**3. Results**

#### *3.1. Cross Sections for the Representative Atom Au and Fullerene Molecule C60*

Figure 1, taken from Ref. [54] presents TCSs for atomic Au and fullerene molecule C60. They typify the TCSs of complex heavy multi-electron atoms and fullerene molecules, respectively. Importantly, they are characterized by dramatically sharp resonances representing negative-ion formation in ground, metastable and excited anionic states, R–T minima and SRs. In both Figs. the red curves represent ground states electron TCSs, while the green curves denote excited state TCSs. Here the ground states anionic BEs in both Au and C60 appearing at the absolute R–T minima matched excellently with the measured EAs, see Figure 1 and Table 1 for comparisons with various measurements. In both systems, the ground states anionic BEs determine their EAs and not the excited anionic BEs (green curves). The data in Table 1 for Pt, At, Ti and Hf atoms as well as for the C70 fullerene were extracted from similar curves, as in Figure 1. Notably, the TCSs for the atoms and fullerene molecules become more complicated as the systems considered become larger as exemplified by the actinide atoms in Ref. [54] and the fullerene molecules in [16].

The availability of excellent measured EAs for the Au and Pt atoms [4–8] and the C60 fullerene molecule [10–12] allowed us to implement the rigorous Regge pole method to complex multi-electron atoms and fullerene molecules. Excellent agreemen<sup>t</sup> with the measurements were obtained for the Au, Pt and At atoms and the fullerene molecules C60 and C70, as demonstrated in Table 1. Thus, the Regge pole-calculated ground state anionic BEs were benchmarked on the measured EAs of both Au and C60. Subsequently, the Regge pole method was implemented in the calculation of low-energy electron elastic TCSs for various complex multi-electron atoms, including atomic Hf and Ti. The calculated ground states anionic BEs for the Au, Pt and At atoms matched excellently with the measured EAs of these atoms. For the fullerene molecules C20 through to C92, the obtained anionic ground states BEs [16,17] generally agreed very well with the measured EAs. Indeed, the Regge pole method accomplished an unprecedented feat in the calculation of the challenging to theoretically calculate EAs of both multi-electron atoms and the fullerene molecules, from C20 through to C92.

**Figure 1.** Total cross sections (a.u.) for electron elastic scattering from atomic Au (**left panel**) and the fullerene molecule C60 (**right panel**) are contrasted. For atomic Au the red, blue and green curves represent TCSs for the ground, metastable and excited states, respectively. For the C60 fullerene the red, blue and pink curves represent TCSs for the ground and the metastable states, respectively, while the green and brown curves denote TCSs for the excited states. The very sharp resonances in both figures Correspond to the Au− and C60− anionic formation. The anionic Bes in the Figures are intended to guide the eye; the complete values are also presented in Table 1 for better comparisons.

Table 1 already demonstrates the ambiguous meaning of the measured EAs in the large atoms Ti and Hf versus the meaning in the Au, Pt and At atoms. The interpretation of the EAs in the former atoms is that they correspond to the anionic BEs of excited states, while in the latter atoms the EAs are identified with the Regge pole BEs of the formed anions in their ground states.

**Table 1.** Negative-ion binding energies (BEs) and ground states Ramsauer–Townsend (R-T) minima, all in eV extracted from TCSs of the atoms Au, Pt, At, Ti and Hf and the fullerene molecules C60 and C70. They are compared with the measured electron affinities (Eas) in eV. GRS, MS-*n* and EXT-*n* (*n* = 1, 2) refer, respectively, to ground, metastable and excited states. Experimental EAs, EXPT and theoretical EAs are also included. The numbers in the square brackets are the references.



**Table 1.** *Cont.*

#### *3.2. Ground State Fullerene Cross Sections*

The main reason for following up with the fullerene molecules immediately after Section 3.1 is that excellent measured EAs from C20 through to C92 are available in the literature. Benchmarked on the measured EAs of C60 and C70, as indicated under Section 3.1, the Regge pole method was used to calculate the ground state electron elastic TCSs of the fullerene molecules from C20 through to C240 [16,17]. It is noted here that in the paper [17] only the ground state anionic BEs were calculated and some of the fullerenes ground state BEs can be found there. The novelty and generality of the Regge pole approach is in the extraction of the anionic BEs from the calculated TCSs of the fullerenes, for ground state collisions these BEs yield the unambiguous and definitive challenge to calculate theoretically EAs. In [16,17], the ground state anionic BEs of the fullerenes C20 through to C92 were found to generally match excellently with the measured EAs. Indeed, these results provided grea<sup>t</sup> credence to the ability of the Regge pole method to extract from the calculated TCSs reliable EAs of the fullerene molecules for the first time. The obtained agreemen<sup>t</sup> represented an unprecedented accomplishment by the Regge pole method, requiring no assistance whatsoever from either experiment or any other theory for the feat. This allowed the interpretation of the EAs of fullerenes as corresponding to the ground state anionic BEs calculated by the Regge pole method. For the fullerenes, other theories continue to struggle to go beyond the theoretically simple C20 and C60 fullerenes.

