On the Structure and Relative Stability of C50 Fullerenes

Electron-vibrational renormalization in fullerenes through ab initio and machine learning methods

The effect of nuclear vibrations on the electronic eigenvalues and the HOMO-LUMO gap is known for several kinds of carbon-based materials, like diamond, diamondoids, carbon nanoclusters, carbon nanotubes and others, like hydrogen-terminated oligoynes and polyyne. However, it has not been widely analysed in another remarkable kind which presents both theoretical and technological interest: fullerenes. In this article we present the study of the HOMO, LUMO and gap renormalizations due to zero-point motion of a relatively large number (163) of fullerenes and fullerene derivatives. We have calculated this renormalization using density-functional theory with the frozen-phonon method, finding that it is non-negligible (above 0.1 eV) for systems with relevant technological applications in photovoltaics and that the strength of the renormalization increases with the size of the gap. In addition, we have applied machine learning methods for classification and regression of the renormalizations, finding that they can be approximately predicted using the output of a computationally cheap ground state calculation. Our conclusions are supported by recent research in other systems.

Element effects in endohedral metal-metal-bonding fullerenes M2@C82 (M = Sc, Y, La, Lu)

Endohedral metal-metal-bonding fullerenes have recently emerged, in which encapsulated metals form a metal-metal bond. However, the physical reasons why some metal elements prefer to form metal-metal bonds inside fullerene are still unclear. Herein, we reported first-principles calculations on electronic structures, bonding properties, dynamics, and thermodynamic stabilities of endohedral metallofullerenes M2@C82 (M = Sc, Y, La, Lu). Multiple bonding analysis approaches unambiguously reveal the existence of one two-center two-electron σ covalent metal-metal bond in M2@C82 (M = Sc, Y, Lu); however, the La-La bonding interaction in La2@C82 is weaker and could not be categorized as one metal-metal covalent bond. The energy decomposition analysis on bonding interactions between an encapsulated metal dimer and fullerene cages suggested that there exist two electron-sharing bonds between a metal dimer and fullerene cages. The reasons why La2 prefers to donate electrons to fullerene cages rather than form a standard σ covalent metal-metal bond are mainly attributed to two following facts: La2 has a lower ionization potential, while the hybridization of ns, (n - 1)d, and np atomic orbitals in La2 is higher. Ab initio molecular dynamic simulations reveal that the M-M bond length at room temperature follows the trend of Sc < Lu < Y. The statistical thermodynamics calculations at different temperatures reveal that the experimentally observed endohedral metal-metal-bonding fullerenes M2@C82 have high concentrations in the endohedral fullerene formation temperature range.

Cage-size effects on the encapsulation of P2 by fullerenes

The classic pnictogen dichotomy stands for the great contrast between triply bonding very stable N₂ molecules and its heavier congeners, which appear as dimers or oligomers. A banner example involves phosphorus as it occurs in nature as P₄ instead of P₂ , given its weak π-bonds or strong σ-bonds. The P₂ synthetic value has brought Lewis bases and metal coordination stabilization strategies. Herein, we discuss the unrealized encapsulation alternative using the well-known fullerenes' capability to form endohedral and stabilize otherwise unstable molecules. We chose the most stable fullerene structures from C_n (n = 50, 60, 70, 80) and experimentally relevant from C_n (n = 90 and 100) to computationally study the thermodynamics and the geometrical consequences of encapsulating P₂ inside the fullerene cages. Given the size differences between P₂ and P₄ , we show that the fullerenes C₇₀ -C₁₀₀ are suitable cages to side exclude P₄ and host only one molecule of P₂ with an intact triple bond. The thermodynamic analysis indicates that the process is favorable, overcoming the dimerization energy. Additionally, we have evaluated the host-guest interaction to explain the origins of their stability using energy decomposition analysis.

Stabilities, Geometries, Electronic Structures, and Conversion Rules of Carbide Cluster Metallofullerenes

Since the discovery of the first carbide cluster metallofullerene (CCMF) Sc₂ C₂ @C₈₄ in 2001, CCMFs have attracted great concerns with variable structures and fascinating characteristics. To date, there are hundreds of studies on CCMFs. Crystallography studies on CCMFs are carried out by single-crystal X-ray diffraction. Theoretical calculations can also be used to study CCMFs in detail without being limited by low experimental yields. This review analyzes the stability of CCMFs reported previously, and indicates that the C₂ unit contributes a lot to their stability. At the same time, the relationship between the structures of inner carbide cluster and cage size is systematically discussed, and the four-electron transfer always occurs. Furthermore, the possible transformation rule between di-EMFs and CCMFs is indicated. Finally, an outlook regarding the future developments and applications of CCMFs is presented.

Sc3N@Cs(39715)–C82: a missing isomer linked to Sc3N@C2v(39718)–C82 by a single step Stone–Wales transformation

Density functional theory combined with statistical mechanics calculations indicate that Sc₃N@C_2v(39718)–C₈₂ and Sc₃N@C_s(39715)–C₈₂ linked by a single Stone–Wales transformation can be obtained at the fullerene formation temperature region.

