Fullerenes Cn Research Articles

Kekulé Counts, Clar Numbers, and ZZ Polynomials for All Isomers of (5,6)-Fullerenes C52-C70.

We report an extensive tabulation of several important topological invariants for all the isomers of carbon (5,6)-fullerenes Cn with n = 52-70. The topological invariants (including Kekulé count, Clar count, and Clar number) are computed and reported in the form of the corresponding Zhang-Zhang (ZZ) polynomials. The ZZ polynomials appear to be distinct for each isomer cage, providing a unique label that allows for differentiation between various isomers. Several chemical applications of the computed invariants are reported. The results suggest rather weak correlation between the Kekulé count, Clar count, Clar number invariants, and isomer stability, calling into doubt the predictive power of these topological invariants in discriminating the most stable isomer of a given fullerene. The only exception is the Clar count/Kekulé count ratio, which seems to be the most important diagnostic discovered from our analysis. Stronger correlations are detected between Pauling bond orders computed from Kekulé structures (or Clar covers) and the corresponding equilibrium bond lengths determined from the optimized DFTB geometries of all 30,579 isomers of C20-C70.

Highly efficient implementation of analytic nonadiabatic derivative couplings within the pseudospectral method.

A pseudospectral implementation of nonadiabatic derivative couplings in the Tamm-Dancoff approximation is reported, and the accuracy and efficiency of the pseudospectral nonadiabatic derivative couplings are studied. Our results demonstrate that the pseudospectral method provides mean absolute errors of 0.2%-1.9%, while providing a significant speedup. Benchmark calculations on fullerenes (Cn, n up to 100) using B3LYP achieved 10- to 15-fold, 8- to 17-fold, and 43- to 75-fold speedups for 6-31G**, 6-31++G**, and cc-pVTZ basis sets, respectively, when compared to the conventional spectral method.

Comparative analysis of thermal characteristics of higher fullerenes when heated in a nitrogen atmosphere

To study the behavior of higher fullerenes Cn (n = 84, 90, 94) under thermal exposure in a nitrogen medium, a thermodynamic modeling method is used, the main task of which is to determine the phase, chemical composition and parameters of the system under study. This method consists of analyzing the system using the TERRA program, which is a further development of the ASTRA program package. For Cn-N2 systems, a comparative study of ongoing chemical reactions was carried out. Intervals of thermal stability of the condensed phase for higher fullerenes are revealed. The periods of thermal destruction in the gas phase in the systems under consideration are determined.

Clar and Fries structures for fullerenes

Fries and Clar numbers are qualitative indicators of stability in conjugated π systems. For a given Kekulé structure, call any hexagon that contains three double bonds benzenoid. The Fries number is the maximum number of benzenoid hexagons, whereas the Clar number is the maximum number of independent benzenoid hexagons, in each case taken over all Kekulé structures. A Kekulé structure that realises the Fries (Clar) number is a Fries (Clar) structure. For benzenoids, it is not known whether every Fries structure is also a Clar structure. For fullerenes Cn, it is known that some Clar structures in large examples correspond to no Fries structure. We show that Fries structures that are not Clar occur early: examples where some Fries structure is not Clar start at C34, and examples where no Fries structure is Clar start at C48. Hence, it is unsafe to use fullerene Fries structures as routes to Clar number. However, Fries structures often describe the neutral fullerene better than a Clar structure, e.g. in rationalising bond lengths in the experimental isomer of C60. Conversely, an extension of Clar sextet theory suggests the notion of anionic Clar number for fullerene anions, where both pentagons and hexagons may support sextets.

Methodological Investigation for Hydrogen Addition to Small Cage Carbon Fullerenes

Hydrogenated small fullerenes (Cn, n < 60) are of interest as potential astrochemical species, and as intermediates in hydrogen-catalysed fullerene growth. However, the computational identification of key stable species is difficult due to the vast configurationally space of structures. In this study, we explored routes to predict stable hydrogenated small fullerenes. We showed that neither local fullerene geometry nor local electronic structure analysis was able to correctly predict subsequent low-energy hydrogenation sites, and sequential stable addition searches also sometimes failed to identify most stable hydrogenated fullerene isomers. Of the empirical and semi-empirical methods tested, GFN2-xTB consistently gave highly accurate energy correlations (r > 0.99) to full DFT-LDA calculations at a fraction of the computational cost. This allowed identification of the most stable hydrogenated fullerenes up to 4H for four fullerenes, namely two isomers of C28 and C40, via “brute force” systematic testing of all symmetry-inequivalent combinations. The approach shows promise for wider systematic studies of smaller hydrogenated fullerenes.

