Theoretical and Quantitative Structural Relationships of the Electrochemical Properties of [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La, Y and n = 20–300) Nanostructure Complexes

ABSTRACT

Carbon nanotubes are either single-walled (SWCNTs) or multiwalled, and the former attract more attention due to their unique electronic, optical, and spectroscopic properties. One of the main recognized structures of carbon nanotubes is the (5,5) armchair single-walled carbon nanotube. Endohedral metalofullerenes of the form M@C _x were introduced as an important class of spherical fullerene group with unique properties. Formation of endohedral metallofullerenes is thought to involve the transfer of electrons from the encapsulated metal atom(s) to the surrounding fullerene cage. Two of these molecules are La@C₈₂ (a) and Y@C₈₂(b). To establish a good understand of structural relationship between the structures of the molecules La@C₈₂, Y@C₈₂, and SWCNT(5,5)-armchair-C_nH₂₀ (n = 20–190) (1–18), the number of carbon atoms (n) of the SWCNTs was used as one of the useful numerical and structural electrochemical properties of the unsaturated compounds. The relationships between this index and the first and second free energies of electron transfer (Δ G _et(n), n = 1, 2), as assessed using the Rehm-Weller equation on the basis of the first and second oxidation potentials ( ^ox E ₁and ^ox E ₂) of La@C₈₂ and Y@C₈₂ for the predicted supramolecular complexes, between 1– 18 and 19– 29, with the endohedral-metalofullerenes La@C₈₂ and Y@C₈₂ as [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La, Y and n = 20–190; 30– 47 and 48– 65) are presented. The results were extended for [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La, Y and n = 200–300; 66– 76 and 77– 87).

KEYWORDS:

• endohedral metalofullerenes
• electron affinity
• free energy of electron transfer
• La@C₈₂ and Y@C₈₂
• molecular topology
• Rehm-Weller equation
• single-walled nantubes

INTRODUCTION

Formation of endohedral metallofullerenes is thought to involve the transfer of electrons from the encapsulated metal atoms to the surrounding fullerene cage. Metallofulerenes are often characterized by the “charge-per-metal” atom encapsulated. This description implies that the oxidation of the metal atom during metallofulerene formation drives the extent of charge transfer to the fullerene cage.^{[ 1 , 2 , 3 , 4 ]} Significantly, C₈₂ is known to form endohedral metallofullerenes. The possibility of a charge transfer reaction during metallofullerene formation is reasonable when considering the relatively large electron affinities of fullerene cages.^{[ 1 , 2 , 3 , 4 , 5 ]} The electronic structure of the fullerene cage is an important parameter in the formation of metallofullerenes. The voltammetry of a series of C₈₂ and C₈₄ metallofullerenes was investigated by Anderson et al. in an attempt to understand metallofullerene behavior in terms of the electronic structure.^{[ 1 ]}

In 1991, Smalley and collaborators showed that fullerenolanthanides can be produced by laser vaporization of graphite and lanthanum oxide and subsequently extracted by toluene.^{[ 2 , 6 , 7 , 8 ]} Johnson et al.^{[ 9 ]} found that La@C₈₂ is a radical species and suggested that the formal oxidation state of lanthanum is 3+. Later, more careful extraction under anaerobic conditions gave five different octet species, including two relatively stable La@C₈₂ isomers.^{[ 9 , 10 , 11 , 12 ]}

Yamamoto et al. obtained and characterized two La@C₈₂ isomers (major and minor isomers).^{[ 8 , 13 , 14 ]} It was concluded, using ultraviolet photoelectron spectroscopy (UPS) and X-ray photoelectron spectroscopy (XPS), that La@C₈₂ is a semiconductor in the solid state.^{[ 8 , 15 , 16 ]} The scan started at the resting potential (−0.20 V versus Fc/Fc⁺ couple) toward the positive potential. Two oxidation peaks at +0.05 and +1.07 V were observed by DPVs (differential pulse voltammograms), and the latter was irreversible by CV (cyclic voltammetery).^{[ 8 ]} The first reversible oxidation potential (+0.07 V) is slightly more positive than that of ferrocene, indicating that La@C₈₂ is a good electron donor.^{[ 8 , 17 , 18 , 19 , 20 ]}

