Pr@C₈₂ Metallofullerene: Calculated Isomeric Populations

Abstract

Relative equilibrium populations of the five lowest-energy isolated-pentagon-rule (IPR) isomeric structures of Pr@C${}_{82}$ under high-temperature fullerene synthesis conditions were calculated with the Gibbs energy terms based on molecular characteristics derived using density functional theory (DFT) treatments (B3LYP/6-31+G${}^{*}$∼SDD energetics and B3LYP/6-31G${}^{*}$∼SDD entropy). Two leading isomers were identified, major Pr@$C_{2v};9$-C${}_{82}$ and minor Pr@$C_s;6$-C${}_{82}$. The calculated isomeric relative equilibrium populations agreed with observations.

1. Introduction

Endohedral metallofullerenes have been pursued as not only targets of fundamental research but also promising species for nanoelectronics, as well as for energy conversions, superconductive properties, and applications in medicine. Such studies have also dealt with praseodymium-encapsulating metallofullerenes, in particular with Pr@C${}_{82}$ [1,2,3,4,5,6,7,8] (a field-effect transistor was also fabricated [7] with one of its isomers). Actually, just two Pr@C${}_{82}$ isomers (Figure 1) have been observed [3,4,5,6], namely the major species with the carbon cage obeying the isolated pentagon rule (IPR), conventionally labeled Pr@$C_{2v};9$-C${}_{82}$, and the minor isomer Pr@$C_s;6$-C${}_{82}$ (or Pr@$C_s(c;6)$-C${}_{82}$). Presently, there are two labeling or coding systems [9] for the IPR C${}_{82}$ cages considered in this research. The older system works only with the symmetries of the cages (namely, the highest topological symmetries, regardless of possible symmetry reductions): $C_{3v}\left(a\right)$, $C_{3v}\left(b\right)$, $C_{2v}$, $C_2\left(a\right)$, $C_2\left(b\right)$, $C_2\left(c\right)$, $C_s\left(a\right)$, $C_s\left(b\right)$, and $C_s\left(c\right)$. The second labeling system employs serial numbers from the enumeration spiral algorithm [10]; these symbols are: $C_{3v}\left(a\right);7$, $C_{3v}\left(b\right);8$, $C_{2v};9$, $C_2\left(a\right);3$, $C_2\left(b\right);1$, $C_2\left(c\right);5$, $C_s\left(a\right);2$, $C_s\left(b\right);4$, and $C_s\left(c\right);6$. Numerous metals can be encapsulated in C${}_{82}$ cages, frequently producing at least two isomers [4,5,6,9,11,12,13,14,15,16]. In the present report, the isomeric relative equilibrium populations (or the relative concentrations) were evaluated for the isomeric system of the five lowest-potential-energy Pr@C${}_{82}$ IPR species, as such stability information is needed for the interpretation of some observations.

The relative stabilities of isomeric nanocarbons are mostly evaluated by simply using potential-energy terms. Nevertheless, several examples have been reported [17,18,19,20,21,22,23,24,25,26] showing that the Gibbs energy should be applied instead, as its entropy component becomes increasingly important at elevated temperatures. Consequently, an isomer that is not the lowest on the potential-energy scale can still exhibit the highest populations at the high temperatures needed for nanocarbon syntheses. Moreover, other higher-energy structures can in some cases exhibit relative stability interchanges with increasing temperature. Clearly enough, it is not possible to predict such relative stability interchanges simply from the potential-energy terms alone. Hence, computations were performed in this report for the high-temperature relative equilibrium populations of the five Pr@C${}_{82}$ IPR isomers with the lowest potential energy, employing both the enthalpy and entropy parts of the Gibbs energy in simulations of the isomeric equilibrium populations under supposed fullerene synthesis conditions.

