Xiang Zhao

Enthalpy–entropy interplay for C36 cages: B3LYP/6-31G* calculations

The recently isolated first lower fullerene C36 is computed primarily at the B3LYP/6-31G* level with some information derived from the semiempirical SAM1 method. Twelve low-energy cages are considered both with fullerene (pentagons and hexagons) and “quasifullerene” (also squares and heptagons) structural pattern. The B3LYP/6-31G* evaluations of the equilibrium isomeric composition at high temperatures agree well with the recently reported solid-state NMR spectra. The fullerene cage with a topological D6h symmetry, that dominates at high temperatures, undergoes a small Jahn–Teller distortion. Some features of C36-related nanotubes are also discussed.

C36 fullerenes and quasi-fullerenes: computational search through 598 cages

C36 is computed at the SAM1 level and partially also at the HF/4-31G and B3LYP/6-31G∗ levels. Altogether 598 cages are generated by a topological Stone–Wales treatment. Three cages contribute by more than 10% to the high-temperature equilibrium isomeric mixture – two conventional fullerenes with D2d and C2v symmetries and a Csquasi-fullerene containing one four-membered ring.

Relative Stabilities of C92 IPR Fullerenes

Abstract The complete set of 86 isolated-pentagon-rule (IPR) isomers of C92 has been described by the SAM1 quantum-chemical method, and their energetics checked by density functional theory at the B3LYP/6-31G* level. Although the lowest-energy cage is not identical in both approaches, it still exhibits D2 symmetry in both cases. As energetics themselves cannot produce reliable relative stabilities at high temperatures, entropy terms are also computed and the relative-stability problem is treated entirely in terms of the Gibbs function. The lowest-energy structure is not the most populated isomer at higher temperatures – it is replaced by a D3 structure. Further stability interchanges are possible at very high temperatures, when C3 and C1 structures are also important. There is a partial agreement of the computations with available observed data.

Computational modeling of the elemental catalysis in the Stone–Wales fullerene rearrangements

Abstract Catalytic effects on the kinetics of the Stone–Wales fullerene transformation are studied computationally. The catalytic agents are represented by free elements, neutral or charged. The computations are performed at semiempirical (PM3) and DFT (B3LYP/6-31G*//PM3) levels on a model bowl-shaped fragment C 34 H 12 . The semiempirical and DFT activation energies agree reasonably well. In all computed cases, the activation barrier is lowered compared with that of the uncatalyzed reaction. The kinetic barriers for the catalyzed rearrangements increase in the following order: N, H, O, P, S, B, Cl, C, F, Li, Se, Fe, Hg, Zn, Si, Sn, Ge, Mg, and Al. Nitrogen atoms are pointed out as especially potent catalytic agents. At the PM3 computational level, the isomerization kinetic barrier is reduced to 193, 110, and 342 kJ mol −1 for the N + , N, and N − species, respectively. If the activation barriers are re-computed at the B3LYP/6-31G*//PM3 level, they are changed to 76, 105, and 323 kJ mol −1 for the N + , N, and N − species, respectively. As small amounts of nitrogen (as well as other elements) are available in virtually any kind of fullerene synthesis, the study offers a computational support for kinetic feasibility of the Stone–Wales fullerene transformation.