One reversible oxidation was observed for Y@C₈₂ by CV. The oxidation state of the yttrium is close to that of the lanthanum and likely 3+.^{[ 21 ]} The second oxidation remained irreversible, and the peak current intensity of the second reduction was twice that of each of the other redox peaks. Furthermore, the shape of the current voltage curve suggested a simultaneous two-electron transfer rather than an overlap of two one-electron transfers.^{[ 8 , 21 ]}

Nanotubes of type (n, n) are called armchair nanotubes because of their “W” shape perpendicular to the tube axis. They are symmetrical along the tube axis, with a short unit cell (0.25 nm or 2.5 Å) that can be repeated along the entire section of a long nanotube.^{[ 22 , 23 , 24 ]} The simplest type of nanotube is a cylindrical structure that, conceptually, could be formed by folding and gluing one pair of opposite sides of a rectangular graphite sheet.^{[ 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 ]} If both ends are capped, it will have at least two pentagons and therefore be a type of fullerene. Nanotubes are large, linear fullerenes with aspect ratios as large as 103 to 105.^{[ 23 ]} The walls of such a tube could have various sizes of polygons.^{[ 38 , 39 , 40 ]} Although many nanoscale fullerene materials occur regularly in applications, controlled production of many fullerenes and nanotubes with well-defined characteristics has not occurred.^{[ 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 ]} Because of the properties of nanotubes, such as the fact that they are hollow, tubular, caged molecules, they can act as lightweight, large-surface-area packing materials for gas storage and hydrocarbon fuel storage devices, nanoscale applications for molecular drug delivery, and casting structures for making nanowires and nanocapsulates. Exceptionally strong nanotubes can be used to make lightweight structural materials. Nanotubes such as “capsulates” can help to store and carry hydrogen and other hydrocarbon-based fuel in engines or aboard spacecraft. Single-walled carbon nanotubes are among the most interesting new carbon allotropes discovered in many years.^{[ 23 , 24 ]}

Carbon nanotubes possess many special properties, such as an open mesoporous structure, high electrical conductivity and chemical stability, as well as extremely high mechanical strength and modulus.^{[ 22 , 26 , 28 , 30 , 31 , 33 ]} These properties, which not only help in the transportation of ions, but also facilitate the charging of the double layer, will confer advantageous attributes in the development of electrochemical capacitors.^{[ 29 ]} Single-walled carbon nanotubes have been recognized as a potential electrode material for electrochemical capacitors.^{[ 27 , 32 ]}

One of the main recognized structures of nanotubes is the (5,5) tube. In the (5,5) tube, the structure can be built up by successively adjoining sections of 10 C atoms. In the infinite tube, the periodic unit cell is two such sections consisting of 20 C atoms.^{[ 24 ]}

The electronic structures and electrical properties of single-walled nanotubes can be simulated from those of a layer of graphite (graphene sheet).^{[ 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 ]}

In Figure 1, the (5,5) armchair form of nanotubes is shown. Figure 1 shows the imaginary structures 1–87 ([M@C₈₂]@[SWCNT(5,5)-Armchair-C_nH₂₀] (M = La, Y, and n = 20–300)). In experiments, nanotubes may contain no hydrogens (there is no hydrogen in the electric-arc technique) and the nanotubes can be easily closed at both ends.

FIGURE 1 The imaginary structures of 1–87. The form of M@C₈₂(a and b) was from Suzuki et al.^{[ 8 ]}

The electronic structures of tubular aromatic molecules derived from the metallic (5,5) armchair single-walled carbon nanotube (SWCNT) for C₂₀H₂₀ up to C₂₁₀H₂₀ (see Figure 1) were reported by Zhou et al. in 2004.^{[ 24 ]} That report considered how the electronic structures of short molecular sections of the (5,5) tube relate to, differ from, and asymptotically approach the infinite metallic tube.^{[ 24 ]} Some of the structural and electronic properties were investigated, such as the ionization potential, electron affinity, Fermi energy (E_F ), chemical hardness, and relative energetic stability.^{[ 24 ]} All of these metrics show the length periodicity seen in the frontier orbitals (FOs; i.e., HOMO-LUMO: highest occupied molecular orbital–lowest unoccupied molecular orbital) gap, in contrast to the optical “charge transfer” transition and the static axial polarizability.^{[ 24 ]} These (5,5) nanotubes have two types of symmetry. For nanotubes with odd identification numbers (1–17), the point group is D _5d , and for nanotubes with even identification numbers (2–18), the point group is D _5h .^{[ 24 ]} In that study, static and TD-DFT (time-dependent density function theory) were used to independently optimize the structure for neutral, cationic, and anionic complexes. The hybrid nonlocal B3LYP (Becke, three-parameter, Lee-Yang-Parr) functional was utilized.^{[ 24 ]}

