A Computational Characterization of CH₄@C₆₀

Abstract

The recently synthetically prepared endohedral CH₄@C₆₀ was characterized here using calculations—namely its structure, energetics, thermodynamics, and vibrational spectrum. The calculations were carried out with DFT (density-functional theory) methods, namely by the DFT M06-2X functional and MP2, as well as B2PLYPD advanced correlated, treatments with the standard 6-31++G** and 6-311++G** basis sets, corrected for the basis set superposition error evaluated using the approximative Boys–Bernardi counterpoise method. The symmetry of the endohedral obtained in the geometry optimizations was tetrahedral T. The energetics of CH₄ encapsulation into C₆₀ was attractive (i.e., with a negative encapsulation-energy term), producing a substantial energy gain of −13.94 kcal/mol at the most advanced computational level, B2PLYPD/6-311++G**. The encapsulation equilibrium constants for CH₄@C₆₀ were somewhat higher than previously found with the CO@C₆₀ system. For example at 500 K, the encapsulation equilibrium constant for CH₄@C₆₀ had a value one order of magnitude larger than for CO@C₆₀. The encapsulation thermodynamic characteristics suggest that high-pressure and high-temperature synthesis could in principle also be possible for CH₄@C₆₀.

1. Introduction

Fullerenes can encapsulate not only single metals and their aggregates but also non-metals and their molecules. Thus, species like N₂@C₆₀, CO@C₆₀, and N₂@C₇₀ have been produced [1], namely by application of high pressures at high temperatures. Moreover, N₂@C₆₀ can also be prepared [2,3] by implantation of the nitrogen ion into the C₆₀ cage, though this technique is preferentially used for the preparation of N@C₆₀ [2,3,4,5,6,7,8,9]. Other non-metallic endohedrals encapsulate noble-gas atoms [10,11,12,13,14], again produced [10] by high pressures at high temperatures and with the help of catalytic action [13]. Moreover, with an entirely different method, an ingenious synthetic approach, hydrogen molecules [15] or even water [16] were first placed inside open fullerene cages that were subsequently closed through synthetic steps [17,18]. More recently, such elegant treatments even enabled [19] preparation of (H₂O)₂@C₇₀. Similarly, carbon monoxide [20,21] as well as hydrogen peroxide [22,23] were also deposited inside derivatives of open C₆₀ cage. Such fullerene encapsulations of non-metals have also been calculated, for example in works [24,25,26,27,28,29,30,31,32,33,34,35] (for a survey, see ref. [36] and references therein) with the first calculations [24] dating from the year 1991. These calculations mostly deal with encapsulations in the C₆₀ cage; however, C₇₀ and C₈₄ [25,36], C₅₉N [33,34], and even C₆₀H_n [25] cases have also been studied.

The present report continues with the latter efforts and describes quantum chemical characterization of the structural and vibrational spectral features, encapsulation energetics, and thermodynamics of the recently prepared [37] CH₄@C₆₀ species.

2. Calculations

The calculation treatment started with structure optimizations carried out with a DFT (density-functional theory) approach, viz the DFT M06-2X functional considered [38] the most reliable technique for various species and aspects, including hydrogen-bonded systems and long-range interactions. The M06-2X functional was combined with the conventional 6-31++G** basis set [39] (the treatment is labeled here as M06-2X/6-31++G**). Subsequently, in the localized M06-2X/6-31++G** hypersurface stationary points, a GF (harmonic) vibrational analysis was performed, which allowed checking the nature of the stationary points; i.e., if local energy minima had really been found (and not activated complexes or even higher types of hypersurface stationary points). For numerical integration with the DFT functional, a so called ultrafine grid was applied. Moreover, the tight SCF (self-consistent field) convergency criterion was used. Finally, the wave function stability was tested throughout, in order to eliminate unstable SCF solutions lacking real physical significance.

