Theoretical studies of the free energies of electron transfer and electron transfer kinetics in nanostructure supramolecular complexes of cis-unsaturated thiocrown ethers and Ce and Gd endohedral metallofullerenes [X–UT–Y][M@C₈₂] (M = Ce, Gd)

1 Introduction

Carbon materials are found in a variety of forms, such as graphite, diamond, carbon fibers, fullerenes, and carbon nanotubes. The shapes of these nano-sized new carbon structures included perfect spheres, ellipsoids, tubes, fibers, polyhedra and further variations, all of them still conforming to the same structural principle as C₆₀ (Psaras and Langford, 1987; Singh Nalwa et al., 2002; Ōsawa, 2002). Since the discovery of fullerenes (C_n), one of the main classes of carbon compounds, and the unusual structures and properties of these molecules, many potential applications and physicochemical properties have been discovered and introduced. At the present time, various empty carbon fullerenes with a different magic number “n” such as C₂₀, C₆₀, C₇₀, C₈₀, C₁₈₀, and C₂₄₀ have been obtained. Endohedral metallofullerenes were first introduced as a new spherical fullerene group with unique properties (Ōsawa, 2002; Shen, 2007). One of the common structural molecules is the M@C₈₂ complex. Metallofullerenes are often characterized by the ‘charge-per-metal’ atom encapsulated. This description implies that the oxidation of the metal atom during metallofullerene formation drives the extent of charge transfer to the fullerene cage. Formation of endohedral metallofullerenes is thought to involve the transfer of electrons from the encapsulated metal atoms to the surrounding fullerene cage (Taherpour, 2008; Anderson et al., 2000; Weaver et al., 1992; Smalley, 1992). Significantly, C₈₂ and C₈₄ are known to form endohedral metallofullerenes. The possibility of a charge transfer reaction during metallofullerene formation is responsible, considering the relatively large electron affinities of the fullerene cages (Taherpour, 2008; Anderson et al., 2000; Weaver et al., 1992; Smalley, 1992; Yannoni et al., 1992). It also suggests that the electronic structure of the fullerene cage is an important parameter for the formation of metallofullerenes. The voltammetry of a series of C₈₂ and C₈₄ metallofullerenes was investigated by Anderson et al. (2000) in an attempt to understand their behavior in terms of the electronic structure.

The square-wave voltammetry for a series of related fullerenes and metallofullerenes measured in pyridine solutions containing 0.10 M tetra-n-butylammonium perchlorate is reported (Taherpour, 2008). Several dimetallic metallofullerenes, M@C_x, are found to have similar voltammetric responses regardless of metal identity or carbon number. Assignment of formal charges to the fullerene cage suggests that these metallofullerenes are isoelectronic and have related molecular orbital structures (Anderson et al., 2000). A variety of metallofullerenes having a C₈₂ cage have been identified where either the type or the number of metal atoms within the C₈₂ cage is altered (like La@C₈₂ and Y@C₈₂) (Anderson et al., 2000; Weaver et al., 1992). The symmetry of these metallofullerenes has not been characterized, but the voltammetry suggests that these species may have related electronic structures (Anderson et al., 2000; Stevenson et al., 1994). It was unexpected, however, to find a C₈₂ or C₈₄ metallofullerene having the appearance of related electronic structures. The (Sc₂)⁺⁴@(C₈₄)⁻⁴ formal charges are assigned to the metallofullerene, in agreement with recent chromatographic results from Dorn (Anderson et al., 2000; Stevenson et al., 1994; Taherpour, 2008). The formal charges for Er₂@C₈₂ were assigned as (Er₂)⁺⁶@ $C_{82}^{-6}$ (Anderson et al., 2000; Slanina et al., 2004; Taherpour, 2008). The voltammetery of Sc₂@C₈₄ is nearly identical to that of Er₂@C₈₂, suggesting that the identity of the encapsulated metal ion does not influence the electronic structure of dimetallic metallofullerenes (DMFs). This result corresponds with reports in the literature for the electrochemistry of monometallic C₈₂ metallofullerenes (e.g., La@C₈₂ and Y@C₈₂) (Nagase and Kobayashi, 1994; Kikuchi et al., 1994; Suzuki et al., 1996, 1993). Hoffman et al. show by emission measurements that Er exists as the “+3” cation in Er₂@C₈₂ (Anderson et al., 2000; Taherpour, 2008). Relative concentrations of nine isomers of Ca@C₈₂ derived from the C₈₂ isolated-pentagon-rule satisfying cages have been computed over a wide temperature interval (Iiduka et al., 2006). The computations are based on the Gibbs energy constructed from partition functions supplied with molecular parameters from density functional theory calculations. Five structures show significant populations at higher temperatures: C_2v > C_s > C₂ > C_3v > C_s. The computed relative stabilities agree well with available observations. As for Er@C₈₂, it may be mentioned that minor isomers are likely analogous to Ca@C₈₂ (Iiduka et al., 2006). Determination of the charge-per-Sc atom of Sc₂@C₈₄ (10) by chromatography, however, suggests that the Sc atoms have a “+2” charge (Anderson et al., 2000; Fuchs et al., 1996).

