Y@C82 metallofullerene: calculated isomeric composition

Abstract Relative populations of the four energy-lowest isolated-pentagon-rule (IPR) isomers of Y@C82 under the high-temperature synthetic conditions are computed using the Gibbs energy based on characteristics from the density functional theory calculations (B3LYP/6–31G SDD entropy, B3LYP/6–31G SDD and B2PLYPD/6–31 + G SDD energetics). Two major species are predicted, Y@ -C82 and Y@ -C82, their calculated equilibrium relative populations agreeing with observations.


Introduction
Various metallofullerenes have been studied as both basicresearch objects and promising agents for single-molecule electronics, as well as for their superconductive properties, medicinal applications or energy conversions. Such studies have also been dealing with yttrium metallofullerenes, in particular [1][2][3][4][5][6][7][8] with Y@C 82 . However, just two of its isomers have been considered, namely the endohedrals with the isolated-pentagon-rule (IPR) obeying carbon cages conventionally labeled as Y@C 2v ; 9-C 82 and Y@C s ; 6-C 82 (or Y@C s ðc; 6Þ-C 82 ). In fact, there are two labeling systems [9] for the C 82 IPR cages employed in the literature. One of them uses the symmetries of the cages (actually, the highest topologically possible symmetries): C 3v ðaÞ, C 3v ðbÞ, C 2v , C 2 ðaÞ, C 2 ðbÞ, C 2 ðcÞ, C s ðaÞ, C s ðbÞ and C s ðcÞ: The other system labels them with serial enumeration numbers, their assignment is: C 3v ðaÞ; 7, C 3v ðbÞ; 8, C 2v ; 9, C 2 ðaÞ; 3, C 2 ðbÞ; 1, C 2 ðcÞ; 5, C s ðaÞ; 2, C s ðbÞ; 4 and C s ðcÞ; 6: The C 82 cages are known to encapsulate also several other metals, frequently yielding at least two isomers. [9][10][11][12][13][14][15][16] In this study, the isomeric interplay is calculated for the set of four energy-lowest Y@C 82 IPR isomers in order to supply still missing information on their relative populations (as it is useful for a more comprehensive interpretation of experiments).
Although stabilities of nanocarbons are mostly studied with the potential energy only, there is a growing group of examples [17][18][19][20][21][22][23][24][25][26] that the entropic part of the Gibbs energy for metallofullerenes becomes important at high temperatures. In some systems, a structure that is not the lowest in potential energy can still be calculated as the most populated one at high synthetic temperatures. Moreover, other higherenergy structures can sometimes undergo relative stability interchanges with increasing temperature. Such relative-stability aspects cannot at all be obtained just from the potential energies. Thus, calculations are carried out in this article for the relative populations of four potential-energy lowest Y@C 82 IPR isomers at elevated temperatures, consistently using both enthalpy and entropy components of the Gibbs energy, in order to understand the isomeric interplay at synthetic conditions.

