La₂C₂@D₅(450)-C₁₀₀: Calculated High Energy Gain in Encapsulation

Abstract

The structure and energetics of the clusterfullerene La₂C₂@$D_5$(450)-C₁₀₀ are calculated at the B3LYP/6-31G*∼SDD level (including counterpoise correction for the basis set superposition error), and the observed features are confirmed. Its stability is explained by substantial energy gain connected with the encapsulation, viz. 140 kcal/mol per atom of the encapsulate, actually higher than previously found for comparable systems.

1. Introduction

In fullerene and metallofullerene research, calculations have from the beginning helped experimental search and understanding of new nanocarbons [1,2,3,4], more recently even with species containing around one hundred carbons [5,6,7]. In the group of such large metallofullerenes [8,9,10,11,12,13,14,15,16,17,18], another giant system has recently been characterized [14] by X-ray crystal diffraction, namely La₂C₂@$D_5$(450)-C₁₀₀. In this study, the giant clusterfullerene is calculated using the density functional theory (DFT) in order to further enhance knowledge of its structure and encapsulation energetics.

2. Calculations

The $D_5$(450)-C₁₀₀ fullerene is one of the 450 isolated-pentagon-rule (IPR) cages of C₁₀₀ [19] (if enantiomers are considered as equivalent, 862 IPR isomers if enantiomers are regarded as distinct). Interestingly, the total number of all C₁₀₀ isomeric cages (i.e., IPR and non-IPR cages) is 285,913 if enantiomers are considered equivalent [4]. The $D_5$(450)-C₁₀₀ fullerene exhibits $D_5$ topological symmetry which is however reduced owing to the Jahn-Teller distortion. At our computational level the topological symmetry is in geometry optimization lowered to the $C_2$ point group of symmetry (Figure 1, one can speak on a near $D_5$-symmetry). Interestingly enough, there are other C₁₀₀ IPR cages with a higher topological symmetry like $D_{5d}$ or T.

The La₂C₂@$D_5$(450)-C₁₀₀ geometrical structure was optimized with the Becke’s three parameter DFT functional with the non-local Lee-Yang-Parr correlation functional [20,21,22], the approach is conventionally labeled as B3LYP. The DFT calculations were at first performed in a combined atomic basis set: the standard 3-21G basis for C atoms while for La atoms the SDD basis set with the SDD effective core potential [23] (B3LYP/3-21G∼SDD). The geometry optimization started from the X-ray structure [14] and was further refined with the B3LYP/6-31G*∼SDD treatment using an extended 6-31G*∼SDD basis set, followed by the GF harmonic vibrational analysis. The vibrational analysis allows for a check of the nature of the stationary points found, i.e., if local energy minima were really found (and not activated complexes or even higher types of hypersurface stationary points). Moreover, the analysis can also be used for simulations of the IR and Raman vibrational spectra. For numerical integrations with the DFT functional the so called ultrafine grid was used. Moreover, the tight SCF (self consistent field) convergency criterion was applied. Finally, the wavefunction stability was tested throughout in order to eliminate unstable SCF solutions (without a real physical significance).

The so-called basis set superposition error (BSSE) was considered accordingly, namely by application of the approximative Boys-Bernardi counterpoise method (also known as CP2, or CPn, depending on the number of the reaction components in the treated association process) [24], since recently considered [25] also with metallofullerenes. The Boys-Bernardi approach to the BSSE correction term is an approximative treatment that ensures that every reaction component of a studied chemical process is formally described by the same number of the atomic basis set functions (so that the calculated energies are directly comparable). The required basis-set consistence is realized by rather formal, artificial atoms (thus also called ghost) possessing no electrons. The BSSE problem essentially originates in the fact that the considered basis sets are always finite. In other words, the BSSE correction would disappear in a situation of a basis set with infinite number of atomic basis functions (of course, such an infinite scheme cannot practically be handled). The BSSE correction term represents an important addition also to the energetics of any encapsulation process. Without the BSSE correction the energy gain created by encapsulation would be exaggerated (so that endohedrals would artificially be over-stabilized). Thus, the BSSE consideration is needed for reliable predictions of energetics, indeed.

