CO₂@C₈₄: DFT Calculations of Structure and Energetics

Abstract

Encapsulations of carbon dioxide into $D_2$(22)-${\mathrm{C}}_{84}$ and $D_{2d}$(23)-$C_{84}$ fullerenes are evaluated. The encapsulation energy is computed with the DFT M06-2X/6-31+${\mathrm{G}}^{*}$ approach corrected for the basis set superposition error evaluated by the counterpoise method. The resulting encapsulation energy for ${\mathrm{CO}}_2$@$D_2$(22)-${\mathrm{C}}_{84}$ and ${\mathrm{CO}}_2$@$D_{2d}$(23)-${\mathrm{C}}_{84}$ amounts to substantial values of −14.5 and −13.9 kcal/mol, respectively. The energy gain is slightly larger than for CO@${\mathrm{C}}_{60}$, already synthesized with a high-temperature and high-pressure treatment—so that a similar preparation of ${\mathrm{CO}}_2{@\mathrm{C}}_{84}$ could be possible. The calculated rotational constants and IR vibrational spectra are presented for possible use in detection. The stability of ${\left({\mathrm{CO}}_2\right)}_2$${@\mathrm{C}}_{84}$ is also briefly discussed.

1. Introduction

Interactions of carbon dioxide with fullerenes have been studied [1,2,3], for example, for boron-based fullerene cage ${\mathrm{B}}_{80}$ [1]. Interestingly, additions of endohedral fullerenes [4,5] or carbon dioxide [6] can improve the efficiency of solar cells. Various non-metallic molecules can also be encapsulated inside fullerene cages. ${\mathrm{N}}_2{@\mathrm{C}}_{60}$ and ${\mathrm{N}}_2{@\mathrm{C}}_{70}$ were the first such non-metal endohedrals, namely prepared [7] by heating under high pressure and with a catalyst. Moreover, there is also an elegant synthetic technique that first places molecules inside open fullerene cages and then the cages are synthetically closed—for example, the preparation [8] of (${\mathrm{H}}_2$O${)}_2$${@\mathrm{C}}_{70}$.

Such fullerene encapsulations of non-metals have also been calculated, e.g., in [9,10,11,12,13,14]. The first such calculations [9] from the year 1991 treated small molecules (${\mathrm{H}}_2$, ${\mathrm{N}}_2$, CO, and LiH) trapped inside the ${\mathrm{C}}_{60}$ cage and predicted a blue shift in the stretching frequencies. Later on, encapsulations of larger non-metal molecules into fullerene cages were also calculated [10,11,12,13,14], for example, ${\mathrm{MH}}_4$ hydrides [10] or water molecules [11,14]. Moreover, the endohedral ${\mathrm{CH}}_4{@\mathrm{C}}_{60}$ was synthesized recently [12], and its stabilization energy was calculated [13] as −13.5 kcal/mol.

As ${\mathrm{CO}@\mathrm{C}}_{60}$ could also be prepared [7] by the high-temperature and high-pressure treatment, there is a question regarding if ${\mathrm{CO}}_2$ may also be encapsulated in some (larger) fullerene cages. In this report, the issue is calculated for the first time—namely for the two lowest-energy and most common ${\mathrm{C}}_{84}$ fullerenes $D_2$(22)${\text{-}\mathrm{C}}_{84}$ and $D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$ [15]. The pristine $D_2$ and $D_{2d}$ ${\mathrm{C}}_{84}$ isomers are produced via high-temperature synthesis [15,16] in a ratio of 2:1. The $D_{2d}$ structure is located [17] only about 0.5 kcal/mol below the $D_2$ species. However, the chirality factor also contributes [15] to the $D_2$/$D_{2d}$ relative isomeric populations.