Pseudospectral implementations of long-range corrected density functional theory.

We have implemented pseudospectral density-functional theory (DFT) with long-range corrected DFT functionals (PS-LRC) in quantum mechanics package Jaguar, and applied it in the calculations of geometry optimizations, dimmer interaction energies, polarizabilities and first-order hyperpolarizabilities, harmonic vibrational frequencies, S1 and T1 excitation energies, singlet-triplet gaps, charge transfer numbers, oscillator strengths, reaction barrier heights, electron-transfer couplings, and charge-transfer excitation energies. From our accuracy benchmark analysis, PS grids, PS dealiasing functions, PS atomic corrections, PS multigrid strategy, PS length scales, and PS cutoff scheme perform well in PS DFT with LRC density functionals with very small and ignorable deviations when compared to the conventional spectral (CS) method. The timing benchmark study of S1 excitation energy calculations of fullerenes (Cn , n up to 540) demonstrates that PS-LRC achieves 1.4-8.4-fold speedups in SCF, 22-32-fold speedups in Tamm-Dancoff approximation, and 6-15-fold speedups in total wall clock time with an average error 0.004 eV of excitation energies compared to the CS method.

Highly efficient implementation of the analytical gradients of pseudospectral time-dependent density functional theory.

The accuracy and efficiency of time-dependent density functional theory (TDDFT) excited state gradient calculations using the pseudospectral method are presented. TDDFT excited state geometry optimizations of the G2 test set molecules, the organic fluorophores with large Stokes shifts, and the Pt-complexes show that the pseudospectral method gives average errors of 0.01-0.1 kcal/mol for the TDDFT excited state energy, 0.02-0.06pm for the bond length, and 0.02-0.12° for the bond angle when compared to the results from conventional TDDFT. TDDFT gradient calculations of fullerenes (Cn, n up to 540) with the B3LYP functional and 6-31G** basis set show that the pseudospectral method provides 8- to 14-fold speedups in the total wall clock time over the conventional methods. The pseudospectral TDDFT gradient calculations with a diffuse basis set give higher speedups than the calculations for the same basis set without diffuse functions included.

Open-shell nature of non-IPR fullerene С40: isomers 29 (C2) and 40 (Td).

It is well-known that the small non-IPR fullerenes Cn (n < 60) are highly unstable and that is why they cannot be obtained as empty cages. However, they become stable as exohedral or endohedral derivatives. In this report, the molecular structures of non-IPR isomers 29 (C2) and 40 (Td) of fullerene C40 are investigated using a semiempirical approach developed earlier for higher fullerenes. Quantum-chemical calculations (DFT) show that isomers 29 (C2) and 40 (Td) have open-shell structures. The distributions of single, double, and delocalized π-bonds in the isomer molecules in question are presented for the first time as well as their molecular formulas. It is found unusual for higher fullerenes chain of π-bonds passing through some cycles. Identified features in the structures of small fullerene molecules can be predictive of the ability to their synthesis as derivatives and will assist in their structure determination.

Structures, stability, electronic, magnetic properties of TM@Cn (n=20, 24, 28) and polymerization of cyclocarbon-24: A DFT study

Small fullerenes Cn (n < 60) can provide clues for the formation mechanism of common fullerenes. However, it is difficult to synthesize and separate the small fullerenes because they violate the Isolated Pentagon Rule. Encapsulation of transition metal (TM) atoms can stabilize the unstable small fullerenes, regulate the physicochemical properties of core-shell clusters, and bring about catalytic and magnetic effects. Herein, the structures, stability, electronic and magnetic properties of TM@Cn were studied by density functional theory. The results showed that the embedded clathrate compounds that can be stabilized by TM atoms include Ti@C24, Sc@C28, Ti@C28, Zr@C28 and Hf@C28. The d orbital of TM atoms in TM@Cn resonates with the s and p orbits near the Fermi level. Fe@C20 has the largest magnetic moment, and the magnetic moments of some structures are 0. For cyclocarbon small fullerene C24, a new tubular strucuture can be formed by end-to-end coalescence. Through polymerization, the stability of C24 is improved significantly. The C24-5 polymer embedded with iodine chain has similar electronic properties with the single-walled carbon nanotube (6,0) with a similar diameter, which provides a new polymerization method for the stabilization of cyclofullerenes similar to C24.