All the reported calculations were performed with the program package Gaussian 09 [26]. The calculations were carried out with computers operating in parallel regime, mostly with 8–24 processors (with computational frequency up 3 GHz each and with the available operational memory up to 60 GB).

3. Results and Discussion

The B3LYP/6-31G*∼SDD optimized structure of La₂C₂@$D_5$(450)-C₁₀₀ shown in Figure 2 basically agrees with the X-ray results [14]. The calculated C-C bond length in the La₂C₂ encapsulate is 1.26 Å while the C-La bond 2.44 Å, the La-C shortest contact with the cage 2.64 Å. The B3LYP/3-21G SDD computed Wiberg bond index for the C-C bond is 2.58 (i.e., between double and triple bond), for C-La bond 0.69 (B3LYP/6-31G*∼SDD values C-C 2.59, C-La 0.63). It should be realized that the calculated values are obtained for the gas-phase free (isolated) endohedral. Moreover, the calculated terms correspond to the equilibrium geometries with motionless nuclei. The X-ray values are however obtained for the crystal-state endohedral.

The Mulliken atomic charges on atoms are (owing to their historical introduction) to be constructed at a simpler level, i.e., B3LYP/3-21G∼SDD. The Mulliken charges on atoms calculated with the 3-21G basis set are known [27] to produce for metallofullerenes a good agreement with the available observed charges [28]. Moreover, there are also general methodological reasons [29,30] why some larger basis sets should not be applied in the calculations of the Mulliken atomic charges on atoms—as they can sometimes produce rather unphysical values. The Mulliken charges on the La atoms are 2.41 and 2.47 (of the electron elementary charge) while on the carbons of the La₂C₂ encapsulate the charges are −0.32 and −0.34. Hence, the total charge on the encapsulate is 4.22. In overall, the charge transfer from the encapsulate to the cage amounts to 4.22 electrons. Thus, energetics of tetra-anions of empty cages could serve for stability estimations with encapsulations of La₂C₂ by other C₁₀₀ cages.

Although the ground state of La₂C₂@$D_5$(450)-C₁₀₀ is singlet electronic state, the first electronic triplet state is rather low lying. At the B3LYP/6-31G*∼SDD level the triplet state is located only about 1.88 kcal/mol above the ground-state singlet.

The energy change connected with the (formal thermodynamic) gas-phase encapsulation process:

$$2La\left(g\right)+2C\left(g\right)+D_5\left(450\right)-C_{100}\left(g\right)=L{a}_2{C}_2@D_5\left(450\right)-C_{100}\left(g\right)$$

(1)

is a useful characteristics known as encapsulation energy. Calculated reaction thermodynamic changes are mostly treated as the reaction potential-energy change $\Delta E_{pot,enc}$. However, a more adequate description refers to the enthalpy change at absolute zero temperature $\Delta H_{0,enc}^o$, i.e., the potential energy change corrected for the vibrational zero-point energies upon encapsulation $\Delta ZP{E}_{enc}$:

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

(2)

The encapsulation potential-energy change corrected $\Delta E_{pot,enc}$ is to be refined by inclusion of the basis set superposition error [24,26] (BSSE/CP5) and so-called steric correction [25] that reflects the cage distortion upon encapsulation [31].

Table 1. The B3LYP/6-31G*∼SDD computed ^a encapsulation energetics for La₂C₂@$D_5$(450)-C₁₀₀.