2. Calculations

The calculations are carried out with density-functional theory (DFT) treatment, namely using the DFT M06-2X functional comprehensively tested [18] for thermochemistry, kinetics, noncovalent interactions, and other applications. The M06-2X functional is employed with the standard 6-31+${\mathrm{G}}^{*}$ basis set [19] M06-2X/6-31+(${\mathrm{G}}^{*}$ approach). The geometry optimizations are performed with the analytical first derivatives of energy. In the localized stationary points, the harmonic vibrational GF analysis is carried out using the analytical second derivatives of energy. The vibrational analysis can confirm that energy minima were indeed localized. For DFT integration, the so-called ultrafine grid is applied (and then refined with the superfine grid). For the SCF (self-consistent field) convergence, the tight criterion is used. The wavefunction stability is systematically tested in order to avoid unphysical unstable SCF solutions.

The encapsulation energy [20] is corrected for the so-called basis set superposition error (BSSE) evaluated by the approximative Boys–Bernardi counterpoise method [21] (also labeled [22] CP2). The counterpoise method guarantees that every component of a treated chemical reaction is formally described by the same number of basis-set functions so that their total energies are comparable. The BSSE correction term constitutes an important refinement of the encapsulation energy. Without the BSSE correction, the energy gain produced by encapsulation process would be overestimated (i.e., products would be over-stabilized).

The reported computations are carried out with the Gaussian program package [22]. The calculations are performed on computers operating in parallel regime, mostly with 8–24 processors (computational frequency up 3 GHz each; available operational memory up to 60 GB).

3. Results and Discussion

The calculated encapsulation energy $\Delta E_{enc}$ means the potential-energy change along the studied gas-phase association process, for example for encapsulation in the $D_2$(22)${\text{-}\mathrm{C}}_{84}$ fullerene cage:

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

(1)

and similarly for the $D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$ cage. The M06-2X/6-31+${\mathrm{G}}^{*}$-calculated encapsulation energy values with and without the BSSE correction are presented in

Table 1. The calculated encapsulation energy $\Delta E_{enc}$ for ${\mathrm{CO}}_2$@$D_2$(22)${{\text{-}\mathrm{C}}_{84}}^a$ and ${\mathrm{CO}}_2$@$D_{2d}$(23)${{\text{-}\mathrm{C}}_{84}}^b$.

Species	Level	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{enc}}$ (kcal/mol)	BSSE
		No BSSE
${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$	M06-2X/6-31${\mathrm{G}}^{*}$	−16.4	−12.1
${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$	M06-2X/6-31+${\mathrm{G}}^{*}$	−17.7	−14.5
${\mathrm{CO}}_2$@$D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$	M06-2X/6-31${\mathrm{G}}^{*}$	−16.2	−11.7
${\mathrm{CO}}_2$@$D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$	M06-2X/6-31+${\mathrm{G}}^{*}$	−17.2	−13.9

^a See Figure 1. ^b See Figure 2.

.

The BSSE-corrected M06-2X/6-31+${\mathrm{G}}^{*}$ encapsulation energy $\Delta E_{enc}$ for ${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$ and ${\mathrm{CO}}_2$@$D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$ amounts to −14.5 and −13.9 kcal/mol, respectively. At this computational level, the BSSE correction reduces the energy gain owing to the encapsulation by about 3 kcal/mol. For illustrative comparison, the encapsulation energies evaluated with smaller basis sets are also presented, i.e., for the M06-2X/6-31${\mathrm{G}}^{*}$ computational approach but still in the same M06-2X/6-31+${\mathrm{G}}^{*}$ optimized structures. As for the smaller basis set, the M06-2X/6-31${\mathrm{G}}^{*}$ energy gain without the BSSE term is reduced by less than 2 kcal/mol compared to the M06-2X/6-31+${\mathrm{G}}^{*}$ approach. On the other hand, the BSSE correction provides a larger reduction in energy gain with the smaller basis (the M06-2X/6-31${\mathrm{G}}^{*}$ approach), namely by about 4 kcal/mol. This finding is in agreement with a general feature of the BSSE correction: the correction term would disappear in a situation of a basis set with an infinite number of basis functions (clearly enough, such an infinite computational scheme cannot practically be handled). Overall, the $\Delta E_{enc}$ terms in Table 1 are somewhat more sensitive to the BSSE correction than to the basis-set choice. The calculated gain in encapsulation energy is slightly larger than in the case of ${\mathrm{CO}@\mathrm{C}}_{60}$, for which the encapsulation energy is calculated to be [23] about −12.5 kcal/mol. The finding suggests that ${\mathrm{CO}}_2{@\mathrm{C}}_{84}$ could possibly be prepared by the high-temperature and high-pressure treatment [7] too.