ZZ Polynomials for Isomers of (5,6)-Fullerenes Cn with n = 20–50

A compilation of ZZ polynomials (aka Zhang–Zhang polynomials or Clar covering polynomials) for all isomers of small (5,6)-fullerenes Cn with n = 20–50 is presented. The ZZ polynomials concisely summarize the most important topological invariants of the fullerene isomers: the number of Kekulé structures K, the Clar number Cl, the first Herndon number h1, the total number of Clar covers C, and the number of Clar structures. The presented results should be useful as benchmark data for designing algorithms and computer programs aiming at topological analysis of fullerenes and at generation of resonance structures for valence-bond quantum-chemical calculations.

Low-energy electron scattering from fullerenes and heavy complex atoms: negative ions formation

Regge poles are generalized bound states. Our robust Regge pole methodology that embeds the crucial electron correlation effects and the vital polarization interaction is used to investigate negative ions formation in low-energy electron elastic scattering from the fullerenes Cn (n = 24, 28, 44, 60, 70, 74, 82, 94, 100, and 140) and the selected heavy lanthanide (Gd and Dy) and actinide (Pa and U) atoms through the elastic total cross sections (TCSs) calculations. All the TCSs are found to be characterized by Ramsauer–Townsend minima, shape resonances, and dramatically sharp resonances manifesting stable ground and metastable anionic formation during the collisions. The ground states anionic binding energies (BEs) for the Cn− (n = 24, 28, 44, 60, 70, and 82) match excellently the measured electron affinities (EAs). The thus benchmarked Regge pole methodology on the ground states anionic BEs for these fullerenes is then used to calculate the ground and the metastable elastic TCSs for the fullerenes, and the heavy atoms wherefrom their anionic BEs are extracted and compared with the measured EAs where they are available. Surprisingly, the C74 fullerene has the largest anionic ground state BE value among the investigated fullerenes in this paper and the ground state of the C140− anion has a large BE as well. Many of these fullerenes could be useful in nanocatalysis, sensor technology and organic solar cells through their ground and metastable anionic BEs. For the heavy atoms the extracted ground and metastable anionic BEs are compared with the measured and calculated EAs. These results particularly the ground states anionic BEs of the fullerenes and heavy atoms are expected to inspire and guide the long overdue experimental and theoretical explorations of electron attachment in low-energy electron scattering in these and related systems.

Giant fullerene formation through thermal treatment of fullerene soot

Coalescence of fullerenes into giant fullerenes (Cn n > 100) has been observed in the gas phase and inside carbon nanotubes. In this work, we demonstrate the formation of giant fullerenes by heating fullerene soot. Extracting the majority of the magic number fullerenes (C60 and C70) allowed the underlying distribution of fullerenes in the solid state to be measured. Upon heating at 800–1000 °C for 30–60 min under vacuum the mass distribution of fullerenes was found to shift toward larger masses. High resolution electron microscopy was used to compare formation of giant fullerenes by thermal heating and electron beam irradiation, showing the former produced more isolated giant fullerene fragments >1 nm in size. The driving force for coalescence and growth by thermal heating was suggested to be vertex strain at pentagonal sites, providing a route towards the synthesis of fullerenes up to C300.

Intersystem Crossing Rates of Isolated Fullerenes: Theoretical Calculations.

Although the triplet states of fullerenes have prosperous applications, it remains unclear how the structural parameters of singlet and triplet states control the intersystem crossing (ISC) rates. Here, electronic structure calculations (reorganization energy, driving force, and spin-orbit coupling) and a rate theory (Marcus formula) are employed to quantitatively predict the ISC rates of isolated fullerenes Cn (n = 60-110). The results demonstrate that the driving force is not the only factor to predict the ISC rates. For instance, although C80, C82, and C110 have the favorable driving force, the ISC rates are close to zero because of small spin obit couplings, whereas small ISC rates of C96 and C100 result from quite small reorganization energies. Meanwhile, in addition to well-known C60 and C70, C92 possesses good ISC property with obviously large ISC rate. C92 also has a higher triplet-state energy than singlet-state oxygen energy; it may thus have a good photoactive property.

