O2 using the catalysts C20− to C136<sup>−</sup>. DFT TS calculations found C52− and C60− anions numerically stable for both. The C136− anion has proved to be the most effective in reducing the energy barrier significantly when catalyzing both water oxidation to peroxide and synthesis.

#### *3.7. Atomic Structure and Dynamics of Bk and Cf: Experiment Versus Theory*

The recent experiment [38] using nanogram material identified a weak spin-orbitcoupling in atomic Bk while a jj coupling scheme described atomic Cf. It concluded that these observations strengthen Cf as a transitional element in the actinide series. Here the Regge pole-calculated low-energy electron elastic TCSs for Bk and Cf atoms are used as novel validation of the experimental observation through the sensitivity of R–T minima and SRs to the electronic structure of these atoms.

In Figure 6, we present the Regge pole-calculated electron elastic TCSs for the Bk (left panel) and Cf (right panel) actinide atoms. As seen the TCSs are characterized generally by dramatically sharp resonances, representing ground, metastable and excited states negative-ion formation, SRs (broad peaks) and R–T minima. Moreover, the highest excited states TCSs (green curves) exhibit fullerene molecular behavior [16]. The energy positions of the sharp resonances, well delineated correspond to the BEs of the formed negative ions during the electron collisions with the Bk and Cf atoms. Each figure consists of BEs of the ground (red curve), metastable (blue and orange curves) and excited (green and brown curves) states TCSs. At first glance these TCSs appear a little complicated. However, they can be readily understood if each curve is discussed separately, see also [93,94] for example. The R–T minima manifest the effects of the polarization interaction [62], while the SRs convey the trapping effect of the centrifugal potential. The underlying physics has already been discussed in [93,94]; it will not be repeated here.

**Figure 6.** Total cross sections (a.u.) for electron elastic scattering from atomic Bk (**left panel**) and Cf (**right panel**). For both Bk and Cf the red and the blue and orange curves represent TCSs for the ground state and the metastable states, respectively. The green and the brown curves denote TCSs for the excited states. The orange curve with a deep R–T minimum in the Bk TCSs is the polarizationinduced TCS due to size. In the Cf TCSs the orange curve has flipped over to a pronounced shape resonance very close to threshold. The dramatically sharp resonances in both figures correspond to the Bk; and Cf; anions formed during the collisions. The labeled BEs are intended to guide the eye; the complete values are presented in Table 3 for clarity.

For our objective here, we focus mainly on the polarization-induced TCSs (orange curves) and the ground state TCSs in both Figs. 3.7. For a better understanding and appreciation of the results, it is appropriate to place in perspective the polarization-induced TCSs that are characterized by a deep R–T minimum near threshold in the TCSs of Bk and a pronounced SR very close to threshold in the Cf TCSs. The polarization-induced TCS with the deep R–T minimum near threshold first appeared in the actinides TCSs through the atomic Pu TCSs [54]. It was attributed to the size effect and the first 6d-orbital collapse impacting the polarization interaction significantly. The first 6d-collapse occurred in the transition Np[Rn]7s25f46d to Pu[Rn]7s25f6. This caused the ground state anionic BE of the Np atom to increase from 3.06 eV to 3.25 eV in Pu. Moreover, the anionic BE of the first metastable state increased from 1.47 eV in Np to 1.57 eV in Pu, see also Table 1 of Ref. [54]. It is the increase in the ground state energy space that facilitated the first appearance of the polarization-induced metastable TCS with the deep R–T minimum near threshold to appear in the Pu TCSs. This R–T minimum in the Pu TCSs continued through the Bk TCSs [77]. Indeed, the increase in the number of polarization-induced TCSs with size has already been clearly demonstrated in the fullerene molecular TCSs [83,92], see also Figure 1 of [54].

The second 6d-orbital collapse occurs in the transition Cm[Rn]7s25f76d to Bk[Rn]7s25f9. To facilitate the discussion, we have included for convenience Table 4; the data have been taken from [54,77,80]. Table 4 shows that the ground state anionic BE increased significantly from 3.32 eV in Cm to 3.55 eV in Bk, thereby widening the energy space for the flip over to take place. Subsequently, the ground state anionic BE dropped from 3.55 eV in Bk to 3.32 eV in Cf. Similarly, for the first metastable states the anionic BEs increased from 1.57 eV in Cm to 1.73 eV in Bk, while it decreased to 1.70 eV in Cf. In Table 4, it is informative to look at mainly the ground state energy values. Briefly, in Bk the ground state anionic BE is 3.55 eV; it decreased to 3.32 eV in Cf after the R–T minimum flipped over to a SR very close to threshold. This is indicative of the smaller energy space required by the SR compared to the R–T minimum. On the other hand, in Pu the ground state anionic BE

is 3.25 eV having increased from 3.06 eV in Np to accommodate the first appearance of the polarization-induced TCS with the deep R–T minimum in Pu. Here we also see the appearance of the 1.22 eV BE as MS-2; the Figures are also quite informative.

