The Calculated Structure and Energetics of (CO₂)₂@C₈₄

Abstract

Calculations are presented for the encapsulation of two CO₂ molecules in the most common C₈₄ fullerenes, producing (CO₂)₂@$D_2$(22)-C₈₄ and (CO₂)₂@$D_{2d}$(23)-C₈₄. The calculations are performed at the DFT M06-2X/6-31+G* level with the BSSE correction. The encapsulation energy for (CO₂)₂@$D_2$(22)-C₈₄ and (CO₂)₂@$D_{2d}$(23)-C₈₄ is calculated as −4.9 and −5.6 kcal/mol, respectively. The encapsulation of two CO₂ molecules is attractive, though the energy gain is, owing to a steric hindrance, smaller than previously found for the encapsulation of one CO₂. The IR vibrational spectra are presented, too.

1. Introduction

The interplay of fullerenes with CO₂ has recently been treated [1,2,3,4], even for boron-based fullerene-like cage B₈₀ [1]. Actually, such species can even have interesting practical applications—additions of endohedral metallofullerenes [5,6] or CO₂ [7] can improve properties of solar cells. In addition to metals, non-metallic species can also be incorporated inside fullerene polyhedrons. First such non-metal endohedrals are represented by N₂@C₆₀ and N₂@C₇₀ produced [8] at high temperatures and high pressures with a catalyst. Another approach first places molecules inside open fullerene cages, and the cage window is then closed synthetically, e.g., preparation [9] of (H₂O)₂@C₇₀. Such fullerene endohedrals with encapsulated non-metals have also been studied by calculations, e.g., [10,11,12,13,14,15]. The first such calculations [10] treated diatomic molecules like H₂ or HF trapped inside the C₆₀ cage and predicted shifts in vibrational spectra. Encapsulations of small polyatomic non-metal species into fullerene cages have also been evaluated, including water, e.g., [11,14,15].

CO@C₆₀ can be produced [7] by the high-pressure and high-temperature treatment, and there is thus a related problem concerning CO₂ encapsulations in some larger fullerene polyhedrons. We have already calculated [4] possibilities of encapsulation of one CO₂ molecule in the two most common C₈₄ fullerenes as well as such encapsulations for one, two, and three water molecules [14,15]. The present report continues with such evaluations for encapsulation of two CO₂ molecules in the two C₈₄ cages, namely $D_2$ and $D_{2d}$ C₈₄, conventionally labeled as $D_2$(22)-C₈₄ and $D_{2d}$(23)-C₈₄. It should be noted for completeness that the two C₈₄ cages are commonly produced in the carbon-arc fullerene synthesis [16] in a ratio [16,17] of about 2:1. Interestingly, the pristine $D_2$ cage is in potential energy located [17,18] about 0.5 kcal/mol above the $D_{2d}$ structure. However, the entropy contributions influence the $D_2$/$D_{2d}$ relative isomeric populations as well.

2. Calculations

The calculations are performed with the DFT (density-functional theory) approach, namely with the M06-2X functional thoroughly tested [19] for various applications including non-covalent interactions. The M06-2X functional is applied with the standard 6-31+G* basis set [20]; the approach is denoted by M06-2X/6-31+G*. The molecular-structure optimizations are carried out with the 1st derivatives of the potential energy constructed analytically (Figure 1 and Figure 2). The stationary points found are checked by the GF harmonic vibrational analysis using the analytical 2nd derivatives of the potential energy. The GF vibrational analysis can prove that the localized stationary points are indeed the local energy minima. Moreover, it can also simulate the vibrational spectra and supply input information for thermodynamic-stability evaluations. The wavefunction stability was tested throughout so that unstable SCF solutions without a physical applicability could be eliminated.

The potential energy changes connected with encapsulations into the carbon cages are corrected [21] with respect to the basis set superposition error (BSSE). The BSSE correction term is evaluated by the counterpoise method introduced by Boys and Bernardi [22]—it is also coded [23,24] CP2 in the case of a dimerization. The counterpoise method ensures that every species participating in a studied chemical process is formally described by the same number of the atomic basis-set functions so that their potential energies can be compared. The BSSE correction term represents a crucial improvement of the encapsulation energy, as without the BSSE term, the potential energy gain yielded by the encapsulation process would be overvalued so that reaction products would be artificially over-stabilized.

The presented evaluations were carried out with the Gaussian program package [23,24]. The calculations were performed on computers operating in parallel regime, with 8–24 processors (computational frequency up to 3 GHz each, operational memory up to 60 GB).