C90 temperature effects on relative stabilities of the IPR isomers

Abstract The complete set of 46 isolated-pentagon-rule (IPR) isomers of C 90 is treated by the SAMI quantum-chemical method (Semi-Ab-Initio Model 1), and their energetics is also checked by ab initio SCF computations (HF/3-21G, HF/4-31G) and the AM1 and PM3 semiempirical methods. All the methods point out a C 2 species as the system state (with an exception at the HF/3-21G level). However, the energetics itself is not able to produce a good agreement with recent observations (one C 2 v , three C 2 , and one C 1 , separeted into three HPLC fractions). The symmetries of the five SAM1 lowest-energy structures are: C 2 , C 2 v , C s , D 5 h , C 1 . In order to respect temperature effects on relative stabilities, entropy contributions are also computed and significant changes are found. The symmetries of the five structures most populated in a high-temperature region are according to the SAM1 computations: two times C 2 , C s , C 2 v , C 1 . One of the recently reported HPLC fractions shows 70 lines in the 13 C NMR spectrum, assigned to a C 2 species (45 lines) and C 2 v species (25 lines, 5 weaker). There is however an alternative interpretation of the spectrum: a C s (46 lines, 2 weaker) and a C 2 v species (24 lines, 3 weaker). The last two structures are indeed present in the SAM1 high-temperature stability set so that agreement between observations and calculations can be achieved.

Computing the relative gas-phase populations of C60 and C70: Beyond the traditional ΔHf,298o scale

Abstract Computations and experiments have shown that the relative heat of formation (i.e., the heat of formation per carbon atom) of C 70 is lower than of C 60 . 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 C 60 is typically more populated than C 70 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 C 60 and C 70 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, C 70 is more populated than C 60 , but at the conditions of a saturated carbon vapor the stability order is reversed in favor of C 60 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, C 60 becomes more populated than C 70 . 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.

Theoretical investigations on relative stabilities of fullerene C94

The complete set of 134 isolated-pentagon-rule isomers of C94 has been first investigated by various quantum-chemical approaches. Twenty low-energy cage structures are computed by the ab initio Hartree–Fock self-consistent field method and hybrid density functional theory treatment. All the methods point out a C2 species as the system ground state. The computed temperature-dependent relative concentrations under the interisomeric thermodynamic equilibrium evaluated by a combined quantum-chemical and statistical-mechanical treatment, predict two distinct structures (Cs,C2) as the most thermodynamically populated isomers in the system. This finding agrees quite well with the recently reported experimental observations.

C98 IPR isomers: Gibbs-energy based relative stabilities

Abstract The complete set of 259 isolated-pentagon-rule (IPR) isomers of C 98 has been treated by full geometry optimizations with the SAM1, PM3, AM1, and MNDO quantum-chemical semi-empirical methods. All the applied methods point out a C 2 species (FM code, 248: C 2 ) as the lowest-energy structure in the IPR set. In order to predict the relative stabilities at elevated temperatures, entropy contributions are also computed and thus, evaluations in terms of the Gibbs energy are possible for this system for the first time. Interesting stability interchanges in the isomeric set are found so that not only the ground-state structure 248: C 2 but also other structures (109: C 1 , 108: C s , 113: C 2 ) are significantly populated at high temperatures. The results are placed into a wider context of higher fullerenes and their computations.

C32: Computations of Low-Energy Cages with Four-Membered Rings

Abstract C32 cages built from four-, five-, six-, and seven-membered rings are computed. The computations are primarily performed with semiempirical quantum-chemical methods (AM1, PM3, SAM1), and altogether 199 cages are optimized. The energetics is further checked through ab initio HF SCF computations with the standard 3-21G basis set, and also by density functional theory at the B3LYP level in the standard 6-31G* basis set. All five levels of theory suggest a D4d cage (two four-membered rings, eight pentagons, eight hexagons) as the lowest-energy structure. Temperature effects are treated in the terms of partition functions so that the entropy contributions are considered accordingly. The thermodynamic treatment points out five cages significantly populated at high temperatures. At very high temperatures the structure lowest in energy is not the most abundant isomer. There are just six conventional fullerenes C32, built exclusively from pentagons and hexagons, however, only two of them show significant populations at high temperatures. The remaining three relatively stable cages contain at least one four-membered ring. No structure with a heptagon shows a non-negligible concentration at high temperatures. The study suggests that in the non-IPR region the quasi-fullerene cages with four-membered rings can in some cases be more important than the conventional fullerenes built from pentagons and hexagons only.