Fullerene Negative Ions: Formation and Catalysis

We first explore negative-ion formation in fullerenes C44 to C136 through low-energy electron elastic scattering total cross sections calculations using our Regge-pole methodology. Then, the formed negative ions C44ˉ to C136ˉ are used to investigate the catalysis of water oxidation to peroxide and water synthesis from H2 and O2. The exploited fundamental mechanism underlying negative-ion catalysis involves hydrogen bond strength-weakening/breaking in the transition state. Density Functional Theory transition state calculations found C60ˉ optimal for both water and peroxide synthesis, C100ˉ increases the energy barrier the most, and C136ˉ the most effective catalyst in both water synthesis and oxidation to H2O2.


Introduction
To celebrate the International Year of the Periodic Table, the Royal Society of Chemistry published the themed collection 'Single Atoms as Active Catalysts' [1]. This has motivated the present investigation of using single fullerene molecular anions as catalysts. Toward this end, we first investigate the formation of negative ions in the fullerene molecules C 44 , C 60 , C 70 , C 98 , C 112 , C 120 , C 132 , and C 136 through low-energy electron elastic scattering total cross sections (TCSs) calculations. Our robust Regge-pole methodology is used for the calculations. The formed anionic fullerenes C 44 − to C 136 − during the collisions are then used to investigate the catalysis of water oxidation to peroxide and water synthesis from H 2 and O 2 . Negative ion catalysis involves anionic molecular complex formation in the transition state, with the atomic negative ion weakening/breaking the hydrogen bond strength. This is the same fundamental mechanism that underlies the well-investigated muon-catalyzed nuclear fusion using a negative muon, a deuteron, and a triton; it has been proposed to drive nanoscale catalysis [2,3]. Specifically, in the experiments [4][5][6], the fundamental atomic mechanism responsible for the oxidation of water to peroxide catalyzed by Au and Pd nanoparticles has been attributed to the interplay between Regge resonances and Ramsauer-Townsend (R-T) minima in the electron elastic TCSs for the Au and Pd atoms, along with their large electron affinities (EAs) [2,3]. The mechanism of negative-ion catalysis has been demonstrated in the oxidation of H 2 O to H 2 O 2 catalyzed using the Au − and Pd − anions to understand the experiments of Hutchings and collaborators [4][5][6], in the catalysis of light, intermediate and heavy water to the corresponding peroxides [7], and in the oxidation of methane to methanol without the CO 2 emission [8] to name a few. Briefly, the experiments [4][5][6] synthesized hydrogen peroxide from H 2 and O 2 using supported on Fe 2 O 3 Au, Pd, and Au-Pd nanoparticles as catalysts. Importantly, these experiments found that the addition of Pd to the Au catalyst increased the rate of H 2 O 2 synthesis significantly as well as the concentration of the formed H 2 O 2 . In [4], it was found that the production of H 2 O 2 increased 7and 30-fold over that of the Au catalyst alone when using the Pd and Au-Pd, respectively. Recently,

Results
In Section 2.1 we first present the variation with the electron impact energy E of the Regge-pole calculated electron elastic scattering TCSs for the fullerene molecules C 44 to C 136 . Section 2.2 demonstrates the utility of the fullerene molecular anions in the catalysis of water oxidation to peroxide and water synthesis from H 2 and O 2 using the anionic fullerene catalysts C 44 − to C 136 − .