The M06-2X/6-31++G** calculated energetics of the encapsulation process was further checked and refined by more advanced, correlated quantum chemical methods, namely by the second-order perturbation Møller-Plesset (MP2) treatment [40], considering all electrons (MP2 = FU) with the conventional 6-31++G** and 6-311++G** basis sets in the previously optimized M06-2X/6-31++G** structure. Moreover, an even more advanced and recently introduced technique known as B2PLYPD = FU [41] (again with all electrons) was also employed for refining the encapsulation energetics. The B2PLYPD technique is based on the MP2 method, however combined with the DFT approach and with inclusion of dispersion correction (D). In addition, the so-called basis set superposition error (BSSE) was also evaluated for the encapsulation energy, namely by the approximative Boys–Bernardi counterpoise method (also known as CP2) [42], since this was recently considered [43] with metallofullerenes. The Boys–Bernardi approach to the BSSE correction term is an approximative scheme that ensures that every reaction component of a studied chemical process is formally described using the same number of basis functions. The required situation is realized by rather artificial (also called ghost) atoms possessing no electrons. The BSSE issue is essentially rooted in the fact that the considered basis sets are always finite. In other words, the BSSE term would disappear in a situation with a basis set with an infinite number of basis functions (of course, such an infinite scheme cannot practically be handled). The BSSE correction term represents an important addition to the energetics of any encapsulation process—without the BSSE correction, the energy gain created by encapsulation would be exaggerated (endohedrals would be artificially over-stabilized). The molecular partition functions used in the thermodynamic treatment are approximated by the rigid rotator and harmonic oscillator (RRHO) description [36], as this treatment represents the only approach applicable to such relatively large systems with the computer resources available at present. All the reported calculations were performed with the program Gaussian 09 [44]. The calculations were carried out with computers operating in parallel regime, mostly with 8–24 processors (with a computational frequency up 3 GHz each and with available operational memory up to 60 GB).

3. Results and Discussion

The presented calculations deal with the encapsulation equilibrium process in the gas-phase:

$$C{H}_4\left(g\right)+C_{60}\left(g\right)=C{H}_4@C_{60}\left(g\right)$$

(1)

connected with the encapsulation change in the potential energy $\Delta E_{enc}$ and also described by the encapsulation equilibrium constant $K_{p,enc}$, defined by the partial pressures p of the individual reaction components:

$$K_{p,enc}=\frac{p_{C{H}_4@C_{60}}}{p_{C{H}_4}{p}_{C_{60}}}.$$

(2)

The equilibrium constant $K_{p,enc}$ for encapsulation (1) is related, as usual, to the standard reaction Gibbs energy change $\Delta G_{T,enc}^o$ (at a temperature T):

$$\Delta G_{T,enc}^o=-RTln{K}_{p,enc}$$

(3)

where R denotes the gas constant. Finally, the standard encapsulation Gibbs energy change $\Delta G_{T,enc}^o$ is related to the standard change of the encapsulation enthalpy at temperature T, $\Delta H_{T,enc}^o$, as well as to the standard encapsulation entropy change $\Delta S_{T,enc}^o$ at temperature T:

$$\Delta G_{T,enc}^o=\Delta H_{T,enc}^o-\Delta T{S}_{T,enc}^o.$$

(4)

Both latter terms were evaluated here through a statistical mechanical treatment based on the RRHO partition functions. As these calculations deal with the species in the gas-phase, no solvent effects needed to be included.

The molecular–structure optimizations began from selected different initial arrangements for the CH₄@C₆₀ complex. The optimized potential energy minimum exhibited tetrahedral T symmetry (Figure 1). The C-H bond length was only slightly influenced by the encapsulation. In the free CH₄, its calculated M06-2X/6-31++G** C-H bond length was equal to 1.091 Å while after the encapsulation, the C-H bond at the M06-2X/6-31++G** level equaled 1.088 Å. Similarly, the changes in the C-C bond lengths on the C₆₀ cage were also small. The 6/6 and 5/6 C-C bond lengths for the empty C₆₀ were calculated at the M06-2X/6-31++G** level as 1.388 and 1.452 Å, respectively. In the CH₄@C₆₀ endohedral at the same level, the 6/6 bonds spanned an interval 1.389–1.390 Å, while for the 5/6 bonds this was 1.451–1.454 Å.