Infinite length SWCNTs are π-bonded aromatic structures that can be either semiconducting or metallic, depending upon the diameter and helical angle of SWCNTs.^{[ 24 ]} In a pioneering 1992 DFT calculation, Mintmire et al.^{[ 32 ]} predicted that the infinite length (5,5) armchair SWCNT (6.70Å diameter) would be metallic, with a very low transition temperature separating the uniform (high-temperature) structure from the Peierls bond alternating (low-temperature) structure.^{[ 27 , 32 , 38 , 39 , 40 , 41 ]} This specific SWCNT is the elongated tube of the C₆₀, C₇₀, etc., molecular family.^{[ 24 ]}

Fullerene peapods are supramolecular complexes, which are formed via filling of SWCNT by fullerenes from the vapor phase.^{[ 42 , 43 ]} Most reports are related to C₆₀@SWCNT and C₇₀@SWCNT.^{[ 36 , 44 , 45 , 46 ]}

The diameter sizes of C₆₀ and [SWCNT(5,5)-armchair-C_nH₂₀]1–18 were reported to be 6.70 and 6.94 Å, respectively.^{[ 47 , 48 ]} With these diameters, it is not possible to encapsulate C₆₀ inside the [SWCNT(5,5)-armchair-C_nH₂₀] in the structure of C₆₀@[SWCNT(5,5)-armchair-C_nH₂₀] (see the imaginary scheme in Figure 1).

Graph theory has been found to be a useful tool in assessing the QSAR (quantitative structure-activity relationship) and QSPR (quantitative structure-property relationship).^{[ 49 , 50 , 51 , 52 , 53 , 54 ]} Numerous studies in the above areas have also used what are called topological indices (TIs).^{[ 55 , 56 , 57 , 58 ]} It is important to use effective mathematical methods to make good correlations between the data corresponding to several chemical properties. One of the useful numerical and structural values in unsaturated compounds like nanotubes is the degree of unsaturation (D_U ). This quantity is a useful index for determining the number of cyclic structures and/or π-bonds in a molecule.^{[ 45 , 46 , 47 , 48 , 59 ]} In previous studies, a relationship between the D_U index and electron affinity and reduction potential ( ^Red. E ₁) of [SWCNT(5,5)-armchair-C_nH₂₀], as well as the free energy of electron transfer (ΔG_et ) between [SWCNT(5,5)-armchair-C_nH₂₀] structures and fullerene C₆₀ as C₆₀@[SWCNT(5,5)-armchair-C_nH₂₀] complexes were investigated.^{[ 45 , 46 ]}

In the study of the structural properties of π-bonds, the relationship between the number of carbon atoms of the SWCNT (C _n) index and electron affinity, reduction potential ( ^Red. E ₁) of [SWCNT(5,5)-armchair-C_nH₂₀]1–18 (and extension the results to 19–29), as well as the first and second free energies of electron transfer (ΔG _et(n), n = 1, 2) as assessed using the Rehm-Weller equation on the basis of the first and second oxidation potentials ( ^ox E ₁ and ^ox E ₂) of La@C₈₂ and Y@C₈₂ for the predicted supramolecular complexes, between 3–20

and the endohedral-metalofullerenes La@C₈₂ and Y@C₈₂ as [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La, Y and n = 20–190), are presented. The results were extended for [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La, Y and n = 200–300).

GRAPHS AND MATHEMATICAL METHOD

The number of carbon atoms of these SWCNT (C _n) was utilized as a structural index for compounds (1–18). All graphing operations were performed using the Microsoft Office Excel 2003 program. The number of carbon atoms of these SWCNTs (C _n) seems to be a useful numerical and structural value for the empty fullerenes. For modeling, both linear (MLRs: multiple linear regressions) and nonlinear (ANN: artificial neural network) models were examined in this study. Some of the other indices were examined and the best results and equations for extending the physicochemical and electrochemical data were chosen.