Energy Term	(kcal/mol/atom)
$\Delta E_{pot,enc}$ without BSSE	−148.7
$\Delta E_{pot,enc}$ with BSSE	−143.2
$\Delta E_{pot,enc}$ with BSSE & steric	−140.2
$\Delta H_{0,enc}^o$	−138.4

^a See process (1), the BSSE correction evaluated with the CP5 approach.

presents the energy terms for reaction (1) considered as reduced or relative values, i.e., per one atom of the La₂C₂ encapsulate (thus in the units of kcal/mol/atom). Such reduced values are convenient for comparisons with encapsulations of species with different numbers of atoms. For example, the reduced $\Delta H_{0,enc}^o$ term of −138.4 kcal/mol/atom from Table 1 represents the encapsulation-energy gain larger by 21.1 kcal/mol/atom than found [18] for Y₂C₂@$C_1$(1660)-C₁₀₈ at the same computational level. Similarly, the reduced $\Delta E_{pot,enc}$ term of −140.2 kcal/mol/atom in Table 1 gives by 3.7 kcal/mol/atom better energy gain than calculated [32] at the same level for Ti₃C₃@$I_h$-C₈₀. In overall, La₂C₂@$D_5$(450)-C₁₀₀ is a new interesting member of the family of metallofullerenes with relatively large energy gains upon encapsulation.

The encapsulation enthalpy $\Delta H_{0,enc}^o$ (enhanced by the related entropic contributions) allows for statistical thermodynamic evaluations of the equilibrium concentrations [33,34], in our case by means of the encapsulation equilibrium constant for process (1):

$$K_{p,enc}\left(1\right)={\displaystyle \frac{p_{L{a}_2{C}_2@D_5\left(450\right)-C_{100}}}{p_{La}^2{p}_C^2{p}_{D_5\left(450\right)-C_{100}}}}$$

(3)

defined by the partial pressures p of the individual reaction components in reaction (1). The encapsulation equilibrium constant from Equation (3) can indeed be expressed [34,35] by a compact formula using the molecular (or atomic) partition functions $q_i^o$ (the partition functions have to correspond to the selected standard state, e.g., ideal gas phase at 1 atm = 101,325 Pa pressure) and the standard encapsulation enthalpy change at the absolute zero temperature $\Delta H_{0,enc}^o$:

$$K_{p,enc}\left(1\right)={\displaystyle \frac{\frac{q_{L{a}_2{C}_2@D_5\left(450\right)-C_{100}}^o}{N_A}}{{\left[\frac{q_{La}^o}{N_A}\right]}^2{\left[\frac{q_C^o}{N_A}\right]}^2\frac{q_{D_5\left(450\right)-C_{100}}^o}{N_A}}}exp[-{\displaystyle \frac{\Delta H_{0,enc}^o}{RT}}],$$

(4)

where symbol $N_A$ stands for the Avogadro number. The molecular partition functions $q_i^o$ primarily consist of the rotational-vibrational partition functions based on the molecular parameters from the GF harmonic vibrational analysis.

In principle, the encapsulation energy can also be defined [34] by a more straightforward encapsulation stoichiometry, namely:

$${La}_2{C}_2\left(g\right)+D_5\left(450\right)-C_{100}\left(g\right)=L{a}_2{C}_2@D_5\left(450\right)-C_{100}\left(g\right)$$

(5)

when the basis set superposition error [24,26] is evaluated by the simpler BSSE/CP2 approach which at the B3LYP/6-31G*∼SDD level yields the reduced energy gain of −58.7 kcal/mol/atom (−62.5 kcal/mol/atom without the BSSE correction). Obviously, the energy gain must be now smaller than with the BSSE/CP5 approach as in the latter case the formation of the bonds of the encapsulate is included.