The focus in this Section is on the ground state anionic BEs of the fullerenes from C20 through to C240 [16,17]. These ground states TCSs are typified by the red curve of the C60 TCSs of Figure 1. In addition to the ground state curve, the revealing metastable and excited state TCSs curves demonstrate the richness in structure of the fullerene TCSs (the larger fullerenes reveal more metastable and excited state TCSs than shown in Figure 1). For the C20, C24, C26, C28 and C44 we have used the data of [16] to demonstrate the reliability of the Regge pole-calculated anionic BEs, since their measured EAs are available. The BE values in Table 2 cover a wider range of the fullerene anionic BEs than those shown in the papers [16,17]. They include values where the experimental EAs are unavailable. In [16], the smaller fullerenes C20, C24, C26, and C28 as well as the larger fullerenes C92 and C112 were studied to assess the extent to which fullerenes behave like "big atoms", as suggested by Amusia [61]. These TCSs were found to be characterized generally by ground, metastable and excited states negative-ion formation, R–T minima and SRs. The ground states anionic BEs correspond to the measured EAs of the fullerenes, see Table 2 as well as Figure 1 for the C60 TCSs. The Regge pole method does not determine the orbital angular momentum of the attached electron. This is particularly important for the C60 fullerene since there is an uncertainty in the literature regarding whether the C60 EA of 2.66 eV corresponds to an s-state or a *p*-state of the attached electron. Since the ground state anionic BE (EA) of C60 is determined here, we believe that a structure-type calculation could use our calculated ground state anionic BE (EA) of C60 to determine the ground-state fine-structure energies, since the metastable energies are also available. Thus, the lingering question could be answered.

The Regge pole-calculated low-energy electron elastic TCSs for the ground and the first (highest) excited states of fullerenes are robust. For C20 (smallest fullerene), the first excited state TCS (highest TCS) resembles that of atomic Au, see Figure 1. Defining R as the value of the ratio of the second to the first R–T minima in the first excited state TCS of C20, in [16] we explored the variation of R from C20 through to C70. We found that for C20 R (~1.4), greater than unity was close to that for Au (~2.) or Th (~1.9), indicative of atomic behavior, while for C24 R was about 1.0. For C70, R was less than 0.5 demonstrating strong departure from atomic behavior due to the significant polarization interaction in C70; which also induces long-lived metastable anions in the C70 TCSs. When probed with low-energy electrons, the results for C20 exhibited fullerene behavior consistent with the view that fullerenes behave like "big atoms" [61]. The atomic behavior quickly disappears with the increase in the fullerene size. As seen from the Figures of [16], the behavior in C28 is no longer atomic because R is less than unity. By C92, the departure from atomic behavior has become significant, due to the increase in the polarization interaction in these larger systems.

For C20, the excited state TCS [16] exhibits a deeper R–T minimum near threshold in comparison with the second R–T minimum, while the ground state TCS ends with a deep R–T minimum, wherein appears the dramatically sharp resonance representing the stable negative ion formed in the ground state during the collision, see also Figure 1. These characteristic R–T minima, also observed in the Dirac R-matrix low-energy electron elastic scattering cross sections calculations for the heavy, alkali-metal atoms Rb, Cs and Fr [62], manifest that the important core-polarization interaction has been accounted for adequately in our calculation, consistent with the conclusion in [63]. The vital importance of the corepolarization interaction in low-energy electron collisions with atoms and molecules was recognized and demonstrated long ago, see [64] for examples and references therein. In C20, the TCSs are characterized by a ground, metastable and excited states TCSs. However, the C24, C26 and C28 TCSs consist of more metastable and excited states TCSs. Suffice to state that the increased energy space determined mainly by the ground states BEs is conducive to the appearance of the polarization-induced metastable TCS in general. Indeed, these results reveal the complicated interplay between the R–T minima and the shape resonances.

Notably, in all the fullerene molecules investigated here, the ground states anionic BEs occur at the absolute R–T minima of the TCSs, see Figure 1 for example. This facilitates considerably the determination of unambiguous and reliable EAs of the fullerene molecules. Noted here also is that generally the sharp resonances of the metastable TCSs lie between the ground states SRs and the dramatically sharp resonances of the ground states. In Table 2, we have presented various fullerene anionic BEs, but mainly ground states BEs and compared them with the measured EAs where these are available. The results demonstrate the power of the Regge pole method to reliably calculate the anionic BEs of fullerene molecules.