2. Calculations

The calculations started with the molecular geometry optimization of the considered isomers, which was performed in a combined atomic basis set—the standard 3-21G basis [27] for carbon atoms and the SDD basis [28] with the SDD effective core potential for praseodymium (labeled 3-21G∼SDD). We applied density functional theory (DFT) treatments employing Becke’s three-parameter functional [29] combined with the Lee–Yang–Parr non-local correlation functional [30]—the unrestricted B3LYP/3-21G∼SDD treatment. Moreover, the B3LYP/3-21G∼SDD pre-optimized structures were improved further with the standard 6-31G${}^{*}$ basis set [31] for carbon atoms—the B3LYP/6-31G${}^{*}$∼SDD approach. The calculations were carried out for the quartet electronic state, as the multiplicity produced significantly lower energy at the considered computational levels than the doublet and sextet electronic states. The geometry optimizations were carried out with the analytical energy gradient. All nine [9] IPR C${}_{82}$ cages were subjected to geometry optimizations. The geometry optimizations with both the B3LYP/3-21G∼SDD and B3LYP/6-31G${}^{*}$∼SDD treatments identified five isomers (

Table 1. Pr@C${}_{82}$ relative potential energies $\Delta E_{pot,rel}$ for the five lowest-energy isomers calculated in the B3LYP/6-31G${}^{*}$∼SDD-optimized structures.

Species	$\mathbf{\Delta }{\mathit{E}}_{\mathbf{pot},\mathbf{rel}}$/kcal mol${}^{-1}$ B3LYP/6-31G${}^{*}$∼SDD	B3LYP/6-31+G${}^{*}$∼SDD
$C_2;5$	17.9	17.3
$C_{3v};7$	13.0	14.2
$C_{3v};8$	7.54	7.37
$C_s;6$${}^a$	3.46	3.47
$C_{2v};9$${}^a$	0.0	0.0

${}^a$ See Figure 1.

) with sufficiently low potential energy. All other Pr@C${}_{82}$ IPR isomers were, according to the B3LYP/6-31G${}^{*}$∼SDD treatment, located 19 kcal/mol or more higher than the lowest-energy species Pr@$C_{2v};9$-C${}_{82}$. Their energy separation (in combination with the entropy terms) produced already insignificant relative equilibrium concentrations. Further on, the inter-isomeric energetics was still improved by applying a yet more advanced B3LYP/6-31+G${}^{*}$∼SDD approach (still using the B3LYP/6-31G${}^{*}$∼SDD geometries). For the optimized B3LYP/6-31G${}^{*}$∼SDD geometries, the vibrational analysis was performed using the harmonic GF scheme with the analytical force–constant matrix F in order to simulate the IR vibrational spectra (Figure 2) and, in particular, to construct the partition functions of the harmonic vibrations for the stability evaluations. The SCF wavefunction stability [32,33] was systematically tested throughout in order to prevent misleading unstable SCF solutions that could produce a physically unsound picture. The calculations were performed with the Gaussian 09 suite of programs [34].

The relative equilibrium concentrations of m isomers are described as mole fractions $x_i$ that can be expressed [35,36] through their partition functions $q_i$ and the enthalpies at a temperature of absolute zero or ground-state energies $\Delta H_{0,i}^o$ (i.e., the relative potential energies corrected for the vibrational zero-point energies) by a master formula:

$$x_i=\frac{q_{i}exp[-\Delta H_{0,i}^o/\left(RT\right)]}{{\sum }_{j=1}^m{q}_{j}exp[-\Delta H_{0,j}^o/\left(RT\right)]},$$

(1)

where R is the gas constant and T is the absolute temperature. Equation (1) is a rigorous formula that can be obtained [35] using statistical thermodynamics with the standard isomeric Gibbs energies under the conditions of the inter-isomeric thermodynamic equilibrium. The rotational-vibrational components of the partition functions were approximated [36] by the conventional rigid-rotor and harmonic-oscillator (RRHO) approach. Frequency scaling was not applied as it was not significant for the $x_i$ values at high temperatures [37]. Moreover, the chirality contribution [38] was considered accordingly (the partition function $q_i$ for a pair of enantiomers was doubled). The temperatures at which fullerene or metallofullerene electric-arc synthesis takes place are not precisely known; however, recent observations [39] suggest that it takes place somewhere around a temperature of 1500 K. Hence, the values discussed in this report were calculated for this particular temperature region.