With respect to the thermodynamic evaluations, it should be mentioned that some common quantum chemical programs can erroneously work with an incorrect symmetry number for the I_h icosahedral C₆₀ cage. In fact, they simply deal with a value of one instead of the proper [36] (and used here) value of sixty. Actually, the I_h symmetry number was only discussed for the first time [36] in the year 1989, soon after the discovery of C₆₀, as previously there was no usage of the term. Before C₆₀ studies, only two other icosahedral molecules were known (dodecahedrane C₂₀H₂₀ and dodecaborate ion B₁₂${\mathrm{H}}_{12}^{2-}$). At present, on the other hand, there is a whole series of (topological) icosahedral fullerenes, like C₂₀, C₈₀, C₁₂₀, C₁₈₀, etc., though some of them undergo Jahn–Teller conditioned symmetry reduction in quantum chemical structural optimizations; for example, C₂₀ and C₈₀. The Jahn–Teller effect in such icosahedral cages can still be suppressed by the charge transfer present in metallofullerenes. Such restoration of symmetry happens, for example, in La₂@C₈₀ derived from the icosahedral C₈₀ cage (which is actually [36] the most populated isomer of La₂@C₈₀—in contrast to the set of the empty C₈₀ cages where the icosahedral cage is, on the contrary, the least populated).

In particular, application of a simple symmetry number of one for C₆₀, instead of the correct value of sixty, would imply a change in equilibrium constant by a huge multiplication factor of 60. Incidentally, a somewhat similar issue also appears [45] with the contribution of chirality to the partition functions; i.e., a factor equal to 2. Both issues are obviously important for thermodynamic stability evaluations. The encapsulation equilibrium constant $K_{p,enc}$ from Equation (2) can be expressed [36] by a compact formula using the molecular partition functions $q_i^o$ (the partition functions also have to correspond to the selected standard state, i.e., here an ideal gas phase at 1 atm = 101,325 Pa pressure) and the standard encapsulation enthalpy change at absolute zero temperature $\Delta H_{0,enc}^o$:

$$K_{p,enc}=\frac{\frac{q_{C{H}_4@C_{60}}^o}{N_A}}{\frac{q_{C{H}_4}^o}{N_A}\frac{q_{C_{60}}^o}{N_A}}exp[-\frac{\Delta H_{0,enc}^o}{RT}],$$

(5)

where the symbol $N_A$ stands for the Avogadro number. The standard encapsulation enthalpy change at the absolute zero temperature $\Delta H_{0,enc}^o$ actually has the meaning of the ground-state energy change; i.e., it contains the encapsulation reaction change $\Delta ZP{E}_{enc}$ in the vibrational zero-point energy (ZPE):

$$\Delta H_{0,enc}^o=\Delta E_{enc}+\Delta ZP{E}_{enc}.$$

(6)

Let us note that the form of Formula (5) should allow some partial cancellation of higher contributions [36] beyond simple RRHO approximation like anharmonicity corrections. As for the symmetry numbers, they have their roots in the rigorous quantum mechanical requirements for the symmetry behavior of wave functions. Practically speaking, the symmetry number is equal to the number of rotations of the molecule that convert it into an identical position; hence, this is given by the order of the rotational subgroup of the related molecular point group of symmetry. The symmetry numbers appear in the rotational contributions to the molecular partition functions $q_i^o$ in Equation (5) and can thus significantly contribute to the values of equilibrium constants.

The situation is more complex with fullerene endohedrals, as their encapsulates can exhibit rotational motions in the cages (or at least large amplitude oscillations). The internal motions can effectively recover [36,37,46,47,48] the original cage symmetry, especially at higher temperatures, thus simplifying the NMR patterns. The collective motions of encapsulates inside the cages yield a time-averaged equalizing environment on the NMR timescale. The aspects of symmetry are reflected in the so-called [36] free, fluctuating, or floating encapsulate model (FEM), originally developed for relative thermodynamic stabilities or populations of isomeric mono-metallofullerenes (where the calculated results consistently agreed [36] with the available observed data). In our situation, the endohedral symmetry should also be treated like the isomeric metallofullerenes, i.e., as the highest (topological) symmetry for the empty cage. Thus, we dealt here with just the dynamic symmetry of CH₄@C₆₀, i.e., I_h, not with its static tetrahedral symmetry T found from the geometry optimizations. The symmetry number for CH₄@C₆₀ (and for similar endohedrals based on C₆₀) was therefore here taken as [36] sixty (generally speaking, the RRHO and FEM treatments could in fact bracket out some bounds for thermodynamic terms). Incidentally, this treatment also removed the chirality feature of the T symmetry in this case. Obviously, the change in the symmetry number (from 12 for T to 60 for I_h) was quite profound and the consequences for the thermodynamic stabilities were substantial.