Table 2. The selected characteristics of ${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$ and ${\mathrm{CO}}_2$@$D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$—the closest O-cage contact ^a $r_{O-C}$, the total charge ^b on cage $q_c$, the lowest vibrational frequency ^a${\omega }_{low}$, and the highest vibrational frequency ^a${\omega }_{high}$.

Species	${\mathit{r}}_{\mathit{O}-\mathit{C}}$/Å	${\mathit{q}}_{\mathit{c}}$	${\mathit{\omega }}_{\mathit{low}}$${\mathrm{/cm}}^{-1}$	${\mathit{\omega }}_{\mathit{high}}$${\mathrm{/cm}}^{-1}$
${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$ c	3.07	−0.012	12.7	2425
${\mathrm{CO}}_2$@$D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$ d	3.05	−0.017	37.2	2431

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

presents four calculated characteristics of the ${\mathrm{CO}}_2{@\mathrm{C}}_{84}$ endohedrals—the closest contact of oxygen atoms with the cage carbons, the total charge on the ${\mathrm{C}}_{84}$ cages, and the lowest and highest vibrational frequency values. The shortest distance of oxygen to the cage is, in both cases, about 3 Å. There is some small charge transfer from the ${\mathrm{CO}}_2$ unit to the cage. The Mulliken atomic charges evaluated at the M06-2X/3-21G level show negative charge transfer to the cage of −0.02 (in elementary charge units). Such rather negligible charge transfer stands in clear contrast to the substantial charge donations associated with [14] metallofullerenes. Let us mention that the Mulliken atomic charges, owing to their construction, are to be evaluated with smaller basis sets, for example, the standard 3–21G basis set applied here. The Mulliken atomic charges computed with the 3–21G basis set are known [14] to yield, for endohedral metallofullerenes, fair agreement with the available observed atomic charges [24]. Moreover, there are also deeper computational arguments [25,26] as to why larger basis sets should not be considered for the evaluation of the Mulliken charges on atoms.

The harmonic vibrational analysis is considered here not only for the confirmation that energy minima were indeed found (no imaginary frequency) but also for simulations of the infrared (IR) spectra as they can be used for identification of the ${\mathrm{CO}}_2{@\mathrm{C}}_{84}$ endohedrals when prepared. Figure 3 shows the M06-2X/6-31+G*-calculated IR vibrational spectra for ${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$ and ${\mathrm{CO}}_2$@$D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$. The lowest vibrational frequency ${\omega }_{low}$ belongs to a rotational motion of ${\mathrm{CO}}_2$ as a whole within the cage. The free ${\mathrm{CO}}_2$ molecule has four vibrational modes; their harmonic frequencies at the M06-2X/6-31+G* level are double-degenerated bending mode $670 {\mathrm{cm}}^{-1}$, symmetric bond stretching $1411 {\mathrm{cm}}^{-1}$, and asymmetric bond stretching $2461 {\mathrm{cm}}^{-1}$. After encapsulation, the ${\mathrm{CO}}_2$ frequencies are somewhat shifted (and the degeneracy removed) owing to the interactions with cages. In the case of ${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$, the M06-2X/6-31+G*-calculated frequencies are 652.2, 652.8, 1404, and $2425 {\mathrm{cm}}^{-1}$ (i.e., ${\omega }_{high}$ in Table 2). Similarly, for ${\mathrm{CO}}_2$@$D_{2d}$${\text{-}\mathrm{C}}_{84}$, the M06-2X/6-31+G* frequencies are 650.2, 650.4, 1408, and $2431 {\mathrm{cm}}^{-1}$. The asymmetric C-O bond stretching is actually the strongest peak in the IR spectra (Figure 3).