Moreover, an adjusted [23,40] RRHO treatment for the description of the metal motions was applied, considering findings [41] suggesting that the encapsulated metal atoms can exercise large-amplitude vibrational motions, in particular at higher temperatures (if the motions are not restricted by derivatizations of cages [42]). It could be expected that if the metal was relatively free, then, at sufficiently elevated temperatures, its motions in different endohedral cages would provide a similar contribution to the overall partition functions. Such similar contributions would consequently be mutually canceled out in Equation (1). This approximation has been called [23,40] the floating, fluctuating, or free encapsulation model (FEM). The FEM treatment consists of two steps. The first step is the suppression of the three lowest vibrational frequencies (which belong to the metal vibrations inside the cage). The second step deals with the symmetries of the cages—they should be taken as the highest topologically possible terms, reflecting the averaging influence of the large-amplitude motions upon the symmetries. For example, with the Pr@C${}_{82}$ IPR isomer related to the $C_{2v};9$ cage (Table 1), the $C_{2v}$ symmetry is indeed applied in the FEM treatment, in spite of the fact that its static [43] symmetry (i.e., derived from the geometry optimization) is only $C_s$ or even $C_1$ (Figure 1). The FEM approach is generally known to produce better agreement [23,40] with observed data than the simple RRHO treatment. The FEM approach was thus also applied in this report.

3. Results and Discussion

The Pr@C${}_{82}$ relative isomeric energetics calculated in the two selected approaches (namely, the relative potential energy without the inclusion of the zero-point vibrational energies) for the five lowest-energy isomers selected from the nine [9,44] IPR-satisfying C${}_{82}$ cages are presented in Table 1. The lowest-energy Pr@C${}_{82}$ isomer was the $C_{2v};9$ species, followed at 3 kcal/mol higher by the $C_s;6$ endohedral species (Figure 1). Then, $C_{3v};8$ and $C_{3v};7$ were located about 7 and 14 kcal/mol above the $C_{2v};9$ stabilomer, respectively. It should be noted that the B3LYP/6-31G${}^{*}$∼SDD and B3LYP/6-31+G${}^{*}$∼SDD relative energies were similar. The five species from Table 1 were subjected to equilibrium thermodynamic analysis.

Table 2. The selected characteristics of the five lowest-energy Pr@C${}_{82}$ isomers—the closest Pr-C contact ${}^a$$r_{Pr-C}$, the Mulliken charge ${}^b$ on Pr $q_{Pr}$, and the lowest vibrational frequency ${}^a$${\omega }_{low}$.

Species	${\mathit{r}}_{\mathbf{Pr}-\mathit{C}}$/Å	${\mathit{q}}_{\mathbf{Pr}}$	${\mathit{\omega }}_{\mathbf{low}}$ / cm${}^{-1}$
$C_2;5$	2.512	2.449	39.3
$C_{3v};7$	2.516	2.293	17.6
$C_{3v};8$	2.476	2.462	22.3
$C_s;6$${}^c$	2.500	2.337	20.5
$C_{2v};9$${}^c$	2.549	2.364	15.8

${}^a$ B3LYP/6-31G${}^{*}$∼SDD terms. ${}^b$ B3LYP/3-21G∼SDD terms. ${}^c$ See Figure 1.

