Equilibrium isomeric mixtures: potential energy hypersurfaces as the origin of the overall thermodynamics and kinetics

Sc3N@C80: Computations on the Two-Isomer Equilibrium at High Temperatures

Reported herein are computations on the relative concentrations of the two experimentally known isomers of Sc3N@C80 , that is, those produced by encapsulation of Sc3N in two particular C80 cages that obey the isolated-pentagon rule, namely, with I(h) and D(5h) symmetries. The calculations are based on density functional methods and have been carried out using the Gibbs energy over a broad temperature interval. It has been computed that, if a relatively free motion of the encapsulate inside the cages is allowed, the observed populations of 10 and 17 % for the D(5h) Sc3N@C80 species are reached at temperatures of 2100 and 2450 K, respectively. The inclusion of the entropy term is essential as, if it is neglected, the D(5h) Sc3N@C80 population at a temperature of 2100 K would be a mere 1 %, owing to the relatively large interisomeric separation potential energy of 19 kcal mol(-1).

On the Structure and Relative Stability of C50 Fullerenes

The complete set of 271 classical fullerene isomers of C50 has been studied by full geometry optimizations at the SAM1, PM3, AM1, and MNDO quantum-chemical levels, and their lower energy structures have also been partially computed at the ab initio levels of theory. A D(5h) species, with the least number of pentagon adjacency, is predicted by all semiempirical methods and the HF/4-31G calculations as the lowest energy structure, but the B3LYP/6-31G* geometry optimizations favor a D3 structure (with the largest HOMO-LUMO gap and the second least number of adjacent pentagons) energetically lower (-2 kcal/mol) than the D(5h) isomer. To clarify the relative stabilities at elevated temperatures, the entropy contributions are taken into account on the basis of the Gibbs energy at the HF/4-31G level for the first time. The computed relative-stability interchanges show that the D3 isomer behaves more thermodynamically stable than the D(5h) species within a wide temperature interval related to fullerene formation. According to a newly reported experimental observation, the structural/energetic properties and relative stabilities of both critical isomers (D(5h) and D3) are analyzed along with the experimentally identified decachlorofullerene C50Cl10 of D(5h) symmetry. Some features of higher symmetry C50 nanotube-type isomers are also discussed.

Ca@C82 isomers: Computed temperature dependency of relative concentrations

Relative concentrations of nine isomers of Ca at C82 derived from the C82 isolated-pentagon-rule satisfying cages are computed in a wide temperature interval. The computations are based on the Gibbs energy constructed from partition functions supplied with molecular parameters from density functional theory calculations. Five structures show significant populations at higher temperatures: C2v > Cs > C2 > C3v > Cs. The computed relative stabilities agree well with available observations.

Computations of model narrow nanotubes closed by fragments of smaller fullerenes and quasi-fullerenes

New type of carbon nanotubes-narrow nanotubes-has recently been observed with diameters of 4-5 A. It has been postulated that the narrow nanotubes are closed by fullerene fragments of C(20) and C(36). This paper presents computational results on related model nanotubes with stoichiometries such as C(80), C(84), C(96), C(108), or C(120). The computations were carried out at the PM3, AM1, SAM1, HF/3-21G, HF/4-31G, and B3LYP/6-31G(*) levels. Two C(36) fullerenes were considered, D(6h) and D(2d). At the PM3 level and with the C(84) nanotube stoichiometry, the D(2d) cage closure gave a lower energy (by 185 kcal/mol and a diameter of 5.42 A). There is another possible candidate, a C(32) cage with D(4d) symmetry (two four-membered rings). At the PM3 level and with the C(96) nanotube stoichiometry, the D(4d) closure (with a diameter of 5.43 A) had energy lower by 210 kcal/mol than that of the D(6h) nanotube closure. On the other hand, four-membered rings should not play a significant role for narrow nanotubes with a diameter of 4 A, where the dodecahedron-related closure should be exclusive. Still narrower nanotubes are briefly discussed.

Computing the relative gas-phase populations of C60 and C70: beyond the traditional delta H(fo),298 scale

Computations and experiments have shown that the relative heat of formation (i.e., the heat of formation per carbon atom) of C70 is lower than of C60. Moreover, various computations suggest that this is actually a general trend among fullerene cages. The relationship is particularly important for gas-phase fullerenes. Experiments have shown that C60 is typically more populated than C70 when produced in high-temperature gas-phase synthesis. It is not immediately obvious how to reconcile those two terms, or whether the relative heats of formation and the relative populations are in conflict or in agreement. This article deals with this problem, treating it as a general task of relative stabilities of gas-phase clusters of different dimensions (i.e., nonisomeric clusters) under different types of thermodynamic equilibria. The results are then applied to C60 and C70 and point out that the conventional standard pressure of 1 atm is considerably different from actual fullerene-synthesis conditions. Apparently, we should expect considerably lower cluster pressures in carbon-arc synthesis. At 1 atm, C70 is more populated than C60, but at the conditions of a saturated carbon vapor the stability order is reversed in favor of C60 so that an agreement with experiment is obtained already within the thermodynamic treatment. The pressure effects are modeled using the MNDO, AM1, PM3, and SAM1 quantum-chemical semi-empirical methods as well as the available experimental data. The computations consistently show that, if the pressure effects are considered, C60 becomes more populated than C70. Relationships of the thermodynamic treatment to more sophisticated but impractical kinetic analysis are also discussed.

Mg@C72 MNDO/d evaluation of the isomeric composition

Temperature development of the relative stabilities of isomers of Mg@C72 (which has not yet been isolated) is computed using the recently introduced MNDO/d method. Four isomers originally considered for the Ca@C72 case are treated: one isolated-pentagon-rule (IPR) structure, two structures with a pair of adjacent pentagons, and one cage with a heptagon. The IPR structure comes as the lowest in MNDO/d potential energy, being rather closely followed by the two structures with a pentagon-pentagon pair. On the other hand, the structure with a heptagon is located too high in potential energy to be of any experimental significance. The entropy contributions are evaluated by the MNDO/d-based partition functions so that the relative concentrations can be treated accordingly. The computations suggest that if Mg@C72 is isolated, it should be a mixture of either two or three isomers. The prediction depends on temperature prehistory. If preparation takes place at temperatures of approximately 1000 K, two isomers should be produced. If temperatures are increased to approximately 2000 K, there will already be three isomers with significant relative concentrations. The study supplies a further interesting example of the profound role of enthalpy-entropy interplay in stabilities of isomeric fullerenic structures.