In 1991, Smalley and his collaborators demonstrated that fullerenolanthanides can be produced by laser vaporization of graphite and lanthanum oxide and extracted by toluene (Fuchs et al., 1996; Bethune et al., 1993; Chai et al., 1991). The physical measurements, such as EPR (Fuchs et al., 1996; Johnson et al., 1992; Suzuki et al., 1992, 1993; Bandow et al., 1992; Hoinkis et al., 1992; Weaver et al., 1992; Shinohara et al., 1992; Kato et al., 1993), mass spectrometry (Fuchs et al., 1996; Alvarez et al., 1991; Ross et al., 1992; Gillan et al., 1992; Moro et al., 1993), extended X-ray absorption fine structure (EXAFS) Fuchs et al., 1996; Soderholm et al., 1992; Park et al., 1993 and X-ray photoelectron spectroscopy (XPS) Fuchs et al., 1996; Weaver et al., 1992, were performed on the extracts containing the mixture of fullerenolanthanides and empty fullerenes.

Unsaturated thiocrown ethers with cis-geometry (1–9) demonstrate interesting properties for physicochemical studies due to their conformational restriction compared to a corresponding saturated system and to the size of their cavities. The presence of sulfur atoms in the structure of crown ethers accounts for the different properties of thiocrown ethers. The cis-unsaturated thiocrown ethers 1–9 were synthesized, and their structures were confirmed (Taherpour, 2008; Tsuchiya et al., 2001, 2006; Cooper, 1988; Blake and Schröder, 1990; Cooper and Rawle, 1990; Parker, 1996; Pedersen, 1971; Murray and Hartley, 1981; Nakayama et al., 1999; Bojkova and Glass, 1998; Diener and Alford, 1998). 1,4-dithiin is the smallest member of compounds 1–9 to have been widely studied (Anderson et al., 2000; Weaver et al., 1992; Smalley, 1992; Yannoni et al., 1992; Ruoff et al., 1995; Fowler and Manolopoulos, 1995; Hoffman et al., 1995; Dennis et al., 1998; Stevenson et al., 1994; Iiduka et al., 2006; Slanina et al., 2004; Nagase and Kobayashi, 1994). In 2001, the structures of [X–UT–Y] (X = 6, 9, 12, 15, 18, 21, 24 and 27; Y = 2–9) 1–9 were reported by Tsuchiya et al. (2001). In that report, ¹H and ¹³C NMR signals, X-ray crystallography and ORTEP drawings, cavity size, and UV spectral data of [X–UT–Y] 1–9 were considered carefully (Tsuchiya et al., 2001). In 2006, the oxidation potential (^oxE₁), cyclic voltammetry (Fc/Fc⁺), and the free energy of electron transfer (ΔG_et) of the supramolecular complex of [X–UT–Y][C₆₀] and [X–UT–Y][La@C₈₂] of cis-unsaturated thiocrown ethers 1–9 were considered by Tsuchiya et al. (2006). The endohedral metallofullerenes and complexes with the thiocrown ethers have shown interesting properties for applications and studies. In 2006, the oxidation potential (^redoxE₁) of Sc₂@C₈₄ (10) and Er₂@C₈₂ (11) were reported by Anderson et al. (2000).