The Rehm-Weller equation estimates the free energy change between an electron donor (D) and an acceptor (A) to be:

wheree is the unit electrical charge, E° _D and E° _A are the reduction potentials of the electron donor and acceptor, respectively,ΔE* is the energy of the singlet or triplet excited state, and ω ₁ is the work required to bring the donor and acceptor to within the electron transfer (ET) distance. The work term in this expression can be considered to be 0 in as much as there exists an electrostatic complex before the electron transfer.^{[ 59 ]}

DISCUSSION

It was assumed that all observed reversible redox processes involve a single electron and the results were explained by the following four reported interpretations^{[ 8 , 60 , 61 ]}: (1) The removal of the unpaired electron corresponds to the first oxidation process and the resulting La@C⁺ ₈₂ should have no radical electrons. (2) Because of the migration of an electron to the HOMO orbital to give the closed-shell species La@C₈₂, the first reduction is relatively easy.^{[ 8 , 60 , 61 ]} (3) The low-lying HOMO⁻¹ could be responsible for the irreversible formation of La@C^– ₈₂. (4) The ab initio calculations on La@C²⁺ ₈₂ show that the LUMO and LUMO⁺¹ derive from the C₈₂ molecular orbitals (MOs), not from the 5d and 6s atomic orbitals (AOs) of the lanthanum metal.^{[ 8 , 60 , 61 ]} Therefore, it was suggested that electrons at least up to the 5-state go to the C₈₂ cage rather than to the metal.^{[ 8 ]}

It was reported that each of the metallofulerene has a remarkably small potential difference between the first oxidation and the first reduction.^{[ 8 ]} 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.^{[ 8 ]} It was found that the ionic radii of Ln³⁺ (La³⁺ and other lanthanide cations) show good linear relationships with the first redox potentials.^{[ 8 ]} The ionization potential and electron affinities of M@C₈₂ (M = lanthanides) were obtained by ab initio calculations.^{[ 8 , 62 ]} The first oxidation and reduction processes take place on the SOMO, which has a higher electron density on the cage near the M³⁺ (M =

lanthanides).^{[ 8 , 60 , 61 , 62 ]} The electrons on the SOMO are bound to the cage stronger when the metal-carbon distance is decreased because the electrostatic interactions between the electrons and the metal increases.^{[ 8 ]} 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.^{[ 1 ]} The potential difference between the oxidation and reduction in these structures is related to the band gap of HOMO-LUMO orbitals. It was suggested that these 4f electrons do not play an important role in fullerenolanthanide chemistry as seen in organic and inorganic lanthanide chemistry.^{[ 8 , 63 , 64 ]}

The energy (E_a ) is released upon attachment of an electron to an atom or a molecule (A), resulting in the formation of the negative ion A⁻, i.e., A + e⁻→ A⁻+E_a . As in the case of the ionization potential, one can define an adiabatic electron affinity (E_aa ) and a vertical electron affinity (E_va ). The adiabatic E_a is equal to the difference between the total energies of a neutral system (A) and the corresponding anion (A⁻). The vertical E_a is equal to the difference between the total energies of A and the anion A⁻ in the equilibrium geometry of “A.”^{[ 65 ]} The free energy of this reaction [ΔE_s (A → A⁻)] corresponds to the absolute redox energy for the above process. The free energy of an electron (e⁻) at rest in the gas phase is set to zero.^{[ 66 , 67 ]} It is possible to calculate the redox energy of the reaction (A + e⁻→ A⁻+E_a ) using the thermodynamic equation (see Equation 2). In this equation, ΔG_s (A) and ΔG_s (A⁻) are the solvation energies of molecule A and its anion A⁻, respectively, and ΔE_g (A → A⁻) is the energy difference between molecule A and its anion (which is defined as the redox energy in the gas phase). On the basis of this thermodynamic cycle, one can obtain ΔE_s (A → A⁻), the absolute redox energy, by^{[ 66 , 67 ]}

Thus, by calculating the gas-phase energies and solvation energies of molecule A and its anion A⁻, one can derive the absolute redox potential (scaled) of molecule A in solution. A scaling coefficient that translates electron affinity to standard redox potentials can be thus extracted.^{[ 66 ]} With respect to the interesting results of reference,^{[ 25 ]} the static and TD-DFT and independently optimized structure were used to calculate the physicochemical and electronic structure of tubular aromatic molecules derived from the metallic (5,5) armchair single-walled carbon nanotubes. The hybrid nonlocal B3LYP functional was employed.^{[ 24 , 68 , 69 , 70 , 71 , 72 ]}