**Table 4.** Negative-ion binding energies (BEs), in eV extracted from the TCSs for the actinide atoms Np through Fm are taken from [54,77,80]. GRS, MS-*n* and EX-*n* (*n* = 1, 2) represent respectively ground, metastable and excited anionic states. The experimental EAs, EXPT, denoted by N/A are unavailable.


The flip over of the near threshold Ramsauer–Townsend minimum from the Bk polarization-induced metastable TCS to a pronounced shape resonance very close to threshold in the Cf metastable TCS provides a sensitive probe of the electronic structure and dynamics of these atoms, thereby permitting the first ever use of the R–T minimum and the SR as novel confirmation of Cf as a transitional element in the actinide series, consistent with the experimental observation [38]. Indeed, the rigorous Regge pole method requires no assistance whatsoever from either experiments or any other theory for the remarkable feat, namely of probing reliably the electronic structure of these complicated actinide atoms.

## **4. Summary and Conclusions**

The Regge pole-calculated low-energy electron elastic total cross sections (TCSs) of complex heavy multi-electron systems are characterized generally by dramatically sharp resonances manifesting negative-ion formation. These yield directly 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 states 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 experiment or any other theory. The EA provides a stringent test of theoretical calculations when their results are compared with those from reliable measurements. For ground states 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 C70 fullerenes. For C20 through to C92 fullerenes our Regge pole-calculated ground-state anionic BEs matched in general excellently 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.

The meaning of the measured EAs of multi-electron atoms and fullerene molecules has also been considered here within the context of two prevailing viewpoints:

(1) The first considers the EA to correspond to the electron BE in the ground state of the formed negative ion during collision; it is exemplified by the measured EAs of Au, Pt and At atoms and the fullerene molecules from C20 through to C92.

(2) The second view identifies the measured EA with the BE of electron attachment in an excited state of the formed anion. The measured EAs of Ti, Hf, lanthanide and actinide atoms provide representative examples of this viewpoint.

This experimental breakthrough [38], including the recent first ever measurements of the EAs of the highly radioactive element At [9], as well as the Th [18] and U [19,20] atoms represent significant advances in the measurements of the challenging highly radioactive elements. In addition, more such measurements of other radioactive atoms can be expected in the near future. Consequently, reliable theoretical predictions are essential for a fundamental understanding of the underlying physics. Here we have presented an entirely new approach to the validation of the experimental observation in [38], namely through the behavior of the R–T minima and the SRs in the metastable electron elastic TCSs of atomic Bk and Cf. Finally, with the available ground, metastable and excited negative-ion BEs calculated here for the multi-electron atoms and the fullerene molecules, sophisticated theoretical methods such as the Dirac R-matrix, Coupled-Cluster method, MCDHF, MCDF-RCI, etc. can now be used to generate reliable EAs, wave functions and fine-structure energies. Indeed, for unambiguous and definitive meaning of the EAs of multi-electron atoms and the fullerene molecules our anionic BEs can be used in sophisticated theoretical methods to carry out careful investigations such as has been done in [15] for the At atom.

**Author Contributions:** Z.F. and A.Z.M. are responsible for the conceptualization, methodology, investigation, formal analysis and writing of the original draft, as well as rewriting and editing. A.Z.M. is also responsible for securing the funding for the research. All authors have read and agreed to the published version of the manuscript.

**Funding:** U.S. DOE, Division of Chemical Sciences, Geosciences and Biosciences, Office of Basic Energy Sciences, Office of Energy Research, Grant: DE-FG02-97ER14743.

**Acknowledgments:** Research was supported by the U.S. DOE, Division of Chemical Sciences, Geosciences and Biosciences, Office of Basic Energy Sciences, Office of Energy Research, Grant: DE-FG02- 97ER14743. The computing facilities of National Energy Research Scientific Computing Center, also funded by U.S. DOE are greatly appreciated.

**Conflicts of Interest:** The authors declare no conflict of interest or state.
