1. 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
44ˉ to C
136ˉ 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
2O to H
2O
2 catalyzed using the Au
ˉ and Pd
ˉ 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
2O
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
2O
2 synthesis significantly as well as the concentration of the formed H
2O
2. In [
4], it was found that the production of H
2O
2 increased 7- and 30-fold over that of the Au catalyst alone when using the Pd and Au-Pd, respectively. Recently, the experiment [
6] used the less expensive atomic Sn catalyst for possible water purification in the developing world. Consequently, here we explore the effectiveness of the fullerene negative ions C
44ˉ to C
136ˉ in the catalysis of water oxidation to peroxide and water synthesis from H
2 and O
2 in search of less expensive catalysts. The focus is particularly on the larger fullerene molecules greater than C
70.
The importance of fullerene molecules in negative ion catalysis, organic solar-cells, sensor technology, drug delivery, catalytic efficiency in fundamental hydrogenation, etc., has motivated us to study the variation of the EA with the fullerene size from C
44 to C
136 and contrast the EAs with that of the standard C
60. Manifesting the existence of long-lived negative ion formation, reliable atomic, and molecular affinities are crucial for understanding the vast number of chemical reactions involving negative ions [
9]. In the formation of fullerene negative ions, it has been demonstrated for the first time that the ground state anionic binding energies (BEs) extracted from our Regge-pole calculated TCSs for the C
20 through C
92 fullerenes matched excellently the measured EAs [
10,
11,
12,
13,
14,
15,
16,
17]. This provides a novel and general approach to the determination of reliable EAs for complex heavy systems. Indeed, the EAs provide a stringent test of theoretical calculations when their results are compared with those from reliable measurements. In addition, the Regge-pole methodology requires no assistance whatsoever from either experiment or other theory for the remarkable feat. The results [
18,
19] provided great credence to the power and ability of the Regge-pole methodology to extract reliable EAs of the fullerene molecules from the calculated ground states electron elastic TCSs. It is noted here that obtaining unambiguous and reliable fullerene EAs is a challenging task for existing theoretical methods. Generally, the Regge-pole calculated low-energy electron elastic TCSs for fullerene molecules are characterized by ground, long-lived polarization-induced metastable, and excited negative ion formation.
Except for the C
60 fullerene, theoretical and/or experimental low-energy electron elastic scattering TCSs for fullerenes are generally sparse. For C
60, low-energy electron scattering cross sections have been investigated theoretically [
20,
21,
22,
23,
24,
25,
26,
27]. Very recently, angle-differential electron elastic scattering from C
60 has been studied [
28]. The investigations of Wigner Time Delay in electron-C
60 elastic scattering [
29] using potential models defined by the fullerene EA and its radius will certainly benefit from this study. Experimentally, low-energy electron elastic scattering differential cross sections for C
60 were measured [
30]. Gas phase fullerenes C
76 and C
78 [
31] and gas phase C
60 and C
70 [
32] have been studied using low-energy electron scattering. In the latter study, several resonant states were identified including the determination of the lifetimes of the formed negative ions. Thermal rate coefficients and cross sections for electron attachment to C
60 have been studied [
33] including their low energy temperature dependence in a crossed electron beam–molecular beam experiment [
34].
The low-energy electron elastic collision TCSs of the fullerene molecules obtained in this paper as well as those of the already studied fullerenes [
18,
19,
35,
36] and the actinide [
37] and the lanthanide [
38,
39] atoms should contribute to a better understanding of the role of the individual atoms/fullerenes in ongoing studies involving endohedral systems [
40,
41,
42,
43,
44,
45,
46]. Additionally, expected to benefit from this study will be the exploration of the
[email protected]60 (M = Ti, Zr,U) fullerene hybrids that have demonstrated catalytic efficiency in fundamental hydrogenation [
47].
3. 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,
,
m being the mass and
E the impact energy,
ρn is the residue of the S-matrix at the
nth 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
r2 in Equation (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 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].
4. 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
44ˉ to C
136ˉ. Transition state calculations using DFT found the C
52ˉ and C
60ˉ anions to be robust (yielding essentially the same transition state energies) for both water and peroxide synthesis and the C
136ˉ 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
2O
2 from H
2O 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.