Graph theory has been found to be a useful tool in QSAR (Quantitative Structure Activity Relationship) and QSPR (Quantitative Structure Property Relationship) (Hansen and Jurs, 1988; Hosoya, 1971; Randić, 1998, 1975; Rücker and Rücker, 1999; Wiener, 1947; Du et al., 2002). Numerous studies have been performed related to the abovementioned fields by using the so-called topological indices (TIs) Randić, 1975; Slanina et al., 1997; Plavšić et al., 1993; Rehm and Weller, 1970; Taherpour, 2008. In 1993 and 1997, a related complex of applications of the Wiener and Harary indices in fullerene science was reported (Slanina et al., 1997; Plavšić et al., 1993).

The use of effective mathematical methods to suitably correlate between several data properties of chemicals is important. The ratio of the summation of the number of carbon atoms (n_c) and the number of sulfur atoms (n_s) to the product of these two numbers (μ_cs) was a useful numerical and structural value of the unsaturated thiocrown ethers 1–9 that were utilized here.

In Fig. 1 was shown the structures of unsaturated thiocrown ethers 1–9 with 10 and 11, to produce supramolecular complexes [X–UT–Y][M@C₈₂] (M = Ce and Gd) 12 and 13.

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Figure 1

The total structures of unsaturated thiocrown ethers 1–9, their supramolecular complexes of [X–UT–Y][Ce@C₈₂] (12) & [X–UT–Y][Gd@C₈₂] (13).

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In this work, the first and second activation free energies of electron transfer and kinetic rate constants of the electron transfers, $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ and k_et (n = 1,2), respectively, on the basis of the previous studies for [X–UT–Y][Ce@C₈₂] (12) and [X–UT–Y][Gd@C₈₂] (13) complexes, were calculated in accordance with the Marcus theory. The QSAR studies of this index relative to some of the structural data of thiocrown ethers 1–9, their maximum wavelength (λ_max), cavity size, and oxidation potential (^oxE₁), as well as the free energy of electron transfer (ΔG_et) between 1–9 and C₆₀ and La@C₈₂, Sc₂@C₈₄, Er₂@C₈₂, Ce@C₈₂ and Gd@C₈₂ were reported previously (Rehm and Weller, 1970; Taherpour, 2007a,b, 2008, 2010).

2 Graphs and mathematical method

All mathematical and graphing operations were performed using MATLAB-7.4.0(R2007a) and Microsoft Office Excel-2003 programs. The ratio of the sum of the number of carbon atoms (n_c) and the number of sulfur atoms (n_s) to the product of these two numbers (μ_cs) is a useful numerical and structural value for the unsaturated thiocrown ethers 1–9 that were investigated (Rehm and Weller, 1970; Taherpour, 2007b, 2008, 2010).

The Rehm–Weller equation estimates the free energy change between an electron donor (D) and an acceptor (A) as:Rehm and Weller, 1970

Marcus theory of electron transfer implies rather weak (<0.05 eV) electronic coupling between the initial (locally excited, LE) and final (electron transfer, CT) states and presumes that the transition state is close to the crossing point of the LE and CT terms. The value of the electron transfer rate constant k_et is controlled by the activation free energy $\mathrm{\Delta }{G}_{\mathit{et}}^{\#}$ , which is a function of the reorganization energy (l/4) and electron transfer driving force ΔG_et (Marcus, 1993, 1965; Andrea, 2008; Barbara, 1996; Newton, 1991; Jortner and Freed, 1970; Marcus and Sutin, 1985; Kuzmin, 2000):