Incidentally, the recent final confirmation with electron spectroscopy [27] of the presence of ${\mathrm{C}}_{60}$ in the interstellar space encourages further cosmic searches [28,29,30]—even for fullerene cages with some encapsulates inside (incidentally, pressure of ${\mathrm{CO}}_2$ at Venus can reach [31] some 90 atm). As rotational spectroscopy could be used for the detection of fullerene species in interstellar space,

Table 3. The M06-2X/6-31+${\mathrm{G}}^{*}$ rotational constants A, B, C [GHz] of the ${\mathrm{CO}}_2{@\mathrm{C}}_{84}$ endohedrals.

Endohedral a	A	B	C
${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$ a	0.044112	0.042604	0.041344
${\mathrm{CO}}_2$@$D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$ b	0.042788	0.042649	0.042555

^a See Figure 1. ^b See Figure 2.

presents the computed rotational constants, A, B, and C, for the ${\mathrm{CO}}_2{@\mathrm{C}}_{84}$ endohedrals (although, in interstellar space, we can rather expect the species in an ionized form).

Similar DFT calculations for related encapsulation in Equation (2), producing (${\mathrm{CO}}_2$${)}_2$${@\mathrm{C}}_{84}$, are now also in progress; for ${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$, they concern the gas-phase process:

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

(2)

and similarly for the $D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$ cage. However, in order to compare the relative reaction populations of the related endohedrals with one and two encapsulated ${\mathrm{CO}}_2$ molecules, even the encapsulation equilibrium constants $K_{p,enc}\left(i\right)$ would be needed. For the process in Equation (1), the first step would be the construction of the encapsulation equilibrium constant $K_{p,enc}\left(1\right)$ (defined by the partial pressures p of the individual reaction components):

$$K_{p,enc}\left(1\right)={\displaystyle \frac{p_{C{O}_2@D_2\left(22\right)-C_{84}}}{p_{C{O}_2}{p}_{D_2\left(22\right)-C_{84}}}}.$$

(3)

Then, the equilibrium constant $K_{p,enc}\left(2\right)$ for encapsulation in Equation (2) should also be calculated:

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

(4)

From Equations (3) and (4), we can obtain the required population ratio of the two endohedrals:

$${\displaystyle \frac{p_{{\left(C{O}_2\right)}_2@D_2\left(22\right)-C_{84}}}{p_{C{O}_2@D_2\left(22\right)-C_{84}}}}=p_{C{O}_2}{\displaystyle \frac{K_{p,enc}\left(2\right)}{K_{p,enc}\left(1\right)}}.$$

(5)

Equation (5) straightforwardly shows how the ${\mathrm{CO}}_2$ pressure $p_{C{O}_2}$ influences the population ratio of (${\mathrm{CO}}_2$${)}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$ and ${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$ endohedrals. The encapsulation equilibrium constants $K_{p,enc}\left(i\right)$ can be calculated using molecular partition functions supplied with rotational and vibrational parameters generated by DFT calculations.

4. Conclusions

Calculations are reported for the encapsulation of carbon dioxide into the two most common ${\mathrm{C}}_{84}$ fullerenes, $D_2$(22)${\text{-}\mathrm{C}}_{84}$ and $D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$, yielding ${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$ and ${\mathrm{CO}}_2$@$D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$. The calculations are carried out with the DFT M06-2X/6-31+${\mathrm{G}}^{*}$ approach. The resulting encapsulation energy for ${\mathrm{CO}}_2$@$D_2$(22)${\text{-}\mathrm{C}}_{84}$ and ${\mathrm{CO}}_2$@$D_{2d}$(23)${\text{-}\mathrm{C}}_{84}$ amounts to substantial values of −14.5 and −13.9 kcal/mol, respectively. The energy gain is slightly larger than for ${\mathrm{CO}@\mathrm{C}}_{60}$. As ${\mathrm{CO}@\mathrm{C}}_{60}$ was already synthesized by heating under high pressure with a catalyst, similar high-temperature and high-pressure preparation of ${\mathrm{CO}}_2{@\mathrm{C}}_{84}$ or even (${\mathrm{CO}}_2$${)}_2$${@\mathrm{C}}_{84}$ could be possible. The computational treatment should be expanded in the future by evaluations of the related encapsulation equilibrium constants. The presented results encourage similar computational studies for still more complex [32,33,34,35,36] endohedral and other nanocarbon systems.