It is possible to calculate the reduction potential ( ^Red E) of 1–18 using the Gibbs equation (ΔG=−nFE) and the definition of adiabatic electron affinity. In this equation, ΔG is equal to the adiabatic electron affinity (the free energy of electron transfer, ΔG_et in J·mol⁻¹, 1 eV=96471 J·mol⁻¹, F = 96495 coulomb, and n = 1). For example, the reduction potentials ( ^Red E) of C₂₀H₂₀ and C₃₀H₂₀ are equal to −0.34 and −0.89 V. The reduction potentials ( ^Red E) of [SWCNT(5,5)-armchair-C_nH₂₀] (n = 20–190) 1–18 have been calculated and are presented in Table 1. As a rapid result, the amount of ^Red E(in V) =−E_aa (in eV), whereE_aa is the adiabatic electron affinity.

TABLE 1 The values of the relative structural coefficients of [SWCNT(5,5)-armchair-C_nH₂₀] (n = 20–190) – and [La@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]– and [Y@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]– ^a

			Adiabatic			[La@C82]@[SWCNT(5,5)-			[Y@C82]@[SWCNT(5,5)-
			electron			armchair-CnH20]30–47			armchair-CnH20]48–65
	Molecular	Point	affinity	^Red E
No.	formula	group	Eaa (eV)	(V)	No.	ΔG et(1)	ΔG et(2)	No.	ΔG et(1)	ΔG et(2)
1	C20H20	D 5d	0.34	−0.34	30	8.76	31.82	48	9.45	31.82
2	C30H20	D 5h	0.89	−0.89	31	21.45	44.51	49	22.14	44.51
3	C40H20	D 5d	0.67	−0.67	32	16.37	39.43	50	17.06	39.43
4	C50H20	D 5h	1.14	−1.14	33	27.21	50.27	51	27.90	50.27
5	C60H20	D 5d	1.56	−1.56	34	36.90	59.96	52	37.59	59.96
6	C70H20	D 5h	1.34	−1.34	35	31.82	54.88	53	32.51	54.88
7	C80H20	D 5d	1.61	−1.61	36	38.05	61.11	54	38.74	61.11
8	C90H20	D 5h	1.98	−1.98	37	46.58	69.64	55	47.27	69.64
9	C100H20	D 5d	1.71	−1.71	38	40.36	63.42	56	41.05	63.42
10	C110H20	D 5h	1.91	−1.91	39	44.97	68.03	57	45.66	68.03
11	C120H20	D 5d	2.24	−2.24	40	52.58	75.64	58	53.27	75.64
12	C130H20	D 5h	2.06	−2.06	41	48.43	71.49	59	49.12	71.49
13	C140H20	D 5d	2.13	−2.13	42	50.04	73.10	60	50.73	73.10
14	C150H20	D 5h	2.43	−2.43	43	56.96	80.02	61	57.65	80.02
15	C160H20	D 5d	2.35	−2.35	44	55.11	78.17	62	55.80	78.17
16	C170H20	D 5h	2.23	−2.23	45	52.35	75.41	63	53.04	75.41
17	C180H20	D 5d	2.53	−2.53	46	59.26	82.32	64	59.96	82.32
18	C190H20	D 5h	2.45	−2.45	47	57.42	80.48	65	58.11	80.48
Note. Values for [SWCNT(5,5)-armchair-CnH20] (n = 20–190) 1–18 were calculated and reported at the 6-31G* level (see Zhou^[ ^24 ^]). ^aThe compounds [M@C60]@[SWCNT(5,5)-armchair-CnH20]30–47 and 48–65 were not synthesized or reported. The data of ΔG et(n)(n = 1 and 2, kcal·mol^−1) for compounds 30–47 and 48–65 were calculated by the Rehm-Weller equation.

The values of the relative structural coefficients of the (5,5) armchair SWCNT for C₂₀H₂₀ up to C₁₉₀H₂₀ ([SWCNT(5,5)-armchair-C_nH₂₀], 1–18), the adiabatic electron affinity (E_aa in eV), and the reduction potentials ( ^Red E in V) of 1–18 are shown in Table 1. The numbers shown in Table 1 for 1–18 (and extended in Table 3 for compounds 19–29, by the use of equations in Table 2 and the Rehm-Weller equation) have some mathematical structures. The absolute value of E_aa or ^Red E increases concomitantly with the number of carbon atoms in 1–18. From C₂₀H₂₀ up to C₁₉₀H₂₀, the point groups alternated between D _5d and D _5h .