Calculations
The molecular-geometry optimization for the isomers started from the structures optimized in a combined basis set, 3-21 G basis [27] for C atoms and Stuttgart/Dresden (SDD) basis [28] with the SDD effective core potential on Y (coded here 3-21 G$SDD), using density functional theory (DFT) approach, namely Becke's three-parameter functional [29] with the non-local Lee-Yang-Parr correlation functional, [30] i.e., the unrestricted B3LYP/3-21G$SDD treatment. Moreover, the structures were further re-optimized using the standard 6-31G Ã basis set [31] for C atoms, i.e., the B3LYP/6-31G Ã $SDD level. The analytical energy gradient was used in the geometry optimizations. In the search for low-energy species, all nine [9] IPR C 82 cages were considered. The optimizations at the B3LYP/3-21G$SDD and B3LYP/6-31G Ã $SDD levels point out four species low in the potential energy (Table 1). All other isomers are at the B3LYP/6-31G Ã $SDD level 20 or more kcal/mol higher than the stabilomer Y@C 2v ; 9-C 82 which in this case (in combination with the entropy factors) renders their relative populations insignificant. Moreover, the inter-isomeric energetics was refined using the B2PLYPD ¼ FU/6-31 þ G Ã $SDD treatment with all electrons (in the optimized B3LYP/6-31G Ã $SDD geometries). The B2PLYPD method [32] is an advanced approach combining the DFT and MP2 approaches (moreover, with addition of a dispersion correction). In the optimized B3LYP/3-21G$SDD and B3LYP/6-31G Ã $SDD geometries, the harmonic vibrational analysis was carried out with the analytical force-constant matrix. The electronic excitation energies were evaluated by means of the time-dependent (TD) DFT response theory [33] at the B3LYP/6-31G Ã $SDD level. The computations are carried out with the Gaussian 09 program package. [34] Relative concentrations (mole fractions) x i of m isomers can be expressed [35,36] through their partition functions q i and the enthalpies at the absolute zero temperature or ground-state energies DH o 0, i (i.e., the relative potential energies corrected for the vibrational zero-point energies) by a compact formula: where R is the gas constant and T the absolute temperature. Equation (1) is an exact formula that can be directly derived [35] from the standard Gibbs energies of the isomers, supposing the conditions of the inter-isomeric thermodynamic equilibrium. Rotational-vibrational partition functions are evaluated [36] here using the conventional rigidrotor and harmonic-oscillator (RRHO) approximation. No frequency scaling is applied as it is not significant [37] for the x i values at high temperatures. Finally, the chirality contribution [38] was included accordingly (for an enantiomeric pair its partition function q i is doubled). Although the temperature region where fullerene or metallofullerene electric-arc synthesis takes place is not yet known, the recent observations [39] supply some arguments to expect it around 1500 K. Thus, the computed results discussed here are also focused on the temperature region. However, a modified [23,40] RRHO approach for description of the encapsulate motions is actually considered here, following findings [41] that the encapsulated atoms can exercise large amplitude motions, especially so at elevated temperatures (unless the motions are restricted by cage derivatizations [42] ). One can expect that if the encapsulate is relatively free then, at sufficiently high temperatures, its motions in different cages will produce about the same contribution to the partition functions. However, such uniform contributions would then cancel out in equation (1). This simplification is called [23,40] free, fluctuating or floating encapsulate model (FEM) and requires two steps. In addition to removal of the three lowest vibrational frequencies (belonging to the metal motions in the cage), the symmetries of the cages should be treated as the highest (topologically) possible, which reflects averaging effects of the large amplitude motions. For example, for the Y@C 82 IPR isomer based on the C 2v ; 9 cage (Table 1), the C 2v symmetry is employed within the FEM scheme though its static [43] symmetry (i.e., after the geometry optimization) is only C s (Figure 1). Generally, the FEM treatment gives a better agreement [23,40] with the available observed data compared to the conventional RRHO approach and it is therefore also preferred here. Table 1 presents the Y@C 82 relative isomeric energetics computed at the two selected levels (namely, the differences in the potential energy without the zero-point vibrational energies) for the four selected low-energy isomers out of nine [9,44] IPR satisfying C 82 topologies. The lowest-energy Y@C 82 isomer is the C 2v ; 9 species (Figure 1), being rather closely followed by the C 3v ; 8 and C s ; 6 species, and then C 2 ; 5: The B3LYP/6-31G Ã $SDD and B2PLYPD/6-31 þ G Ã Table 1. Y@C 82 relative potential energies DE pot, rel for the four energy-lowest isomers calculated in the B3LYP/6-31G Ã $ SDD optimized structures.