Theoretical study of electron transfer process between fullerenes and neurotransmitters; acetylcholine, dopamine, serotonin and epinephrine in nanostructures [neurotransmitters].C n complexes.

Neurotransmitters are the compounds which allow the transmission of signals from one neuron to the next across synapses. They are the brain chemicals that communicate information throughout brain and body. Fullerenes are a family of carbonallotropes, molecules composed entirely of carbon, that take the forms of spheres, ellipsoids, and cylinders. Various empty carbon fullerenes (Cn) with different carbon atoms have been obtained and investigated. Topological indices have been successfully used to construct effective and useful mathematical methods to establish clear relationships between structural data and the physical properties of these materials. In this study, the number of carbon atoms in the fullerenes was used as an index to establish a relationship between the structures of neurotransmitters (NTs) acetylcholine (AC) 1, dopamine (DP) 2, serotonin (SE) 3, and epinephrine (EP) 4 as the well-known redox systems and fullerenes C n (n = 60, 70, 76, 82, and 86) which create [NT].Cn; A-1 to A-5 up to D-1 to D-5. The relationship between the number of carbon atoms and the free energy of electron transfer (ΔG et(n); n = 1-4) is assessed using the Rehm-Weller equation for A-1 to A-5 up to D-1 to D-5 supramolecular [NT].Cn complexes. The calculations are presented for the four reduction potentials ( (Red.) E 1 to (Red.) E 4 ) of fullerenes C n . The results were used to calculate the four free energy values of electron transfer (ΔG et(1) to ΔG et(4)) of the supramolecular complexes A-1 to A-8 up to D-1 to D-8 for fullerenes C60 to C120. The first to fourth free activation energy values of electron transfer and the maximum wavelength of the electron transfers, ΔG (#) et(n) and λ et (n = 1-4), respectively, were also calculated in this study for A-1 to A-8 up to D-1 to D-8 in accordance with the Marcus theory.

Structural relationships and theoretical study of the free energies of electron transfer, electrochemical properties, and electron transfer kinetic of cephalosporin antibiotics derivatives with fullerenes in nanostructure of [R]·C n (R = cefadroxil, cefepime, cephalexin, cefotaxime, cefoperazone and ceftriaxone) supramolecular complexes

Various empty carbon fullerenes (C n ) with different carbon atoms have been obtained and investigated. Cephalosporin antibiotics and its derivatives have important medicinal properties. The β-lactam class of antibiotics has a broad spectrum of antimicrobial properties. Their antibacterial and pharmacokinetic properties have wide therapeutic applications. Topological indices have been successfully used to construct effective and useful mathematical methods to establish clear relationships between structural data and the physical properties of these materials. In this study, the number of carbon atoms in fullerenes was used as an index to establish a relationship between the structures of cefadroxil, cefepime, cephalexin, cefotaxime, cefoperazone and ceftriaxone (β-lactam antibiotics) and fullerenes (C n , n = 60, 70, 76, 82 and 86), which create [cefadroxil]·C n , [cefepime]·C n , [cephalexin]·C n , [cefotaxime]·C n , [cefoperazone]·C n and [ceftriaxone]·C n . The relationship between the number of fullerene carbon atoms and the free energies of electron transfer (ΔG et(1)–ΔG et(4)) are assessed using the Rehm-Weller equation for A-1 to A-5, B1 to B-5, C-1 to C-5, D-1 to D-5, E-1 to E-5 and F-1 to F-5 of the supramolecular complexes [R]·C n (where R = cefadroxil, cefepime, cephalexin, cefotaxime, cefoperazone and ceftriaxone) complexes. The calculations are presented for the four reduction potentials (Red. E 1–Red. E 4 ) of fullerenes C n . The results were used to calculate the four free energies of electron transfer (ΔG et(1)–ΔG et(4)) of the cephalosporin-fullerene supramolecular complexes A-1 to A-5, B1 to B-5, C-1 to C-5, D-1 to D-5, E-1 to E-5 and F-1 to F-5 for fullerenes C60–C120. The free energies of activation for electron transfer, Δ $$ G_{{_{{{\text{et}}\left( n \right)}} }}^{{^{\# } }} $$ (n = 1–4) were also calculated for these complexes in accordance with the Marcus theory. In this study, was presented the calculated wavelengths (λ(n); n = 1–4; in nm) of the photoelectron transfer process as well in the nanostructure complexes.