TABLE 2 The linear relationships between the values of the adiabatic electron affinity (E_aa in eV) and the values of the reduction potential ( ^Red E in V) of [SWCNT(5,5)-armchair-C_nH₂₀] (n = 20–190) – versus the first and second free energies of electron transfer (ΔG _et(n),n = 1,2 in kcal·mol⁻¹) of [La@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]– and [Y@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]–

			ΔG et(n)= aE +b
Equation
no.	E	n	a	b	R ^2
8	^Red E	1	−23.0610	0.9220	1.00
9	^Red E	1	−23.0620	1.6106	1.00
10	^Red E	2	−23.0610	23.9820	1.00
11	Eaa	1	23.0610	0.9220	1.00
12	Eaa	1	23.0620	1.6106	1.00
13	Eaa	2	23.0610	23.9820	1.00

Equations 3 and 4 show the relationship between the values of the number of carbon atoms (n) of these [SWCNT(5,5)-armchair] versus the adiabatic electron affinity (E_aa in eV) and the reduction potential ( ^Red. E in V) of [SWCNT(5,5)-armchair-C_nH₂₀] (n = 20–190) 1–18, respectively. Equation 5, similar to Equation 4, shows the Nieperian logarithmic behavior of the relationship. The R-squared value (R ²) for the graphs was found to be .9461.

Using the equations, it is possible to achieve a good approximation for extending the determination of the adiabatic electron affinity (E_aa ) and the reduction potential ( ^Red. E) for the other [SWCNT(5,5)-armchair-C_nH₂₀] (n = 200–300) 19–29.

Table 1 also shows the values of the relative structural coefficients, the adiabatic electron affinity (E_aa in eV), and the reduction potential ( ^Red. E in V) of [SWCNT(5,5)-armchair-C_nH₂₀] (n = 20–190) 1–18, [La@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]30–47, and [Y@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]48–65, by the Rehm-Weller equation.

The results of the reduction potentials ( ^Red E in V) were extended for C₂₁₀H₂₀ up to C₃₀₀H₂₀ ([SWCNT(5,5)-armchair-C_nH₂₀], 19–29). The calculated results for ^Red E, as well as the free energies of electron transfer (ΔG _et(n), n = 1, 2, in kcal·mol⁻¹) according to the Rehm-Weller equation between La@C₈₂ (a) and Y@C₈₂(b) with [SWCNT(5,5)-armchair-C_nH₂₀], 19–29 in the structures [La@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]66-76 and [Y@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]77–87 (n = 210–300) are presented in Table 3.

TABLE 3 The values of the relative structural coefficients of [SWCNT(5,5)-armchair-C_nH₂₀] (n = 20–190) –, [La@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]–, and [Y@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]– ^a

						[La@C82]@[SWCNT(5,5)-armchair-CnH20]30–47					[Y@C82]@[SWCNT(5,5)-armchair-CnH20]48–65
						ΔG et(1)(kcal·mol^−1)		ΔG et(2)(kcal·mol^−1)			ΔG et(1)(kcal·mol^−1)		ΔG et(2)(kcal·mol^−1)
			Adiabatic electron
	Molecular	Point	affinity	^Red E		Eqs. 8	Rehm-	Eqs. 10	Rehm-		Eqs. 9	Rehm-	Eqs. 10	Rehm-
No.	formula	group	Eaa (eV)	(V)	No.	and 11	Weller	and 13	Weller	No.	and 12	Weller	and 13	Weller
19	C200H20	D 5d	2.49	−2.49	66	58.34	58.34	81.40	81.40	77	59.03	59.03	81.40	81.40
20	C210H20	D 5h	2.53	−2.53	67	59.26	59.26	82.32	82.32	78	59.96	59.96	82.32	82.32
21	C220H20	D 5d	2.57	−2.57	68	60.19	60.19	83.25	83.25	79	60.88	60.88	83.25	83.25
22	C230H20	D 5h	2.60	−2.60	69	60.88	60.88	83.94	83.94	80	61.57	61.57	83.94	83.94
23	C240H20	D 5d	2.64	−2.64	70	61.80	61.80	84.86	84.86	81	62.49	62.49	84.86	84.86
24	C250H20	D 5h	2.67	−2.67	71	62.49	62.49	85.55	85.55	82	63.18	63.18	85.55	85.55
25	C260H20	D 5d	2.71	−2.71	72	63.42	63.42	86.48	86.48	83	64.11	64.11	86.48	86.48
26	C270H20	D 5h	2.74	−2.74	73	64.11	64.11	87.17	87.17	84	64.80	64.80	87.17	87.17
27	C280H20	D 5d	2.77	−2.77	74	64.80	64.80	87.86	87.86	85	65.49	65.49	87.86	87.86
28	C290H20	D 5h	2.80	−2.80	75	65.49	65.49	88.55	88.55	86	66.18	66.18	88.55	88.55
29	C300H20	D 5d	2.83	−2.83	76	66.18	66.18	89.24	89.24	87	66.87	66.87	89.24	89.24
^aThe compounds 66–76 and77–87 were not synthesized and reported before.