Clearly, the Regge pole approach, entirely new in the field of electron-cluster/fullerene collisions, implemented here represents a theoretical breakthrough in low-energy electron scattering investigations of fullerenes/clusters and complex heavy atoms. Its implementation should speed up the long overdue fundamental theoretical understanding of the mechanism underlying low-energy electron scattering from fullerenes, including heavy and complex atoms, leading to negative ion formation. These results should also help in the construction of the popular square-well potentials for the investigated fullerenes. Most importantly, its grea<sup>t</sup> strength is in the ability to produce reliable data without assistance from experiments and/or other theories.

**Table 2.** Fullerene ground (GR-S), metastable (MS-*n, n* = 1, 2, 3) and first excited (EXT-1), second excited (EXT-2) and third excited (EXT-3) anionic states binding energies (BEs). The measured EAs are represented as EXPT and the other theoretical values are denoted as Theory. All the energies are in eV and the numbers in the square brackets are the references. For most of the fullerenes the parameters "α" and "β" of the potential, Equation (2) are tabulated in [17].


#### *3.3. Cross Sections for the Large Atoms Hf, Pt, Au, Ti and At*

In the context of the viewpoints (1) and (2) of the Introduction, it is appropriate to discuss the measured EAs of the large atoms Hf(72), Pt(78), Au(79), Ti(81) and the radioactive At (85) in an attempt to understand the meaning of the measured EAs of the Hf and Ti atoms (the numbers within the brackets are the Zs). That is, do their EAs correspond to electron BEs in the ground, the metastable or the excited states of the formed anions during the collision? As seen from Table 1, the meaning of the EAs of Au, Pt and At is clear, namely it corresponds to the ground state anionic BEs of the formed anions during the collisions. However, for Hf and Ti the meaning lacks definitiveness.

For clarity, Figure 2 shows the TCSs for the Hf atom; a similar Figure was obtained for the Ti atom. As seen from the Figure, it is difficult to understand any selection of the

anionic BEs other than the ground state anionic BE as the EA of Hf. A similar argumen<sup>t</sup> applies to the Ti atom. The measured EA of Hf at 0.178 eV [26], the Regge pole-calculated SR of 0.232 eV, the RCI EA of 0.114 eV [27] and the Regge pole-calculated second excited state anionic BE of 0.113 eV [28] are reasonably close together. The highest excited state BE of Hf is at 0.017 eV [29]. The TCSs for Hf presented in Figure 2 demonstrate the additional presence to the above discussed anionic BEs, a metastable TCS (green curve) and a ground state TCS (pink curve) with anionic BEs of 0.525 eV and 1.68 eV, respectively. Indeed, here we are faced with the problem of interpretation of what is meant by the EA.

**Figure 2.** Total cross sections (a.u.) for electron elastic scattering from Hf. The pink, green, orange and purple curves represent the TCSs for the ground, metastable and the two excited states, respectively. The dramatically sharp resonances correspond to the Hf; anionic formation during the collisions.

As indicated in the Introduction, for the Ti atom two measurements obtained its 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 anionic state, namely 0.281 eV [56], see comparisons in Table 1. However, the measured value of 0.075 eV [22] and the Regge pole-calculated BE of the highest excited state of the formed Ti; anion, 0.0664 eV are close together. We note that the Regge pole-calculated ground state anionic BE of Ti is 2.42 eV [56], very close to that of the At atom. Clearly, the results of the Hf and Ti atoms are difficult to interpret without a rigorous theoretical data, as discussed under Section 3.4 dealing with the lanthanide atoms.

#### *3.4. Cross Sections for the Lanthanide Atoms*

The EA provides a stringent test of theory when the theoretical EAs are compared with those from reliable measurements. This statement holds strongly in the case of the lanthanide atoms. The general problem of interpretation of the measured EAs of the lanthanide atoms has been exposed in [29], as well as elucidated through the rigorous Regge pole method [73]. Appropriately, we begin this section by placing in perspective the existing measurements/calculations of the EAs of the lanthanide atoms. Low-energy electron elastic collision cross sections for the lanthanide atoms, La through Lu were first investigated using the CAM (Regge pole) method [33]. Unfortunately, the investigation was limited to the near threshold energy region, 0.0 ≤ E ≤ 1.0 eV and focused upon the comparison with the existing measured and theoretical EAs and never questioned the meaning of the EAs. The CAM calculated TCSs were found to be characterized generally by dramatically sharp resonances whose energy positions were identified with the measured/calculated EAs of the lanthanide atoms. The extracted EAs from the TCSs varied from a low value of 0.016 eV for Tm to a high value of 0.631 eV for atomic Pr. In that paper [33], one sees the

effective use of the Regge Trajectories and the Im L (L is the complex angular momentum) in analyzing and interpreting the results. Moreover, the lanthanide parameters "α" and "β" of the potential, Equation (2) are tabulated in that paper [33].