The Mulliken charges on atoms calculated at the M06-2X/3-21G level indicated that there was a (small) negative charge transfer of −0.069 (in elementary charge units) from CH₄ to the C₆₀ cage (so that there was an electrostatic contribution to the stabilization, cf. H₂O@C₅₉N [33,34]). Such a rather negligible charge transfer, however, stands in clear contrast to the substantial charge transfers known [36] for metallofullerenes. Mulliken charges on the cage carbons have values between −0.007 and +0.003. The Mulliken charge on the carbon in the free CH₄ at the same level was equal to −0.787, while in the encapsulated CH₄ this amounted to −0.764 (on H atoms of the encapsulated CH₄ it was +0.208). Let us mention for completeness that the Mulliken charges on atoms, due to their historical construction, were calculated using smaller sets of basis functions, like for example the standard 3–21 G basis set used here. Mulliken charges on atoms calculated with the 3–21 G basis set are known [49] to produce a good agreement with the available observed charges [50]. In addition, there are also more general methodological reasons [51,52] why some larger basis sets should not be applied in the calculation of the Mulliken atomic charge on atoms.

Table 1. The encapsulation energy $\Delta E_{enc}$ for CH₄@C₆₀ calculated with the selected approaches ^a and with inclusion of BSSE correction.

Calc. Level	$\Delta {\mathit{E}}_{\mathit{enc}}$ (kcal.mol−1)
M06-2X/6-31++G**	−16.35
M06-2X/6-311++G**	−16.41
MP2/6-311++G**	−14.22
B2PLYPD/6-311++G**	−13.94

^a In the M06-2X/6-31++G** optimized geometry, see Figure 1.

presents the encapsulation energy $\Delta E_{enc}$ for CH₄@C₆₀ evaluated with the four selected approaches. The $\Delta E_{enc}$ term was always negative—there was an energy gain upon the encapsulation, responsible for the complex stabilization. At the DFT M06-2X level, both considered basis sets produced about the same values. Similarly, the correlated MP2 and B2PLYPD encapsulation energies were quite close. As the B2PLYPD treatment is the most sophisticated among the four approaches considered in Table 1 (and also very close to the value in [35]), the $\Delta E_{enc}$ value of −13.94 kcal/mol was used in the subsequent evaluations of the thermodynamic stability. After the addition of the M06-2X/6-31++G** $\Delta ZP{E}_{enc}$ term, the corresponding encapsulation enthalpy change at absolute zero temperature $\Delta H_{0,enc}^o$ amounted to −11.38 kcal/mol. The encapsulation energy $\Delta E_{enc}$ of −13.94 kcal/mol was comparable with the energy gain found in [36] for the encapsulation producing CO@C₆₀ (−12.5 kcal/mol) or for the encapsulation yielding [36] N₂@C₆₀. Moreover, CH₄@C₆₀ had a significantly larger energy gain than the endohedrals [36] H₂@C₆₀, Ne@C₆₀, or [36] NH₃@C₆₀ (though not as high as that found in [36] for (H₂O)₂@C₇₀, (H₂O)₂@C₈₄). For example, the formation of NH₃@C₆₀ was calculated [36] to be connected with an encapsulation energy of just −5.23 kcal/mol at the MP2 = FC/6-31G** level.

The temperature interplay between the standard change of the encapsulation enthalpy $\Delta H_{T,enc}^o$ and the standard entropy change $T\Delta S_{T,enc}^o$ yielded the encapsulation equilibrium constants $K_{p,enc}$ (

Table 2. The standard ^a enthalpy $\Delta H_{T,enc}^o$, entropy $T\Delta S_{T,enc}^o$, Gibbs energy $\Delta G_{T,enc}^o$ changes, and the equilibrium constants $K_{p,enc}$ for the gas-phase equilibrium formation (1) of CH₄@C₆₀ evaluated ^b at the selected temperatures T.