3. Results and Discussion

This study deals with the encapsulation energy $\Delta E_{enc}$, representing the potential-energy change in the gas-phase process involving the $D_2$(22)-C₈₄ fullerene cage:

$$2{\mathrm{CO}}_2\left(\mathrm{g}\right)+{\mathit{D}}_2\left(22\right)-{\mathrm{C}}_{84}\left(\mathrm{g}\right)={\left({\mathrm{CO}}_2\right)}_2@{\mathit{D}}_2\left(22\right)-{\mathrm{C}}_{84}\left(\mathrm{g}\right),$$

(1)

as well as with similar encapsulation in the $D_{2d}$(23)-C₈₄ cage.

Table 1. The calculated encapsulation energy $\Delta E_{enc}$ for (CO₂)₂@$D_2$(22)-C₈₄ ^a and (CO₂)₂@$D_{2d}$(23)-C₈₄ ^b.

Species	Level	Δ${\mathit{E}}_{\mathit{e}\mathit{n}\mathit{c}}$ (kcal/mol)
		No BSSE	BSSE
(CO2)2@$D_2$(22)-C84	M06-2X/6-31G*	−14.0	−2.1
(CO2)2@$D_2$(22)-C84	M06-2X/6-31+G*	−12.7	−4.9 c
(CO2)2@$D_{2d}$(23)-C84	M06-2X/6-31G*	−14.6	−2.8
(CO2)2@$D_{2d}$(23)-C84	M06-2X/6-31+G*	−13.4	−5.6 d

^a See Figure 1. ^b See Figure 2. ^c The value for CO₂@$D_2$(22)-C₈₄ is −14.5 kcal/mol [4]. ^d The value for CO₂@$D_{2d}$(23)-C₈₄ is −13.9 kcal/mol [4].

presents the M06-2X/6-31+G* encapsulation energetics with and also without the BSSE correction.

The M06-2X/6-31+G* encapsulation energy $\Delta E_{enc}$ corrected with the BSSE term is for (CO₂)₂@$D_2$(22)-C₈₄ and (CO₂)₂@$D_{2d}$(23)-C₈₄ equal to −4.9 and −5.6 kcal/mol, respectively. The encapsulation of two CO₂ molecules is attractive though the energy gain is owing to a steric hindrance smaller than found [4] for the encapsulation of just one CO₂. The BSSE term decreases the energy gain significantly—by about 8 kcal/mol. Table 1 presents for methodological comparison also the values obtained with a simpler approach, viz. M06-2X/6-31G* in the M06-2X/6-31+G* optimized geometry. The encapsulation of two CO₂ molecules is attractive at the simpler level as well, though the stabilization is reduced.

Table 2. The selected characteristics of (CO₂)₂@$D_2$(22)-C₈₄ and (CO₂)₂@$D_{2d}$(23)-C₈₄—the closest C-cage contact ^a $r_{C-C}$, the total charge ^b on the cage $q_c$, the lowest vibrational frequency ^a ${\omega }_{low}$, the highest vibrational frequency ^a ${\omega }_{high}$.

Species	${\mathit{r}}_{\mathit{C}-\mathit{C}}$/Å	${\mathit{q}}_{\mathit{c}}$	${\mathit{\omega }}_{\mathit{l}\mathit{o}\mathit{w}}$/cm−1	${\mathit{\omega }}_{\mathit{h}\mathit{i}\mathit{g}\mathit{h}}$/cm−1
(CO2)2@$D_2$(22)-C84 c	3.13	−0.079	38.3	2471
(CO2)2@$D_{2d}$(23)-C84 d	3.06	−0.074	29.9	2457

^a M06-2X/6-31+G* values. ^b M06-2X/3-21G values. ^c See Figure 1. ^d See Figure 2.

reports some computed molecular parameters of the (CO₂)₂@C₈₄ endohedrals—the closest contact $r_{C-C}$ of the carbon atom in CO₂ with the cage carbons, the total charge $q_c$ on the C₈₄ cage, and the lowest/highest vibrational frequency. The closest contact $r_{C-C}$ is in both species about 3.1 Å. The total charge $q_c$ on the cage evaluated using the M06-2X/3-21G Mulliken atomic charges is only about −0.08 in elementary charge units. Such a small charge transfer is considerably different from substantial charge transfers calculated for metallofullerenes [14]. It should be mentioned that the Mulliken charges on atoms should be evaluated owing to their definition [25,26] with smaller basis sets like the standard 3–21G basis set considered in this work. Such small basis sets are indeed known [14] to produce for metallofullerenes a good agreement with the atomic charges obtained from observations [27]. As vibrational spectra can be useful for identification of nanocarbon species [28,29], Figure 3 presents the M06-2X/6-31+G* simulated IR vibrational spectra of (CO₂)₂@$D_2$(22)-C₈₄ and (CO₂)₂@$D_{2d}$(23)-C₈₄.