Dimetallic sulfide endohedral metallofullerene Sc2S@C76: density functional theory characterization

In terms of density functional theory combined with statistic mechanics computations, we investigated a dimetallic sulfide endohedral fullerene Sc2S@C76 which has been synthesized without any characterization in experiments. Our theoretical study reveals that Sc2S@Td(19151)-C76 which satisfies the isolated-pentagon rule (IPR) possesses the lowest energy, followed by three non-IPR structures (Sc2S@C2v(19138)-C76, Sc2S@Cs(17490)-C76, and Sc2S@C1(17459)-C76). To clarify the relative stabilities of those isomers at high temperatures, enthalpy-entropy interplay has been taken into consideration. Calculation results indicate that three species Sc2S@Td(19151)-C76, Sc2S@C2v(19138)-C76, and Sc2S@C1(17459)-C76 have noticeable molar fractions at the fullerene-formation temperature region (500-3000K), and the Sc2S@C1(17459)-C76 with one pentagon pair becomes the most predominant isomer above 1800 K, suggesting that the unexpected non-IPR structure is thermodynamically favorable at elevated temperatures. In addition, the structural characteristics, electron features, UV-vis-NIR adsorptions, and (13)C NMR spectra of those three stable structures are introduced to assist experimental identification and characterization in future.

Sc2S@C68: an obtuse di-scandium sulfide cluster trapped in a C2v fullerene cage

Density functional theory calculations combined with statistical thermodynamics treatments revealed that the mass-spectrum detected Sc₂S@C₆₈ should possess the C_2v(6073)-C₆₈ cage.

Regioselective chlorine-addition reaction toward C54Cl8 and role of chlorine atoms in Stone-Wales rearrangement

By means of density functional theory, detailed studies of regioselective chlorine-addition reactions of two C(54)Cl(8) isomers disclose a highly competitive advantage of (#540)C(54)Cl(8) in the chlorofullerene formation process. The regioselectivity of the addition pattern in (#540)C(54)Cl(8) is found to be dependent on both local and general factors. Special structural relationships reveal that the pristine cage of (#540)C(54)Cl(8) can transform to that of (#864)C(56)Cl(10) and (#913)C(56)Cl(12) through both C(2) addition and Stone-Wales rearrangement. It is found that Stone-Wales rearrangement, which is believed to be a high energy barrier reaction, can be facilitated remarkably well if chlorine atoms participate in the rearrangement process. Furthermore, investigation into the electronic properties of C(54) exohedral fullerenes reveal the different impacts of halogen and hydrogen atoms.

C₇₄ endohedral metallofullerenes violating the isolated pentagon rule: a density functional theory study

Precise studies on M(2)@C(74) (M = Sc, La) series by means of DFT methods have disclosed that certain non-IPR isomers are more stable than the IPR structure. M(2)@C(2)(13295)-C(74) and M(2)@C(2)(13333)-C(74), both of which have two pentagon adjacencies (PA), present excellent thermodynamic stability with very small energy differences. Statistical mechanics calculations on the M(2)@C(74) series reveal that M(2)@C(2)(13295)-C(74) and M(2)@C(2)(13333)-C(74) are quite favoured by entropy effects below 3000 K. Sc(2)@C(74) and La(2)@C(74) series are found to have similar electronic transfer but different electronic structures due to the distinct properties of scandium and lanthanum elements according to Natural Bond Orbital (NBO) analysis in conjunction with orbital interaction diagrams. Investigations of bonding energies reflect quite different influences of the two types of metal atoms to C(74) metallo-fullerenes. Further examinations on C(74) metallo-fullerenes uncover significant stabilization effects of metal atoms acting on PA fragments. Geometrical structures of certain non-IPR cages (from C(72) to C(76)), which exhibit splendid stabilities when encapsulating metallo-clusters, are found to be related by Stone-Wales transformation and C(2) addition. Furthermore, IR spectra and (13)C NMR spectra of M(2)@C(2)(13295)-C(74) and M(2)@C(2)(13333)-C(74) have been simulated to assist further experimental characterization.

Nonclassical fullerenes with a heptagon violating the pentagon adjacency penalty rule

Nonclassical fullerenes with heptagon(s) and their derivatives have attracted increasing attention, and the studies on them are performing to enrich the chemistry of carbon. Density functional theory calculations are performed on nonclassical fullerenes C(n) (n = 46, 48, 50, and 52) to give insight into their structures and stability. The calculated results demonstrate that the classical isomers generally satisfy the pentagon adjacency penalty rule. However, the nonclassical isomers with a heptagon are more energetically favorable than the classical ones with the same number of pentagon-pentagon bonds (B(55) bonds), and many of them are even more stable than some classical isomers with fewer B(55) bonds. The nonclassical isomers with the lowest energy are higher in energy than the classical ones with the lowest energy, because they have more B(55) bonds. Generally, the HOMO-LUMO gaps of the former are larger than those of the latter. The sphericity and asphericity are unable to rationalize the unique stability of the nonclassical fullerenes with a heptagon. The pyramidization angles of the vertices shared by two pentagons and one heptagon are smaller than those of the vertices shared by two pentagons and one hexagon. It is concluded that the strain in the fused pentagons can be released by the adjacent heptagons partly, and consequently, it is a common phenomenon for nonclassical fullerenes to violate the pentagon adjacent penalty rule. These findings are heuristic and conducive to search energetically favorable isomers of C(n), especially as n is 62, 64, 66, and 68, respectively.