Subsequently, when the energy range was increased from 1.0 eV to 10.0 eV, ground, metastable and excited states anionic BEs were clearly revealed and delineated. Then, the question persisted: do the measured EAs of the lanthanide atoms correspond to the anionic BEs of electron attachment in the ground, metastable or excited states of the formed anions during the collision? For atomic Eu, the resonance at E = 0.116 eV with Im L = 7.6 × 10−<sup>6</sup> of the Figure 6 of Ref. [33] should be compared with the results of Figure 2 of Ref. [30] reproduced here for convenience as Figure 3 (left panel). In Figure 13 of Ref. [33] depicting the TCSs of Tm the dramatically sharp resonance at E = 0.016 eV with Im L = 3.4 × 10−<sup>5</sup> should be viewed in the context of the recent Figure 2 of Ref. [30], also presented here as Figure 3 (right panel).

**Figure 3.** Total cross sections (a.u.) for electron elastic scattering from atomic Eu (**left panel**) and Tm (**right panel**). For Eu the pink, red and blue curves represent the TCSs for the ground, metastable and the excited states, respectively. For Tm atom the red, green, pink and black curves represent the TCSs for the ground, metastable and the two excited states, respectively. The dramatically sharp resonances in both figures correspond to the Eu; and Tm; negative-ions formed during the collision.

The lanthanide and the Hf atoms provide clear cases of the ambiguous and confusing measured and/or calculated EA values. As examples, for Eu we focus on the ground state, pink curve with the BE value of 2.63 eV and the blue curve with the BE of 0.116 eV, corresponding to an excited state TCS. The measured EA (0.116 eV) [3] is in outstanding agreemen<sup>t</sup> with the excited state BE value above and the RCI calculated EA (0.117 eV) [36], see Table 1 of [30]. The metastable BE value of 1.08 eV, red curve in Figure 3 (left panel) agrees excellently with the measured EA (1.053 eV) [35]. This clearly demonstrates the ambiguous and confusing meaning of the measured EA of Eu by Refs. [3,35]. Does the EA of Eu correspond to the BE of electron attachment in the metastable state or in the excited state of the formed anion during the collision? Similarly with the case of the Tm atom, the Regge pole calculated ground and excited states BEs are, respectively, 3.36 eV and 0.274 eV. The measured EA of Tm is 1.029 eV [37] and agrees excellently with the Regge pole calculated metastable state BE value of 1.02 eV, green curve in the Figure 3 (right panel). Accordingly, here the meaning of the measured EA of Tm corresponds to the BE of the metastable state. In both Eu and Tm atoms, the meaning of the measured EAs is ambiguous and confusing as well.

A comment is appropriate here regarding the importance or unimportance of Relativistic effects in the calculation of the EAs using the Regge pole method. With a relatively high Z of 63, but a small measured EA of 0.116 eV [3], the Eu atom provides a stringent test of the nonrelativistic CAM method when its prediction (EA = 0.116 eV) [33] is compared

with that calculated using the MCDF-RCI (EA = 0.117 eV) [36]. The interpretation aside, the results demonstrate the unimportance of relativistic effects in the calculation of the EA of Eu. Indeed, the EAs calculated using structure-based theoretical methods tend to be riddled with uncertainty and lack definitiveness for complex multi-electron systems and fullerene molecules. For instance, relativistic effects in gold chemistry were investigated by Wesendrup et al. [74] who performed large-scale fully Relativistic Dirac–Hartree–Fock and MP2 calculations as well as nonrelativistic pseudopotential calculations and obtained the EAs of 2.19 eV and 1.17 eV, respectively. These values should be contrasted with the CAM calculated value of 2.263 eV [25] and compared with the measured EAs of Au in Table 1. Of importance, also, is the review on relativistic effects in homogeneous gold catalysis [75].