However, as the energy gain after encapsulation of two CO₂ molecules in C₈₄ is smaller than in the case of just one CO₂, and also smaller than for two water molecules [15], the C₈₄-based endohedrals containing two CO₂ molecules are less likely to be prepared. Still, an application of high CO₂ pressures can improve [4] the yield. Let us consider (2) the equilibrium constant $K_{p,enc}\left(d\right)$ for process (1)—$K_{p,enc}\left(d\right)$ expressed using the partial pressures p of the reaction components is as follows:

$$K_{p,enc}\left(d\right)={\displaystyle \frac{p_{{\left(C{O}_2\right)}_2@D_2\left(22\right)-C_{84}}}{p_{C{O}_2}^2{p}_{D_2\left(22\right)-C_{84}}}},$$

(2)

and also (4) the equilibrium constant $K_{p,enc}\left(m\right)$ for the encapsulation (3) of just one carbon dioxide:

$${\mathrm{CO}}_2\left(\mathrm{g}\right)+{\mathit{D}}_2\left(22\right)-{\mathrm{C}}_{84}\left(\mathrm{g}\right)={\mathrm{CO}}_2@{\mathit{D}}_2\left(22\right)-{\mathrm{C}}_{84}\left(\mathrm{g}\right)$$

(3)

$$K_{p,enc}\left(m\right)={\displaystyle \frac{p_{C{O}_2@D_2\left(22\right)-C_{84}}}{p_{C{O}_2}{p}_{D_2\left(22\right)-C_{84}}}}.$$

(4)

The two equilibrium constants yield [4] the ratio of the endohedrals in the following form:

$${\displaystyle \frac{p_{{\left(C{O}_2\right)}_2@D_2\left(22\right)-C_{84}}}{p_{C{O}_2@D_2\left(22\right)-C_{84}}}}=p_{C{O}_2}{\displaystyle \frac{K_{p,enc}\left(d\right)}{K_{p,enc}\left(m\right)}}$$

(5)

which straightforwardly shows the influence of the CO₂ pressure $p_{C{O}_2}$ on the endohedral population ratio. The related detailed thermodynamic-stability evaluations are to be supplied later on. Incidentally, the finding of C₆₀ in the cosmic space [30] suggests astrophysical searches [31,32,33] also for fullerene endohedrals—for example, the pressure of CO₂ at Venus can reach [15] some 90 atm.

There is an interesting related issue, namely isomerism of the free, i.e., not encapsulated, gas-phase (CO₂)₂ species described by quantum-chemical calculations [34]. Two different local energy minima have been located and named [34] parallel P and T shape structures. Recently, we have studied [15] the water-dimer thermodynamics using a high-level energetics G3 & MP2 = FC/6-311+G* and anharmonic-thermodynamic treatment. As the combined treatment produces a very good agreement with observed data in the water-dimer case, it can also be applied to (CO₂)₂. At the advanced G3 & MP2 = FC/6-311+G* level, the P form is lower in energy, though only by some 0.14 kcal/mol. The rotational and vibrational terms needed for the anharmonic treatment originate [15] from the MP2 = FC/6-311+G* calculations. At very low temperatures the lower-energy isomer must be prevailing, i.e., the P form. However, with increasing temperature, the relative isomeric populations are mutually approaching; finally, the stability order is interchanged, and at high temperatures the T isomer becomes prevailing. The equal populations of both isomers are obtained at a temperature of about 120 K; at room temperature the T isomer then represents 77% of the equilibrium mixture. Obviously, with the exception of the lowest temperatures, one has to expect the presence of both isomeric forms, which should be taken into account with interpretations of observations, including planetary atmospheres.

4. Conclusions

The recent theoretical treatments of rather complex endohedral and other nanocarbon species [35,36,37,38,39,40,41,42,43,44,45,46] are extended here to encapsulation of two CO₂ molecules in the most common C₈₄ fullerenes, producing endohedrals (CO₂)₂@$D_2$(22)-C₈₄ and (CO₂)₂@$D_{2d}$(23)-C₈₄. The calculations are performed at the DFT M06-2X/6-31+G* level with the BSSE correction. The encapsulation energy for (CO₂)₂@$D_2$(22)-C₈₄ and (CO₂)₂@$D_{2d}$(23)-C₈₄ is calculated as −4.9 and −5.6 kcal/mol, respectively. The encapsulation of two CO₂ molecules is attractive, though the energy gain is, owing to a steric hindrance, smaller than previously found for the encapsulation of one CO₂ molecule.