Insertion of C50 into single-walled carbon nanotubes: Selectivity in interwall spacing and C50 isomers

The structures and electronic properties of nanoscale "peapods," i.e., C(50) fullerenes inside single-walled carbon nanotubes (SWCNTs), were computationally investigated by density functional theory (DFT). Both zigzag and armchair SWCNTs with diameters larger than 1.17 nm can encapsulate C(50) fullerenes exothermically. Among the SWCNTs considered, (9,9) and (16,0) SWCNTs are the best sheaths for both D(3) and D(5h) isomers of C(50), corresponding to 0.32-0.34 nm tube-C50 distances. The orientation of C(50) inside nanotubes also affects the insertion energies, which depend on the actual tube-fullerene distances. The insertion of D(3) and D(5h) isomers of C(50) is somewhat selective; the less stable D(5h) isomer can be encapsulated more favorably inside SWCNTs at same tube-C(50) spacing. Because of the weak tube-C(50) interaction, the geometric and electronic structures of the peapods do not change greatly for the most stable configurations, but the selectivity in the interwall spacing and isomer encapsulation can be useful in separating various carbon fullerenes and their isomers.

Structures, stabilities, and electronic and optical properties of C52 fullerene, ions, and metallofullerenes

The 437 classical isomers of fullerene C52 have been studied by PM3, HCTH/3-21G, and B3LYP6-31G(d). C(2):029 with the least number of adjacent pentagons is predicted to be the most stable isomer. The investigations show that both the number of adjacent pentagons and the degree of aromaticity play important roles in the relative stabilities of fullerene isomers. To clarify the relative stabilities of the C52 isomers in a wide range of temperatures, the entropy contributions are taken into account on the basis of the Gibbs energy at the B3LYP6-31G(d) level. C(2):029 prevails in a wide temperature range. In addition, the electronic spectra and second-order hyperpolarizabilities are determined by means of ZINDO and sum-over-states model. The static second-order hyperpolarizability of C(2):029 is 51% larger than that of C60. Furthermore, intensity-dependent refractive index gamma (-omega;omega,omega,-omega) (omega=1.1653 eV) of C(2):029 is 13 times larger than that of C60. The encapsulation of Ca atom in C52 fullerene is exothermic and the metallofullerene Ca-C52 is described as Ca2+-C52(2-).

Valence of D5h C50 fullerene

The energetic and electronic properties of D5h C50 before and after passivation by H or Cl are investigated using first-principle computational method of density functuional theory with generalized gradient approximation and local density approximation functionals. The results show that H or Cl addition can lead to energetic stabilization. Additions also increase the highest occupied molecular orbit-lowest unoccupied molecular orbital (HOMO-LUMO) gaps of C50 fullerides and make them chemically more stable. In the series of C50H2m (m = 0 approximately 7), the Saturn-shaped D5h C50H10 has the largest HOMO-LUMO gap, which suggests that such a structure of C50H10 is a "magic-number" stable one of C50 adducts, and ten is a pseudovalence or effective valence of C50 fullerene pseudoatom. This point also is supported by the energetic properties of C50H2m series such as binding energies, etc. A minimal energy reaction pathway is constructed to get C50H10 and C50H14. Some useful experience for determining the favorable addition sites was summarized. A simple steric method is developed to predict the effective valences of classical fullerenes.

Search for suitable approximation methods for fullerene structure and relative stability studies: Case study with C50

Local density approximation (LDA), several popular general gradient approximation (GGA), hybrid module based density functional theoretical methods: SVWN, BLYP, PBE, HCTH, B3LYP, PBE1PBE, B1LYP, and BHandHLYP, and some nonstandard hybrid methods are applied in geometry prediction for C60 and C70. HCTH with 3-21G basis set is found to be one of the best methods for fullerene structural prediction. In the predictions of relative stability of C50 isomers, PM3 is an efficient method in the first step for sorting out the most stable isomers. HCTH with 3-21G predicts very good geometries for C50, similar to the performance of B3LYP6-31G(d). The gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital from the predictions of all the density functional theory methods has the following descending order: E(gap)(half-and-half hybrid)>E(gap)(B3LYP)>E(gap)(HCTH)(GGA)>E(gap)(SVWN)(LDA).