3 Discussion

The relationships between “μ_cs” index and oxidation potential (^oxE₁) of 1–9, as well as the first and second free energies of electron transfer (ΔG_et(n), for n = 1,2; which is given by the Rehm–Weller equation) between 1–9 and Ce@C₈₂ (10) and Gd@C₈₂ (11) and their supramolecular complex derivatives as [X–UT–Y][M@C₈₂] (M = Ce and Gd) 12 and 13 were presented and investigated before (Taherpour, 2010). It was reported that each of the metallofullerenes has a remarkably small potential difference between the first oxidation and the first reduction (Suzuki et al., 1996). This may suggest that the HOMO of M@C₈₂, originally the LUMO⁺ of the C₈₂, is singly occupied (i.e., SOMO) as purposed for La@C₈₂ (Suzuki et al., 1996). The ionization potential, electron affinity, oxidation and reduction potentials of Ce@C₈₂ and Gd@C₈₂ (E in volts vs. Fc/Fc⁺ and radii of lanthanides) were reported previously. It was found that the ionic radii of Ln³⁺ show good linear relationships with the first redox potentials (Suzuki et al., 1996). The ionization potentials and electron affinities of M@C₈₂ (M = lanthanides) were reported as obtained by ab initio calculations (Suzuki et al., 1996; Nagase and Kobayashi, 1994). The first oxidation and reduction processes occur on the SOMO whose electron density is higher on the cage close to the M³⁺ (M = lanthanides) Suzuki et al., 1996; Nagase and Kobayashi, 1994; Laasonen et al., 1992. The electrons on the SOMO are bound to the cage with a higher energy when the metal–carbon distance decreases because the electrostatic interactions between electrons and the metal intensify (Suzuki et al., 1996). Several important metallofullerenes, M@C₈₂ (where M = Ce and Gd), were found to have similar voltammetric responses regardless of metal identity or carbon number. Assignment of formal charges to the fullerene cage (that they are characterized by the ‘charge-per-metal’ atom encapsulated) suggests that these metallofullerenes are isoelectronic and have related molecular orbital structures (Anderson et al., 2000). The predicted complex structures of unsaturated thiocrown ethers (1–9) with 10 and 11 were introduced here as [X–UT–Y][Ce@C₈₂] (12) and [X–UT–Y][Gd@C₈₂] (13) complexes. The potential difference between the oxidation and reduction in these structures is related to the band gap of HOMO–LUMO orbitals. The electronic properties of Ce@C₈₂ and Gd@C₈₂ are very similar to those of La@C₈₂, although both Ce and Gd have 4ƒ level electrons. It was suggested that these 4ƒ electrons do not play an important role in fullerenolanthanide chemistry as seen in organic and inorganic lanthanide chemistry (Suzuki et al., 1996; Nagase and Kobayashi, 1993; Marks and Ernst, 1982; Cotton and Wilkinson, 1988).

The X-ray crystal structures and ORTEP drawings for some of the structures of 1–9 [X–UT–Y] (X = 15, 18, 21, 24 and 27; Y = 5–9) were determined. These show there are cavities in these molecules, and also that the sulfur atoms are nearly coplanar (Tsuchiya et al., 2001). The average radii of the cavity sizes for 4–8 are 1.76, 2.34, 3.48, 4.43 and 5.36 Å, respectively (Fowler and Manolopoulos, 1995; Tsuchiya et al., 2001, 2006). The ¹³C and ¹H NMR results (in CDCl₃) were reported previously. These results show that compound 4 has the highest chemical shifts in both ¹H and ¹³C NMR. The electron density of the C⚌C increases with the size of the rings from 4 → 9 and diminishes from 4 → 1 with decreasing ring size of [X–UT–Y] (Tsuchiya et al., 2001, 2006). The first and second reduction potentials (^redE₁) of Ce@C₈₂ (10) are −0.41 and −1.41 V, and, for Gd@C₈₂ (11), they are to −0.39 and −1.38 V, respectively (Suzuki et al., 1996). The oxidation potentials (^oxE₁) of 4–7 were measured to be 0.82, 0.79, 0.73 and 0.69 V, respectively (Anderson et al., 2000; Tsuchiya et al., 2001, 2006). The free energies of electron transfer (ΔG_et) between 1–9 with 10 and/or 11 to produce [X–UT–Y][Ce@C₈₂] (12) and [X–UT–Y][(Gd@C₈₂] (13) complexes were calculated by the Rehm–Weller equation (Anderson et al., 2000; Weaver et al., 1992; Park et al., 1993; Rehm and Weller, 1970; Taherpour, 2007b, 2008).

Table 1 has shown the Nieperian logarithmic behavior equations (6–9) that indicate the relationship between the index μ_cs and the first and second free energies of the free activated electron transfer ( $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ , n = 1,2) between unsaturated thiocrown ethers 1–9 with 10 and 11 in the structures [X–UT–Y][M@C₈₀-R] (M = Ce & Gd) 12 and 13. In Table 1 was shown the very good R² of the relationships.