Figure 2 shows the curve of the relationship between the values of the number of carbon atoms (n) of the [SWCNT(5,5)-armchair] versus the first free energy of electron transfer (ΔG _et(1), kcal·mol⁻¹) of [La@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]30–47, as presented in Table 1. Equation 5 applies to Figure 2 and shows the Nieperian logarithmic behavior of the relationship. Using this equation, it is possible to achieve a good approximation for extending the first free energy of electron transfer (ΔG _et(1)) for the other [La@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]66–76 (n = 200–300). R ² for this graph was found to be .9393.

FIGURE 2 The relationship between the number of carbon atoms “n” and the first free energy of electron transfer (ΔG _et(1), kcal·mol⁻¹) of La@C₈₂@[SWCNT(5,5)-armchair-C_nH₂₀]30–47.

The predicted values of ΔG _et(1) for [La@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]66–76 (n = 200–300) were calculated using Equation 5 (see Table 3).

Equation 6 shows the relationship between the values of the number of carbon atoms (n) of these SWCN(5,5)-armchair versus the first free energy of electron transfer (ΔG _et(1), kcal·mol⁻¹) of [Y@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]48–65. Equation 6, similar to Equation 5, shows the Nieperian logarithmic behavior of the relationship. R ² for this graph was found to be .9461.

Using this equation, it is possible to achieve a good approximation for extending the determination of the first free energy of electron transfer (ΔG _et(1)) for the other [Y@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (n = 200–300)77–87.

The relationship between the values of the number of carbon atoms (n) of [SWCNT(5,5)-armchair-C_nH₂₀] (n = 20–190) 1–18 versus the second free energy of electron transfer (ΔG _et(2), kcal·mol⁻¹) of [La@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]30–47 and [Y@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]48–65 are shown in Equation 7. Equation 7 also exhibits Nieperian logarithmic behavior for ΔG _et(2) of the both groups of [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La and Y, n = 20–190). R ² for this graph was found to be .9461.

Using this equation, it is possible to extend the calculation of the second free energy of electron transfer (ΔG _et(2) in kcal·mol⁻¹) for some of the other supramolecular structures of [La@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]30–47 and [Y@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]48–65.

There is a very good linear relationship between the values of the adiabatic electron affinity (E_aa in eV) and the values of the reduction potential ( ^Red. E in V) of [SWCNT(5,5)-armchair-C_nH₂₀] (n = 20–190) 1–18 versus the first and second free energies of electron transfer (ΔG _et(n), n = 1, 2 in kcal·mol⁻¹) of [La@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]30–47 and [Y@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀]48–65, respectively. Equations 8–13 are used to apply the correlations giving the R ² for these relationships of 1.00. See Table 2.

In light of the good linear correlations between ΔG _et(n) (n = 1, 2), E_aa and ^Red E of

1–18 with 30–47 and48–65, respectively, it is possible to use the values of E_aa and ^Red E to calculate the free energies of electron transfer (ΔG_et in kcal·mol⁻¹) of [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La, Y, n = 20–300) 66–76 and 77–87. The electron affinity and the reduction potential are

in fact the same magnitude with the sign reversed, whereas the free energy of electron transfer is calculated with the Rehm-Weller equation, which can be straightforwardly proven to be linearly dependent on the electron affinity of the compounds studied here. In Table 3, the values of the free energies of electron transfer obtained for compounds 66–76 and 77–87 from the equations in Table 2 and those obtained with the Rehm-Weller equation are compared. Obviously, these numbers are equal because those calculated with the Rehm-Weller equation were calculated using electron affinities obtained from Equations 5–7, which are in fact equivalent to the equations in Table 2.