Results and discussion
DE pot, rel /kcal mol À1 Species B3LYP/6-31G Ã $SDD B2PLYPD/6-31 þ G Ã $SDD C 2 ; 5 14.3 13.9 C s ; 6 a 4.90 4.14 C 3v ; 8 4.57 3.21 C 2v ; 9 a 0.0 0.0 a See Figure 1. $SDD isomeric separation energies are quite similar. The four isomers from Table 1 are considered in the following thermodynamic treatment. Table 2 presents selected calculated characteristics for the four potential-energy-lowest Y@C 82 isomers. The B3LYP/6-31G Ã $SDD calculated closest contacts r YÀC between Y and the cages are similar to those found previously with other C 82 -based metallofullerenes. [9,15,44] Let us mention that the calculated position of the yttrium atom in Y@C 2v ; 9-C 82 (i.e., under a hexagonal ring along the symmetry axis) agrees with the previous experimental findings. [5] The lowest vibrational frequencies x low in Table 2 are in agreement with the generally known relatively-free motions of encapsulates in metallofullerenes. On the other hand, the lowest electronic excited states exhibit relatively high excitation energies X low : The B3LYP/3-21G$SDD computed Mulliken atomic charges q Y on Y are close to 3 while the Mulliken spin densities r Y on Y are about 0.02. Let us stress that the Mulliken charges from the 3-21G$SDD basis, in contrast to, e.g., the 6-31G Ã $SDD level, give [25] for metallofullerenes a good agreement with the observed charge densities. [45] Moreover, there are more general arguments [46][47][48][49] why larger basis sets should be avoided for the Mulliken chargesas sometimes they can, in particular, produce truly unphysical values. [47] Figure 2 presents the main output of this study -temperature development of the relative equilibrium populations for the four potential-energy-lowest Y@C 82 isomers in a wide temperature region. The relative populations (in other words, thermodynamic stabilities) are evaluated in the FEM treatment with the B3LYP/6-31G Ã $SDD entropy, and with the B3LYP/6-31G Ã $SDD or B2PLYPD/6-31 þ G Ã $SDD energetics. At very low temperatures the structure lowest in the DH o 0, i scale must be prevailing, i.e., the Y@C 2v ; 9-C 82 isomer. However, the second (Y@C 3v ; 8-C 82 ) and the third (Y@C s ; 6-C 82 ) lowest species in the potential energy exhibit rather fast increase of their relative population. Interestingly, at a temperature of 265 K (not important from a practical point of view) Y@C s ; 6-C 82 becomes the second most populated isomer (for the B3LYP/6-31G Ã $SDD energetics). Moreover, at a very high temperature of 3303 K the equimolarity of the two most populated isomers is reached for the energetics. At the suggested [39] synthetic temperature of 1500 K, the equilibrium populations are 61.3%, 25.1%, 11.8%, and 1.8 % for the C 2v ; 9, C s ; 6, C 3v ; 8, and C 2 ; 5 isomers, respectively, with the B3LYP/6-31G Ã $SDD energetics (while 53.6%, 28.3%, 16.3% and 1.8 % with the B2PLYPD/6-31 þ G Ã $SDD energetics). Hence, both sets of inter-isomeric separation energies agree with the observations [1][2][3][4][5][6][7][8] of two isomers, major Y@C 2v ; 9-C 82 and minor Y@C s ; 6-C 82 . Their population ratio deduced from the observed HPLC peaks ½6 is about 6.2 which would suggest with the B3LYP/6-31G Ã $SDD energetics a quite reasonable Table 2. The selected characteristics of the four energy-lowest Y@C 82 isomers the closest Y-C contact a r YÀC , the Mulliken charge b on Y q Y , the Mulliken spin density b on Y r Y , the lowest vibrational frequency a x low and the lowest electronic excited state a X low .
Species  Figure 1. synthetic temperature of 955 K (or with the B2PLYPD/6-31 þ G Ã $SDD energetics a temperature of 805 K). However, it is not clear how close is the observation to the inter-isomeric equilibrium supposed in the calculations. From the experimental point of view, it is interesting that the calculations predict one or two still less populated minor species, too.
The presented results are somewhat similar to the previous calculations [12,23,[50][51][52][53] for the C 82 -based metallofullerenes, for example [13,23] La@C 82 or Sm@C 82 isomers. [9] While in Y@C 82 the charge transfer to the cage amounts to about 3 electrons, it is about two electrons [9] in Sm@C 82 . In both isomeric systems the C 2v ; 9-C 82 endohedral prevails at lower temperatures. However, at higher temperatures the C s ; 6-C 82 endohedral approaches the C 2v ; 9-C 82 population rather fast. Their equimolarity is reached at a temperature of 2450 and 3303 K in the Sm@C 82 and Y@C 82 system, respectively. While in the Y@C 82 set the Y@C 3v ; 8-C 82 species is the third most abundant, with Sm@C 82 the third most stable isomer is Sm@C 2 ;5-C 82 . Metallofullerenes are not formed via some new covalent bond but rather stabilized by just an ionic bond [54][55][56][57][58] as documented by the topological AIM concept. [54] An important factor is a low metal ionization potential as demonstrated on homologous series Z@C 82 (Z ¼ Al, Sc, Y, La) where the computed relative potential-energy changes upon encapsulation and the relative observed ionization potentials of the free atoms are well correlated. [56] The La@C 82 formation is connected with the highest energy gain [56] and the La ionization potential is the lowest in the series.
Let us add that observed populations can depend on the applied metal sources [59] that could be related to kinetic and catalytic aspects, [60][61][62] i.e., to different degrees to which the expected inter-isomeric thermodynamic equilibrium could be achieved in the synthesis. Another issue is solubility [63][64][65] of different isomers in the applied extraction solventsthey are likely not exactly the same (as inherently supposed in the theory-experiment comparisons). In overall, the Y@C 82 results represent another example of reliability of the Gibbsenergy treatment of the isomeric populations, encouraging further applications to still more complex nanocarbon systems. [66][67][68][69][70] It is thus possible to apply the treatment to further systems to get still more data. Later on, such larger series of results could be analyzed in order to get some more general rules similarly to already established relationships [71][72][73] or even to more formal information-entropy approaches. [74,75] Before reaching the stage, however, detailed molecular description of further metallofullerene systems is to be step by step accumulated.

Disclosure statement
No potential conflict of interest was reported by the authors.