T (K)	$\Delta {\mathit{H}}_{\mathit{T},\mathit{enc}}^{\mathit{o}}$ (kcal/mol)	$\mathit{T}\Delta {\mathit{S}}_{\mathit{T},\mathit{enc}}^{\mathit{o}}$ (kcal/mol)	$\Delta {\mathit{G}}_{\mathit{T},\mathit{enc}}^{\mathit{o}}$ (kcal/mol)	${\mathit{K}}_{\mathit{p},\mathit{enc}}$ (atm−1)
298.15	−11.87	−9.82	−2.05	3.18 × 10+1
500	−11.29	−15.74	4.45	1.14 × 10−2
1000	−9.47	−28.97	19.50	5.47 × 10−5
1500	−7.58	−41.15	33.58	1.28 × 10−5

^a The standard state—ideal gas phase at 1 atm = 101325 Pa pressure. ^b Calculated with the B2PLYPD/6-311++G** energetics and the M06-2X/6-31++G** partition functions.

). The presented $K_{p,enc}$ values were higher than those found in [36] for the CO@C₆₀ formation. For example, at 500 K, the encapsulation equilibrium constant $K_{p,enc}$ for CH₄ had a value of 0.0114 atm⁻¹, while in the case of CO@C₆₀ its value in [36] was lower by one order of magnitude, namely 0.00118 atm⁻¹. The finding suggests that production of CH₄@C₆₀ by applying high pressures at high temperature could in principle be possible. However, there is not only thermodynamic, but also kinetic, aspects. An efficient reaction mechanism is needed for encapsulation with high pressure at high temperature [1,7], which should temporarily open a window in the fullerene cage. In fact, a catalytic (or autocatalytic) action [13,53,54] is required for the window opening (and also for its closure). The standard encapsulation changes in enthalpy $\Delta H_{T,enc}^o$ for process (1) consistently exhibit a negative sign. In the view of the van’t Hoff relationship,

$$\frac{dln{K}_{p,enc}}{dT}=\frac{\Delta H_{T,enc}^o}{R{T}^2}$$

(7)

the encapsulation equilibrium constant $K_{p,enc}$ must exhibit a decreasing course with increasing temperature—as indeed seen in both Table 2 and Figure 2.

GF vibrational analysis was used here for confirmation that the energy minimum was indeed found (no imaginary frequency) and as input for the RRHO partition functions. Figure 3 shows the M06-2X/6-31++G** calculated IR and Raman harmonic vibrational spectrum of CH₄@C₆₀. The high I_h symmetry [55] of the free C₆₀ grossly simplifies its IR vibrational spectrum [56]. The symmetry representation ${\Gamma }_{C_{60}}$ of the 174 vibrational modes of C₆₀ was as follows [57]:

$${\Gamma }_{C_{60}}=2A_g+A_u+3T_{1g}+4T_{2g}+4T_{1u}+5T_{2u}+6G_u+6G_g+8H_g+7H_u,$$

(8)

where A, T, G, and H denote one-, three-, four-, and five-fold degenerate modes, respectively (so that we deal with a number of modes 3 × 1 + 16 × 3 + 12 × 4 + 15 × 5 = 174 = 60 × 3 − 6). This high symmetry actually led to only 46 distinct vibrational frequencies [57], with four of them ($T_{1u}$) active in the IR and ten of them ($A_g$, $H_g$) in the Raman spectra. However, if the high symmetry of the cage was reduced by encapsulation, the spectral symmetry selection rules should also be relaxed. Nevertheless, even the CH₄@C₆₀ IR spectrum primarily consisted of four (broader) bands. The CH₄ vibrational modes appeared in the CH₄@C₆₀ spectrum, with low intensities in the regions around 1300 cm⁻¹ (the bond-angle deformation, in the free CH₄ at the M06-2X/6-31++G** level amounted to 1349 cm⁻¹) and around 3200 cm⁻¹ (bond stretching, in the free CH₄ 3192 cm⁻¹). The CH₄@C₆₀ endohedral had 189 vibrational modes—174 of them actually originated from the vibrational modes of the free C₆₀ and 9 from the vibrational modes of the free CH₄, though their frequencies were somewhat shifted and the exact degeneracies had been removed. The remaining six vibrational modes had the nature of hindered rotational motions or hindered translational motions of CH₄ in the C₆₀ cage (the translational motions can also be viewed as the encapsulate vibrations between the cage walls). The hindered collective motions actually possessed the lowest frequencies in the vibrational spectrum, about 173 cm⁻¹ for the hindered rotations and about 247 cm⁻¹ for the hindered translations (though their IR intensities were negligible, while the modes around 247 cm⁻¹ should have still been a bit more active in the Raman spectra). In contrast to the straightforward FEM treatment of the relative thermodynamic stabilities or populations in sets of isomeric monometallofullerenes [36], the lowest vibrational modes in our case were important for the description of the overall encapsulation thermodynamics. The six lowest vibrational modes were also primary contributors to the encapsulation reaction change $\Delta ZP{E}_{enc}$ of the vibrational zero-point energy in Equation (6).