However, if one is interested in the thermodynamic-equilibrium populations of non-isomeric endohedrals [33], approach according to Equation (1) should be preferred. Incidentally, in addition to our starting system La₂C₂@$D_5$(450)-C₁₀₀ another pertinent species is known [10], viz. La₂@$D_5$(450)-C₁₀₀. The relative populations of the two non-isomeric endohedrals can be described using the equilibrium constant $K_{p,enc}\left(1\right)$ defined by Equation (3) and its equivalent $K_{p,enc}\left(6\right)$ for parent process in Equation (6):

$$2La\left(g\right)+D_5\left(450\right)-C_{100}\left(g\right)=L{a}_2@D_5\left(450\right)-C_{100}\left(g\right)$$

(6)

The population ratio of the two endohedrals is given by formula [33]:

$${\displaystyle \frac{p_{L{a}_2{C}_2@D_5\left(450\right)-C_{100}}}{p_{L{a}_2@D_5\left(450\right)-C_{100}}}}=p_C^2{\displaystyle \frac{K_{p,enc}\left(1\right)}{K_{p,enc}\left(6\right)}}.$$

(7)

Obviously, the relative equilibrium populations of such non-isomeric nanocarbons represent distinctly more complex problem compared to the situation of just isomeric species [34,35,36,37]. The equilibrium constant $K_{p,enc}\left(6\right)$ can again be evaluated by the partition functions and calculated molecular parameters as in Equation (4). The partial pressure of carbon $p_C$ can be approximated by the carbon saturated pressure [34]. However, the actual carbon pressure $p_C$ depends on particular experimental setup and thus varies from case to case. This aspect is one of the reasons why the population ratio of the two endohedrals can be different in different experiments, too. The interesting relative stability problem for non-isomeric nanocarbons should further be calculated in future.

The encapsulation of La₂C₂ into the $D_5$(450)-C₁₀₀ cage creates six new vibrational modes, actually with rather low frequencies—at the B3LYP/6-31G*∼SDD level between 24 and 73 cm⁻¹. The low frequencies basically belong to motions of the whole La₂C₂ encapsulate inside the cage, they could roughly be viewed as relatively free rotations. Such low frequency motions would particularly be important for the endohedral thermodynamics [35,36]. The new vibrational frequencies created by the encapsulation also contribute to the reaction change $\Delta ZP{E}_{enc}$ of the vibrational zero-point energy for process (1). On the other hand, the highest computed vibrational frequency has a value of 1926 cm⁻¹ and belongs to the bond stretching of the C-C bond in La₂C₂ (though the vibrational mode exhibits rather negligible IR intensity as well as Raman activity). Let us note that the gas-phase C₂ molecule has [38] in the singlet ground state observed vibrational frequency of 1855 cm⁻¹ (the C₂ lowest triplet state lies higher only by 2.05 kcal/mol). For another comparison, the highest vibrational frequency of the empty $D_5$(450)-C₁₀₀ cage at the B3LYP/6-31G* level has value of 1615 cm⁻¹. Figure 3 presents the B3LYP/6-31G*∼SDD calculated harmonic IR and Raman spectra of La₂C₂@$D_5$(450)-C₁₀₀. The spectra should be useful in identification of the species.

4. Conclusions

The structure and energetics of the metallofullerene (or clusterfullerene) La₂C₂@$D_5$(450)-C₁₀₀ are evaluated using the DFT approach at the B3LYP/6-31G*∼SDD level with inclusion of the correction for the basis set superposition error calculated by the counterpoise method. The calculated encapsulation potential-energy change $\Delta E_{pot,enc}$ amounts to a sizeable value of −140.2 kcal/mol/atom (i.e., per atom of the encapsulate), or at the $\Delta H_{0,enc}^o$ level to −138.4 kcal/mol/atom. The found energy gains connected with encapsulation are larger than previously reported for other clusterfullerene systems like Y₂C₂@$C_1$(1660)-C₁₀₈ or Ti₃C₃@$I_h$-C₈₀. Possible descriptions of encapsulation energetics are discussed and a thermodynamically consistent evaluation of the relative equilibrium populations of non-isomeric metallofullerenes is suggested with an aim to compare the La₂C₂@$D_5$(450)-C₁₀₀ equilibrium population with that of a related observed species La₂@$D_5$(450)-C₁₀₀. The approach can be of a use also for studies of other interesting complex nanocarbon systems [39,40,41,42,43,44,45].