Fullerene Electron Scattering Total Cross Sections
In fullerene negative ion formation, it has been demonstrated for the first time that the ground state anionic BEs extracted from our Regge-pole calculated electron elastic scattering TCSs for the C 20 through C 92 fullerenes matched excellently the measured EAs of these fullerenes [18,19]. This provides a novel and general approach to the determination of unambiguous and reliable EAs for complex heavy systems. The Regge-pole methodology requires no assistance whatsoever from either experiment or  other theory to achieve the remarkable feat.  Figures 1 and 2 present the elastic TCSs for the fullerene molecules C 44 through C 136 and Table 1 summarizes the essential data. Indeed, the Regge-pole calculated low-energy electron elastic TCSs for the fullerene molecules considered here are found to be characterized generally by ground, polarization-induced metastable and excited negative ion formation. For ground state collisions the resultant anionic BEs yield the theoretically challenging to calculate EAs and demonstrate their wide variation from fullerene to fullerene. The results here are consistent with the observation that low-energy electron-fullerene interactions are generally characterized by rich resonance structures [32,[48][49][50] and that the experimentally detected fullerene isomers correspond to the metastable states [51]. They also support the conclusion that the EAs of fullerene molecules are relatively large [52]. This should satisfy part of the requirement to increase fullerene acceptor resistance to degradation by the photo-oxidation mechanism as well as improve the understanding of the degradation mechanism in organic solar cells [53]. The determined EAs here could also be employed to construct the widely used simple model potentials for the fullerene shells, including endohedral fullerenes [54]. The resonance-rich structures of the fullerene TCSs and their large EAs explain the tendency of fullerenes to form compounds with electron-donor anions and their vast applications as well. These TCSs require careful delineation and identification of the attendant resonance structures for reliable interpretation as well as extraction of the EAs. provides a novel and general approach to the determination of unambiguous and reliable EAs for complex heavy systems. The Regge-pole methodology requires no assistance whatsoever from either experiment or other theory to achieve the remarkable feat. Figures 1 and 2 present the elastic TCSs for the fullerene molecules C44 through C136 and Table  1 summarizes the essential data. Indeed, the Regge-pole calculated low-energy electron elastic TCSs for the fullerene molecules considered here are found to be characterized generally by ground, polarization-induced metastable and excited negative ion formation. For ground state collisions the resultant anionic BEs yield the theoretically challenging to calculate EAs and demonstrate their wide variation from fullerene to fullerene. The results here are consistent with the observation that low-energy electron-fullerene interactions are generally characterized by rich resonance structures [32,[48][49][50] and that the experimentally detected fullerene isomers correspond to the metastable states [51]. They also support the conclusion that the EAs of fullerene molecules are relatively large [52]. This should satisfy part of the requirement to increase fullerene acceptor resistance to degradation by the photo-oxidation mechanism as well as improve the understanding of the degradation mechanism in organic solar cells [53]. The determined EAs here could also be employed to construct the widely used simple model potentials for the fullerene shells, including endohedral fullerenes [54]. The resonance-rich structures of the fullerene TCSs and their large EAs explain the tendency of fullerenes to form compounds with electron-donor anions and their vast applications as well. These TCSs require careful delineation and identification of the attendant resonance structures for reliable interpretation as well as extraction of the EAs.      For a better appreciation of the physics underlying the resonance-rich TCSs for the various fullerenes presented in the Figures 1 and 2, we first discuss briefly the TCSs for the C44 fullerene. With less structure, the TCSs were first calculated in [18]; here they have been recalculated to expose more resonances. It is noted that generally the internal region of zero potential provided by the hollow cage structure of the fullerenes is conducive to metastable anionic formation during the collisions. This is clearly manifested through the appearance of additional resonances in the TCSs as  Table 1. 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). R-T refers to the energy position of the ground state R-T minimum. The measured EAs are represented as Expt. All the energies are in eV. For a better appreciation of the physics underlying the resonance-rich TCSs for the various fullerenes presented in the Figures 1 and 2, we first discuss briefly the TCSs for the C 44 fullerene. With less structure, the TCSs were first calculated in [18]; here they have been recalculated to expose more resonances. It is noted that generally the internal region of zero potential provided by the hollow cage structure of the fullerenes is conducive to metastable anionic formation during the collisions. This is clearly manifested through the appearance of additional resonances in the TCSs as the fullerene size increases from C 44 through C 136 . Also, this explains the existence of the two series of resonances, the first is associated with the ground state TCS while the second belongs to the highest excited state TCS (green curve).