The values of the relative data of 1–9 are shown in Table 2. The values shown in Table 2 demonstrate that μ_cs decreases with increasing molecular size of the compounds 1–9. In Table 2, the related values for the supramolecular complexes of [X–UT–Y] 1–9 with Ce@C₈₄ (10) and Gd@C₈₂ (11) are also shown. Table 2 shows the values of oxidation potential (^oxE₁), as well as the free energy of electron transfer ( $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ ; n = 1,2) between some of the [X–UT–Y] and their complexes with 10 and 11.

In Table 3 was shown the calculated values of the first and second kinetic rate constants of the electron transfers (k_et) between unsaturated thiocrown ethers 1–9 with 10 and 11 in the structures of [X–UT–Y][Ce@C₈₂] (12) and [X–UT–Y][Gd@C₈₂] (13) complexes.

In Fig. 2, two-dimensional diagram shows the relationships between the main values that were demonstrated in Table 1.

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Figure 2

The relationship of μ_cs versus the first and second free activated energies of electron transfer ( $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ ; n = 1,2 in kcal mol⁻¹) between 1–9 and the Ce@C₈₂ (10) for [X–UT–Y][Ce@C₈₂] (12) supramolecular complexes are shown. The structure of the relationship of μ_cs versus the first and second free activated energies of electron transfer ( $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ ; n = 1,2 in kcal mol⁻¹) between 1–9 and the Gd@C₈₂ (11) for [X–UT–Y][Gd@C₈₂] (13) supramolecular complexes have same behavior as shown in Fig. 2.

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The ratio of summation of the number of carbon atoms (n_c) and the number of sulfur atoms (n_s) to the product of these two numbers (μ_cs) shows a high correlation with the physicochemical and structural values of the unsaturated thiocrown ethers 1–9. The results show the calculated values of free energy of electron transfer (ΔG_et) of the [X–UT–Y][Ce@C₈₂] (12) and [X–UT–Y][Gd@C₈₂] (13) supramolecular complexes on the basis of the first and second reduction potentials (^redE₁ and ^redE₂) of Ce@C₈₂ (10) and Gd@C₈₂ (11). The data for the compounds and their complexes were previously reported (Taherpour, 2010).

Marcus theory is currently the dominant theory of electron transfer in chemistry. Marcus theory is so widely accepted because it makes surprising predictions about electron transfer rates that have been nonetheless supported experimentally over the last several decades (Marcus, 1993, 1965; Andrea, 2008; Barbara, 1996; Newton, 1991; Jortner and Freed, 1970; Marcus and Sutin, 1985; Kuzmin, 2000).

Electron transfer (ET) is one of the most important chemical processes in nature, playing a central role in many biological, physical and chemical (both organic and inorganic) systems. Solid state electronics depend on the control of the ET in semiconductors, and the new area of molecular electronics depends critically on the understanding and the control of the transfer of electrons in and between molecules and nanostructures. The other reason to study electron transfer is that it is a very simple kind of chemical reaction and in understanding it, one can gain insight into other kinds of chemistry and biochemistry. After all, what is important is the chemistry of the transfer of electrons from one place to another (Marcus, 1993, 1965; Andrea, 2008; Barbara, 1996; Newton, 1991; Jortner and Freed, 1970; Marcus and Sutin, 1985; Kuzmin, 2000).

The free energy of electron transfer ΔG_et is the difference between the reactants on the left and the products on the right, and $\mathrm{\Delta }{G}_{\mathit{et}}^{\#}$ ; is the activation energy. The reorganization energy is the energy it would take to force the reactants to have the same nuclear configuration as the products without permitting the electron transfer. If the entropy changes are ignored, the free energy becomes energy or potential energy (Marcus, 1993, 1965; Andrea, 2008; Barbara, 1996; Newton, 1991; Jortner and Freed, 1970; Marcus and Sutin, 1985; Kuzmin, 2000).