In addition to C₆₀, the C₈₄ cages could also be of interest for the CH₄ accommodation. C₈₄ is a relatively well known system of isomeric fullerenes, in particular the two most common isomers, $D_2$ and $D_{2d}$, conventionally also labeled [58] as 22 and 23, $D_2$(22)-C₈₄ and $D_{2d}$(23)-C₈₄. The $D_{2d}$ structure is the lowest-energy isomer, but it is located only about 0.5 kcal/mol below the $D_2$ species [59]. The M06-2X/6-31++G** water dimer encapsulation energy into the $D_2$(22)-C₈₄ cage was calculated [36] as −17.4 kcal/mol, while for $D_{2d}$(23)-C₈₄, it was −14.4 kcal/mol. On the other hand, the inclusion of the water dimer into C₆₀ was repulsive [26]. Incidentally, our ongoing calculations on CO₂@$D_{2d}$(23)-C₈₄ at the M06-2X/6-31+G* level preliminary gave an encapsulation energy of −13.6 kcal/mol. Hence, CH₄@C₈₄ should also be of computational interest in further studies.

In future, still higher methodological levels for the molecular structure optimization and energetics, as well as more advanced descriptions of the motions of the encapsulate in the cage, with related modifications of the RRHO approximation (including the combination [36] of the RRHO and FEM treatments), should be considered. Another more advanced approach [60] treats the motions of a monoatomic encapsulate (for example in He@C₆₀) as the quantum problem of a particle in a sphere (a similar approach [60] can also be developed for encapsulate motions in nanotubes using a model cylinder). This approach produces contributions to thermodynamic functions, showing for example an interesting temperature development [60,61] of the heat capacity. Further generalizations of this approach could lead to a more realistic description of the encapsulation thermodynamics. A generalization of the classical Pitzer description [62] of the hindered internal rotations in molecules should also be studied [55,63] for this purpose. Moreover, there have been some promising results [36] for anharmonicity considerations based on quantum chemical calculations, too. Finally, let as mention for completeness that topological, non-numerical approaches [64,65,66,67] can also be useful in understanding some encapsulation issues.

Such expected future progress should in turn encourage studies of still more complex nanocarbon systems [68,69,70,71] via applications of Gibbs energy treatments. The results of such more advanced calculations with an even higher accuracy could also be used in the search for non-metallic fullerene endohedrals in the interstellar spectra (diffuse interstellar bands, DIBs). This would be a subsequent step after the discovery [72,73,74] of the empty C₆₀ and C₇₀ in interstellar space and hypotheses [75,76,77] of their origin there. One of the possible endohedral candidates is [78] H₂O@${\mathrm{C}}_{60}^{+}$. For the application, the calculations should also be adjusted for charged species (including electronic spectra).

4. Conclusions

The previously synthetically prepared endohedral CH₄@C₆₀ was characterized through calculations. The symmetry of the endohedral was found to be tetrahedral T symmetry. The calculated encapsulation energetics were attractive, with a substantial energy gain of −13.94 kcal/mol at the most advanced computational level. The encapsulation equilibrium constants for CH₄@C₆₀ were somewhat higher than previously found with the CO@C₆₀ system—this finding suggests that high-pressure and high-temperature synthesis could in principle be possible for CH₄@C₆₀. In future, these computational approaches should be developed further, for both encapsulation energetics and description of the motions of the encapsulate in the cage. The presented results encourage studies of still more complex endohedral and other nanocarbon systems.