Full. Bes GR-S
Focusing specifically on the C 44 TCSs, Figure 1a, the red, blue, pink, brown and green curves represent respectively the TCSs of the ground; the first & the second metastable and the two excited states. The fundamental physics underlying these curves can be readily understood if we focus on each color-coded TCS. For the analysis we select the ground state TCS curve, the red curve. Near threshold the TCS exhibits the characteristic shape resonance (SR), broad maximum. As the electron energy is increased, the fullerene becomes polarized and reaches maximum polarization manifested through the appearance of the first R-T minimum at about 1.01 eV, indicative that the polarization interaction has been accounted for adequately in the calculation [60]. With further increase in the electron impact energy, the electron becomes trapped by the centrifugal potential, demonstrated by the appearance of the SR at 1.41 eV. As the electron leaks out of the centrifugal potential, the C 44 shell, due to its strong polarizability, becomes significantly polarized leading to the generation of the second deep R-T minimum in the TCS at 3.13 eV. At the absolute minimum the long-lived ground state of the C 44 − anion is formed with the BE of 3.15 eV. At the R-T minimum the electron spends many angular rotations about the C 44 as it decays; the angular life is determined by 1/[Im λ n (E)] → ∞, since for the ground state resonance Im λ n (E) → 0, see Equation (1). Notably, at the R-T minimum new molecules can be created from fermions. The analysis is also applicable to the other fullerene TCSs presented in Figure 1 as well as in Figure 2. The extracted BEs of the negative ions formed during the collisions are summarized in Table 1 where they are compared with available EAs. Indeed, for the ground state collisions the extracted from the TCSs anionic BEs correspond to the EAs of the fullerenes. The Regge-pole calculated TCSs for the C 60 fullerene presented in Figure 1b is taken from [39]. The TCSs, typical of those calculated in this paper, are found to be characterized generally by dramatically sharp resonances manifesting ground, metastable and excited anionic formation during the collisions, Ramsauer-Townsend (R-T) minima and shape resonances. Indeed, the ground state TCS (red curve) yields the anionic BE, located at its absolute R-T minimum; it has been identified with the C 60 EA [19]. Viewed as presented in the Figure 1b the C 60 TCSs appear complicated as well. However, they are readily understood and interpreted as was done in [19,39]. This ground state TCS is clearly shown alone in Figure 1 of [19] and the underlying physics is also presented there.
Figures 1b and 2 demonstrate the variation of the electron TCSs with E for the C 60 , C 70 , C 98 , C 112 , C 120 , C 132 and C 136 fullerene molecules. Clearly, these TCSs are characterized as in the C 44 case by ground, metastable and excited anionic formation, R-T minima and shape resonances. The extracted anionic BEs from the ground states TCSs correspond to the EAs of the fullerene molecules. These BEs, presented in Table 1 demonstrate their wide variation from fullerene to fullerene. The various dramatically sharp resonances in the TCSs represent negative ion formation in the ground, metastable and excited states.