Using Eqs. (4) and (5), it is possible to calculate the first and second activation free energies of electron transfer and kinetic rate constants of the electron transfers, $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ and k_et(n) (n = 1–2), respectively, for 12 and 13 in accordance with the Marcus theory. Fig. 3 shows the surfaces of the free energies of electron transfer ΔG_et(n) and $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ (n = 1–2) between [X–UT–Y] 1–9 with 10 and 11, to produce supramolecular complexes [X–UT–Y][M@C₈₂] (M = Ce and Gd) 12 and 13. The values of the first and second activation free energies of electron transfer, $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ (n = 1–2) for 12 and 13, decrease with ΔG_et(n) and μ_cs descriptor, while the kinetic rate constants of the electron transfers k_et(n) (n = 1–2), increase with decreasing ΔG_et(n) and $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ (n = 1–2) for 12 and 13 complexes. See Tables 2 and 3 and Fig. 3.

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Figure 3

The surfaces of the free energies of electron transfer ΔGet_(n) and $\mathrm{\Delta }{\mathit{Get}}_{(n)}^{\#}$ (n = 1,2) for 1–9 with 10 and 11 in the structures of [X–UT–Y][M@C82] (M = Ce and Gd) 12 and 13. For the data of ΔGet_(n) (n = 1,2) see reference (Taherpour, 2010).

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Fig. 3 shows the surfaces of the free energies of electron transfer ΔGet_(n) and $\mathrm{\Delta }{\mathit{Get}}_{(n)}^{\#}$ (n = 1 and 2) between 1–9 with 10 and 11 in the structures 12 and 13 complexes. With the appropriate equations and in light of the good correlations (see Figs. 1–3 and equations 1–9), it is possible to calculate the values of the first and second free energies of electron transfer (ΔG_et in kcal mol⁻¹), the first and second activation free energies of electron transfer and kinetic rate constants of the electron transfers, $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ and k_et(n) (n = 1–2), respectively, for 12 and 13, in close accordance with the results of Marcus theory. The supramolecular complexes of unsaturated thiocrown ethers 1–9 with 10 and 11 as [X–UT–Y][M@C₈₂] (M = Ce and Gd) 12 and 13 and their electrochemical data $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ and k_et(n) (n = 1,2) have neither been synthesized nor reported before. But the first and second free energies of electron transfer (ΔG_et(n), for n = 1,2; which were given by the Rehm–Weller equation) between 1–9 and Ce@C₈₂ (10) and Gd@C₈₂ (11) and their supramolecular complexes derivatives as [X–UT–Y][M@C₈₂] (M = Ce and Gd) 12 and 13 were presented and investigated before (Taherpour, 2010).

4 Conclusion

Fullerene C₈₂ is known to readily form endohedral metallofullerenes. Formation of endohedral metallofullerenes is thought to involve the transfer of electrons from the encapsulated metal atom(s) to the surrounding fullerene cage. The cis-UT ethers 1–9 have important physicochemical properties. The electrochemical behaviors, oxidation potential (^oxE₁) and the free energies of electron transfer (ΔG_et) on the basis of the first and second reduction potentials (^redE₁ and ^redE₂) of Ce@C₈₂ (10) and Gd@C₈₂ (11) for the predicted supramolecular complexes of [X–UT–Y][Ce@C₈₂] (12) and [X–UT–Y][Gd@C₈₂] (13) were reported before. The predicted values of ΔG_et for 12 and 13 were calculated by using the Rehm–Weller equation. Using the ratio of summation of the number of carbon atoms (n_c) and the number of sulfur atoms (n_s) to the product of these two numbers (μ_cs), the equations of the model can derive sound structural relationships between the aforementioned physicochemical data. By utilizing the model, one can calculate the values ΔG_et(n) (n = 1–2), $\mathrm{\Delta }{G}_{\mathit{et}(n)}^{\#}$ and k_et(n) (n = 1–2) on the basis of the first and second reduction potentials (^redE₁ and ^redE₂) of 10 and 11 for the 12 and 13 supramolecular complex groups using the Rehm–Weller equation and Eqs. (2), (3) concerning the Marcus theory. The compounds [X–UT–Y][Ce@C₈₂] (12) and [X–UT–Y][Gd@C₈₂] (13) supramolecular complexes were previously neither synthesized nor reported.