#### *3.5. Cross Sections for the Actinide Atoms*

In [54], we investigated the low-energy electron scattering from the radioactive actinide atoms Th, Pa, U, Np and Pu through the elastic TCSs calculations. The objective was to delineate and identify the characteristic resonance structures, as well as to understand and assess the reliability of the existing theoretical EAs. The recent measurement of the EA value of Th warrants some remark. There is no reason whatsoever for the selective comparison of data by the experiment; there are calculated EAs in the literature [36,76]. Particularly interesting in the study above [54], is the finding for the first time that the TCSs for atomic Pu exhibited fullerene molecular behavior near threshold through the TCS of the highest excited state, while maintaining the atomic character through the ground state TCS. Also, the first appearance of the near threshold deep R–T minimum in the actinide TCSs was first identified in the TCSs of atomic Pu, see Figure 5 of [54].

Figure 4, taken from Ref. [54] with a slight modification due to recalculation presents the TCSs for atomic Th (top figure) and U (bottom figure). They typify the TCSs of the complex multi-electron actinide atoms. Importantly, they are characterized by dramatically sharp resonances representing negative-ion formation in the ground, metastable and excited anionic states, R–T minima and SRs. In both Figures, the red curves represent electron attachment in the ground states while the pink curves denote the highest excited states. For Th, Figure 4 (top) the measured and the calculated EA values are 0.608 eV and 0.599 eV [18], respectively. These values are close to the Regge pole-calculated anionic BE of the second excited state, pink curve (0.549 eV). Close to this value there is a SR at 0.61 eV defined by the blue curve; the ground state anionic BE is at 3.09 eV. Not shown is the highest excited state curve with anionic BE value of 0.149 eV. Importantly, here we note the clear atomic behavior exhibited by the Th TCSs (pink curve) as expected [16]. However, the pink curve in the U TCSs shows strong fullerene behavior [16]. The EAs of U have been measured very recently to be 0.315 eV [19] and 0.309 eV [20] as well as calculated to be 0.232 eV [20]. These values are close to the Regge pole anionic BE value of 0.220 eV for the highest excited state, see Table 3 and Ref. [54] for additional comparisons. Here we do not understand the inconsistency in the meaning of the EAs in Figures 1 and 4, namely as corresponding to the BEs of electron attachment in the ground and the excited anionic states, respectively.

Of particular interest and importance here are the contrasted TCSs for atomic Am and Lr in [77]. In that paper, the polarization-induced TCS (brown curve) of Am exhibited a deep R–T minimum near the threshold. This brown curve behaves similarly to that in the TCSs of Pu, while in Lr it has already flipped over to a SR. Where does the actual flipping take place? This is the subject of Section 3.7. To understand the measurement [38], we need the previously unavailable data for the Cf, Fm and Md atoms to determine where the actual flipping takes place. There are no measured EAs for the actinide atoms beyond U to compare our BEs. However, theoretical EAs are available [36,76,78,79] and these have been compared with our data for the actinide atoms [54,77,80], see also Table 3. For atomic Lr the EA values of 0.310 eV [78] and 0.295 eV [36] are very close to the Regge pole BE of the highest excited state, namely 0.321 eV. These values should guide the reliable measurement of the EA of Lr. We have brought together the data for most of the actinide atoms in Table 3 mainly for convenience of analysis. Hitherto fore, they were scattered all over the literature. Importantly, the recent data from [80] are crucial for understanding Section 3.7.

**Figure 4.** Total cross sections (TCSs) for atomic Th (**top figure**) and U (**bottom figure**) Figures. In this paper the relevant curves in both TCSs are the ground states (red curves) and the excited states (pink curves). The dramatically sharp resonances in both Figures with attendant BEs represent electron attachment. These BE values are intended to guide the eye; the complete values are presented in Table 3. There are also shape resonances next to these sharp peaks.

**Table 3.** Negative ion binding energies (BEs) in eV and energy positions of ground-state Ramsauer– Townsend (R-T) minima, in eV obtained from the TCSs for the actinide atoms from Pu through Lr. Additionally, included for comparison are the data for Au, Th and U. GRS, MS-*n* and EXT-*n* (*n* = 1, 2) represent ground, metastable and excited states, respectively. The experimental EAs, EXPT and the theoretical EAs, including RCI [36] and GW [76] are also presented. The numbers in the square brackets are the references.



**Table 3.** *Cont.*

It is now clear why many existing experimental measurements and sophisticated theoretical calculations have considered the anionic BEs of the stable metastable and/or excited negative ion formation to correspond to the EAs of the considered lanthanide and actinide atoms. This is contrary to the usual meaning of the EAs found in the standard measurement of the EAs of such complex systems as atomic Au, Pt and the radioactive At, as well as of the fullerene molecules. In these systems, the EAs correspond to the ground state BEs of the formed negative ions. The negative ions obtained here are also important in catalysis.

## *3.6. Fullerene Negative-Ion Catalysis*