Fullerene Transition State Barriers
The utility of the fullerene negative ions has been demonstrated in the catalysis of water oxidation to peroxide and water synthesis from H 2 and O 2 using the anionic fullerene catalysts C 44 − to C 136 − .
The reactions of interest are: Water Oxidation to Peroxide Reaction: Water Synthesis Reaction: Reaction (1) is similar to Equation (1) of Ref. [2] where the active catalyst is the Au − anion. The processes considered here are exactly similar to that, except that here the Au − anion catalyst is replaced by the C 44 − to C 136 − anion catalysts. We will therefore use the familiar Au − anion catalyst to explain and demonstrate the importance of the transition state (TS) in the reactions. Additionally, in the end, we will simply replace the Au − with the C 44 − to C 136 − anion catalysts. Since the final product, viz. Equation (4) of Ref [2] is devoid of the catalyst as it should, we look at the transition states, Equation DFT transition state calculations found the C 52 − and C 60 − anions to be numerically stable for both water oxidation and water synthesis and the C 100 − anion to increase the energy barrier the most in the water synthesis from H 2 and O 2 . When catalyzing both water oxidation to peroxide and synthesis from H 2 and O 2 , the C 136 − anion has proved to be the most effective in reducing the energy barrier significantly. Importantly, a single large fullerene such as the C 136 , C 120 , or even the C 70 could replace the Au, Pd, and Sn atoms in the catalysis of H 2 O 2 from H 2 O in the experiments of Hutchings and collaborators [4][5][6] acting as a multiple-functionalized catalyst. These fullerenes have their metastable BEs close to the EAs of the used atoms in the experiments. Thus, an inexpensive dynamic water purification system for the developing world could be realized [6]. replaced by the C44ˉ to C136ˉ anion catalysts. We will therefore use the familiar Auˉ anion catalyst to explain and demonstrate the importance of the transition state (TS) in the reactions. Additionally, in the end, we will simply replace the Auˉ with the C44ˉ to C136ˉ anion catalysts. Since the final product, viz. Equation (4) of Ref [2] is devoid of the catalyst as it should, we look at the transition states, Equation (2), and Equation (1). In the oxidation of H2O to H2O2 catalyzed by the Auˉ anion, the anion-molecular complex Auˉ(H2O)1,2 is formed in the TS. This complex subsequently breaks up into Auˉ and (H2O)1 and (H2O)2. The large EA of atomic Au played an essential role in the process. It is important in the dissociation energy of the complex Auˉ(H2O)1,2 into the above products. The need in negative ion catalysis for systems with reliable EAs is now evident. In the present calculation, we simply replace the Auˉ anion catalyst with the fullerene anion catalysts. Figures 3 and 4 demonstrate the Density Functional Theory (DFT) calculated transition states. DFT and dispersion corrected DFT approaches have been employed for the transition state evaluations. Geometry optimization of the structural molecular conformation utilized the gradient-corrected Perdew-Burke-Ernzerhof parameterizations [61] of exchange-correlation as implemented in DMol3 [62]. A tolerance of 1x10 -3 Ha was used with a smearing value of 0.005 Ha. DFT calculated energy barriers reduction in the oxidation of H2O to H2O2 catalyzed using the anionic fullerene catalysts C44ˉ to C136ˉ are shown in Figure 3. Results in Figure 4 are for the water synthesis from H2 and O2 catalyzed using the anionic fullerene catalysts C44ˉ to C136ˉ as well.
DFT transition state calculations found the C52ˉ and C60ˉ anions to be numerically stable for both water oxidation and water synthesis and the C100ˉ anion to increase the energy barrier the most in the water synthesis from H2 and O2. When catalyzing both water oxidation to peroxide and synthesis from H2 and O2, the C136ˉ anion has proved to be the most effective in reducing the energy barrier significantly. Importantly, a single large fullerene such as the C136, C120, or even the C70 could replace the Au, Pd, and Sn atoms in the catalysis of H2O2 from H2O in the experiments of Hutchings and collaborators [4][5][6] acting as a multiple-functionalized catalyst. These fullerenes have their metastable BEs close to the EAs of the used atoms in the experiments. Thus, an inexpensive dynamic water purification system for the developing world could be realized [6].

Method of Calculation
In [63] it was confirmed that Regge poles formed during low-energy electron elastic scattering become stable bound states. Here we adopt the Regge-pole methodology, also known as the complex angular momentum (CAM) method for the calculation of the electron scattering TCSs. Regge poles, singularities of the S-matrix, rigorously define resonances [64,65]. Being generalized bound states, they can be used to calculate reliably the anionic BEs of the ground, metastable and excited states of complex heavy systems through the TCSs calculations. The Mulholland formula [66] is used here to calculate the near-threshold electron-fullerene collision TCS resulting in negative ion formation as resonances. In the form below, the TCS fully embeds the essential electron-electron correlation effects [67,68]  , m being the mass and E the impact energy, ρn is the residue of the S-matrix at the n th pole, λn and I(E) contains the contributions from the integrals along the imaginary λ-axis; its contribution has been demonstrated to be negligible [69].
As in [26] the complicated details of the electronic structure of the fullerene itself are not considered here. The incident electron is assumed to interact with the complex atom/fullerene through the Thomas-Fermi type potential, known as the Avdonina, Belov and Felfli (ABF) potential [70] which accounts for the vital core-polarization interaction In Eq. (4) Z is the nuclear charge, α and β are variation parameters. This potential has the appropriate asymptotic behavior, viz. ~ −1/(αβr 4 ) and accounts properly for the polarization interaction at low energies. This potential, extensively studied [71], 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 Eq. (4) ensures that spherical and non-spherical atoms/fullerenes are correctly treated.
The effective potential 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 [68] and further details of the calculations may be found in [72].
In the calculations, the optimal value of α was determined to be 0.2. When the TCS as a function of β has a dramatically sharp resonance [69], corresponding to the formation of a stable negative ion, this resonance is longest lived for a given value of the energy, which corresponds to