The values of the number of carbon atoms (n), E_aa , ^Red E, and ΔG _et(n) (n = 1, 2) of [SWCNT(5,5)-armchair-C_nH₂₀] (n = 200–300) 19–29 and their complexes with La@C₈₂ (a) and Y@C₈₂(b) ([M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La, Y, n = 20–300) 66–76 and 77–87) are shown in Table 3. In Figures 2–4, the periodicity of the points plotted appears to be 3, which is quite common among benzenoids. By using Equations 1 (Rehm-Weller equation) and Equations 3–13, the values ofE_aa , ^Red E, and ΔG _et(n) (n = 1, 2) were calculated for 19–29, 66–76, and 77–87. The values of the number of carbon atoms (n) show a good relationship with the values of the adiabatic electron affinity (E_aa in eV), the values of the reduction potential ( ^Red E in V) of [SWCNT(5,5)-armchair-C_nH₂₀] (n = 20–190) 1–18 and 19–29 and the free energy of electron transfer (ΔG_et in kcal·mol⁻¹) in the [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La, Y, n = 20–300) 30–47, 48–65, 66–76, and 77–87. With the above equations, it is possible to calculate the adiabatic electron affinity (E_aa in eV), the values of the reduction potential ( ^Red E in V) in 1–18 and19–29 and the first and second free energies of electron transfer (ΔG_et in kcal·mol⁻¹) in the 30–47, 48–65, 66–76, and 77–87 with good approximation. The armchair single-walled nanotubes [SWCNT(5,5)-armchair-C_nH₂₀] (n = 20–300) 1–18, 19–29, and their supramolecular complexes with La@C₈₂ and Y@C₈₂, i.e., [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La, Y, n = 20–300) 30–47, 48–65, 66–76, and 77–87 were neither synthesized nor reported.

FIGURE 3 The relationship between the number of carbon atoms “n” and the first free energy of electron transfer (ΔG _et(1), kcal·mol⁻¹) of Y@C₈₂@[SWCNT(5,5)-armchair-C_nH₂₀]48–65.

FIGURE 4 The relationship between the number of carbon atoms “n” and the second free energy of electron transfer (ΔG _et(2), kcal·mol⁻¹) of La@C₈₂@[SWCNT(5,5)-armchair-C_nH₂₀]30–47 and Y@C₈₂@[SWCNT(5,5)-armchair-C_nH₂₀]48–65.

CONCLUSIONS

Nanoscale structures of carbon display an attractive variation of structural characteristics, and many useful forms have been synthesized and identified. Two of these structures are the SW carbon nanotubes and endohedral metalofullerenes (M@C _x ). Nanoscale materials have been applied in many areas of sciences such as computers, microchips, sensors, actuators, and machines. Graph theory applications have provided different mathematical methods for finding the relationships between several numerical properties of the materials. In this study, the structural relationships between the number of carbon atoms (n) index and the adiabatic electron affinity (E_aa in eV), the values of the reduction potential ( ^Red E in V) of [SWCNT(5,5)-armchair-C_nH₂₀] (n = 20–190) 1–18 and 19–29 and the free energies of electron transfer (ΔG _et(n), n = 1, 2 in kcal·mol⁻¹) in the [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La, Y, n = 20–300) 30–47, 48–65, 66–76, and 77–87 were demonstrated. The number of carbon atoms shows strong correlation with the values of E_aa and ^Red E in the (5,5) armchair SWCNTs 1–18 and 19–29. The number of carbon atoms is an important factor in these materials. The values of ΔG_et was calculated using the Rehm-Weller equation in [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La, Y, n = 20–300) 30–47 and 48–65. Using this model and the associated equations, it is possible to calculate in a simple manner and with good approximation the E_aa , ^Red E, and ΔG _et(n) (n = 1, 2 in kcal·mol⁻¹) of this family of compounds 19–29 and the complexes with La@C₈₂ and Y@C₈₂ 30–47, 48–65, 66–76, and 77–87. The model can be used to study structural properties, the electron affinity, the reduction potential, and the first and second free energies of electron transfer properties of these types of nanotubes (5,5), armchair SWCNTs and their supramolecular structures [M@C₈₂]@[SWCNT(5,5)-armchair-C_nH₂₀] (M = La and Y).

Acknowledgments

The author gratefully acknowledges his colleagues in the Chemistry Department of The University of Queensland, Australia, for their useful suggestions.