presents selected calculated molecular parameters for the five lowest-potential-energy Pr@C${}_{82}$ species. The B3LYP/6-31G${}^{*}$∼SDD calculated shortest distances $r_{Pr-C}$ between Pr and the carbon cages at around 2.5 Å, which were thus similar to those previously reported for other C${}_{82}$-based endohedrals [9,15,44]. Interestingly, the calculated position of the Pr atom in Pr@$C_{2v};9$-C${}_{82}$ (i.e., under a hexagonal ring) is also known for other systems [9]. The lowest harmonic vibrational frequencies ${\omega }_{low}$ in Table 1 reflect the relatively free motions of the encapsulated metals in metallofullerenes. B3LYP/3–21G∼SDD produced Mulliken atomic charges $q_{Pr}$ on Pr are close to 2.4 (electron elementary charge). It should be stressed that the Mulliken atomic charges from the 3–21G∼SDD basis are known to produce [25] for metallofullerenes good agreement with experimental charges [45] (which is not the case for, e.g., the 6-31G${}^{*}$∼SDD level). In addition, there are deeper physical reasons [46,47,48,49] why extended bases should not be used for Mulliken charges (as the latter can sometimes produce rather unphysical charges [47]). As for yet another basis-set effect, let us also mention that, as we were only dealing with isomers, there was no need to consider the basis set superposition error (BSSE) [49]. The BSSE correction is indeed important for association processes in order to assure that all the components are treated in the same basis set. In a set of isomers, however, the basis set used is the same for all the species, and therefore the total energies do not require a correction of the BSSE type. If such a treatment was still performed, there would be a relatively constant additive term that would then cancel out in Equation (1).

Figure 3 gives the key results of this report—the temperature dependencies of the relative equilibrium concentrations for the five lowest-potential-energy Pr@C${}_{82}$ isomers at moderate and high temperatures. The presented relative isomeric populations were obtained with the FEM approach. In any case, at very low temperatures, the species with the lowest $\Delta H_{0,i}^o$ had to be the most populated, i.e., the Pr@$C_{2v};9$-C${}_{82}$ isomer. However, the second (Pr@$C_s;6$-C${}_{82}$) and, somewhat, the third (Pr@$C_{3v};8$-C${}_{82}$) lowest-potential-energy isomers showed a rapid increase in their relative populations, while the remaining isomers were negligible. At the suggested [39] representative temperature of 1500 K for fullerene synthesis, the relative equilibrium populations were 59.3, 34.9, 4.8, 0.3, and 0.8% for the $C_{2v};9$, $C_s;6$, $C_{3v};8$, $C_{3v};7$, and $C_2;5$ isomers, respectively. Interestingly, in a stability evaluation without the entropic part (i.e., when just the simple Boltzmann factors [35,36] were considered), the relative isomeric populations were rather different: 71.0, 22.2, 6.0, 0.6, and 0.2% for the $C_{2v};9$, $C_s;6$, $C_{3v};8$, $C_{3v};7$, and $C_2;5$ species, respectively.

Overall, the calculations agreed with the observations [3,4,5,6] for two isomers, major Pr@$C_{2v};9$-C${}_{82}$ and minor Pr@$C_s;6$-C${}_{82}$. Their observed [6] population ratio has been reported as Pr@$C_{2v};9$-C${}_{82}$: Pr@$C_s;6$-C${}_{82}$ = 2.5. In our calculations, the population ratio was achieved at a quite reasonable synthetic temperature of 1130 K. Still, it is not known how close the experiments were to the inter-isomeric equilibrium predicted by the thermodynamic treatment. For the interpretation of the observations, it is also of interest that the calculations predicted just two significant isomers, while the populations of the other species were rather negligible. The calculated IR spectra (Figure 2) should also be of use for the identification of the two isomers.

The reported results presented similarities with the previous calculations [12,23,50,51,52,53,54] for other C${}_{82}$-based metallofullerenes with a similar metal-to-cage charge transfer—in particular [12,23], La@C${}_{82}$. The similarities originated from the fact that metallofullerenes are not stabilized by some new covalent bond but instead formed by an ionic bond [55,56,57,58,59]. It should be mentioned that experimental populations can depend on the metal sources used [60]. This aspect can be related to catalytic and kinetic factors [61,62,63] and indicates the variable levels to which the expected inter-isomeric thermodynamic equilibrium could actually be obtained in the synthesis. Yet another factor is the solubility [64,65,66] of individual isomers in the selected extraction solvents. In conclusion, the Pr@C${}_{82}$ results provide more evidence of the good applicability of the Gibbs energy approach to isomeric populations, thus supporting further studies of this type into yet more complex nanocarbon isomeric sets [67,68,69,70,71].