Method of Calculation
In [63] it was confirmed that Regge poles formed during low-energy electron elastic scattering become stable bound states. Here we adopt the Regge-pole methodology, also known as the complex angular momentum (CAM) method for the calculation of the electron scattering TCSs. Regge poles, singularities of the S-matrix, rigorously define resonances [64,65]. Being generalized bound states, they can be used to calculate reliably the anionic BEs of the ground, metastable and excited states of complex heavy systems through the TCSs calculations. The Mulholland formula [66] is used here to calculate the near-threshold electron-fullerene collision TCS resulting in negative ion formation as resonances. In the form below, the TCS fully embeds the essential electron-electron correlation effects [67,68] (atomic units are used throughout): In Equation (3) S(λ) and λ are respectively the S-matrix and the CAM, k = √ 2mE, m being the mass and E the impact energy, ρ n 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; its contribution has been demonstrated to be negligible [69].
As in [26] the complicated details of the electronic structure of the fullerene itself are not considered here. The incident electron is assumed to interact with the complex atom/fullerene through the Thomas-Fermi type potential, known as the Avdonina, Belov and Felfli (ABF) potential [70] which accounts for the vital core-polarization interaction In Equation (4) Z is the nuclear charge, α and β are variation parameters. This potential has the appropriate asymptotic behavior, viz.~−1/(αβr 4 ) and accounts properly for the polarization interaction at low energies. This potential, extensively studied [71], 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 (4) ensures that spherical and non-spherical atoms/fullerenes are correctly treated. The effective potential V(r) = U(r) + λ(λ + 1)/2r 2 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 [68] and further details of the calculations may be found in [72].
In the calculations, the optimal value of α was determined to be 0.2. When the TCS as a function of β has a dramatically sharp resonance [69], corresponding to the formation of a stable negative ion, this resonance is longest lived for a given value of the energy, which corresponds to the EA of the system (for ground state collisions) or the BE of the metastable/excited anion. Also calculated in the CAM methods are the Regge Trajectories, viz. Im λ n (E) versus Re λ n (E); they have been used to demonstrate that at low energy relativistic and non-relativistic calculations yield the same results [73].

Conclusions
The Regge-pole calculated low-energy electron elastic TCSs for the fullerene molecules considered here are found to be characterized generally by ground, metastable, and excited negative ion formation. Indeed, the rich resonance structures of 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 formed negative ions has been demonstrated in the catalysis of water oxidation to peroxide and water synthesis from H 2 and O 2 using the anionic fullerene catalysts C 44 − to C 136 − .
Transition state calculations using DFT found the C 52 − and C 60 − anions to be robust (yielding essentially the same transition state energies) for both water and peroxide synthesis and the C 136 − to be the most effective in reducing the energy barrier significantly. Importantly, a single large fullerene such as the C 136 , C 120 , or even the C 60 could replace the Au, Pd, and Sn atoms in the catalysis of H 2 O 2 from H 2 O in the experiments [4][5][6] acting as a multiple-functionalized catalyst. Thus, an inexpensive dynamic water purification system could be realized through the use of fullerene anions as catalysts. Furthermore, these fullerenes could also be used as catalysts in the production of methanol from methane without carbon dioxide emission with significant impact on the environment.
Author Contributions: Conceptualization, methodology, investigation, formal analysis and writing of the original draft as well as rewriting and editing were carried out by A.Z.M. He is also responsible for securing the funding for the research. Z.F. was responsible for the theoretical calculations of the electron scattering total cross sections for the fullerene molecules, acquisition of the corresponding data and plotting as well as tabulating them. K.S. conceptualized the fullerene negative ion catalysis and calculated the transition state energy barriers of the fullerene molecules. He also plotted the relevant figures. N.N. assisted in collecting the references and organized them. He also helped in the calculation of the transition states of the fullerenes. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.