Distributed curvature and stability of fullerenes.

Energies of non-planar conjugated π systems are typically described qualitatively in terms of the balance of π stabilisation and the steric strain associated with geometric curvature. Curvature also has a purely graph-theoretical description: combinatorial curvature at a vertex of a polyhedral graph is defined as one minus half the vertex degree plus the sum of reciprocal sizes of the faces meeting at that vertex. Prisms and antiprisms have positive combinatorial vertex curvature at every vertex. Excluding these two infinite families, we call any other polyhedron with everywhere positive combinatorial curvature a PCC polyhedron. Cubic PCC polyhedra are initially common, but must eventually die out with increasing vertex count; the largest example constructed so far has 132 vertices. The fullerenes Cn have cubic polyhedral molecular graphs with n vertices, 12 pentagonal and (n/2 - 10) hexagonal faces. We show that there are exactly 39 PCC fullerenes, all in the range 20 ≤n≤ 60. In this range, there is only partial correlation between PCC status and stability as defined by minimum pentagon adjacency. The sum of vertex curvatures is 2 for any polyhedron; for fullerenes the sum of squared vertex curvatures is linearly related to the number of pentagon adjacencies and hence is a direct measure of relative stability of the lower (n≤ 60) fullerenes. For n≥ 62, non-PCC fullerenes with a minimum number of pentagon adjacencies minimise mean-square curvature. For n≥ 70, minimum mean-square curvature implies isolation of pentagons, which is the strongest indicator of stability for a bare fullerene.

Study of electron transfer process between fullerenes and membrane cells of Escherichia coli in the presence of dinitrophenol and dicyclohexylcarbodiimide

Microorganisms are the basis of all known biological activities. The electrical potential difference (ΔΨ) across the membrane of Escherichia coli was utilized for theoretical study of electron transfer (ET) process between fullerenes and the membrane cells of E. coli bacteria. The strains of E. coli which were selected in this study are: FRAG-5, TK-1207 and TK-1208. Dinitrophenol (2,4-DNP) and dicyclohexylcarbodiimide (DCC) are the compounds in the reported experimental conditions which were selected in this study. Various empty fullerenes (Cn) with different carbon atoms have been obtained and investigated before. Topological indices have been successfully used to construct effective and useful mathematical methods to establish clear relationships between structural data and the physical properties of these materials. The relationship between the number of carbon atoms of the fullerenes and the free energies of electron transfer (ΔGet) were assessed using the Rehm–Weller equation for ET-process between fullerenes and the membrane cells of E. coli with strains FRAG-5, TK-1207 and TK-1208 and TK-1235 in the presence of DNP and DCC. The calculations are presented on the basis of the oxidation potential (Ox.E1) of fullerenes Cn. In this study, the results were used to calculate the free-energies of electron transfer (ΔGet) of the ET-process for the fullerenes. The first free activation energies of electron transfer (ΔGet#) and the maximum wave length of the electron transfers (λet), were also calculated for the selected conditions in accordance with the Marcus theory.

Theoretical Study and Structural Relationships of Free Energies of Electron Transfer, Electrochemical Properties and Electron Transfer Kinetics of Ferrocene Derivatives with Fullerenes in Nanostructures of [(R)2Cp2-Fe].Cn Supramolecular Complexes

Ferrocene (Cp2-Fe) and its derivatives are part of a broad-spectrum of organometallic compounds that are commonly used in electronics and semiconductors. Fullerenes are a family of carbon allotropes composed entirely of carbon in the form of spheres, ellipsoids and cylinders. Various empty carbon fullerenes (Cn) have been obtained and investigated. The Cp2-Fe derivatives have exhibited important electron-transfer properties. Topological indices have been proposed to correlate structures with properties. They have been successfully employed to construct effective and useful mathematical methods to establish clear relationships between structural data and the physical properties of these materials. In this study, the number of carbon atoms in the fullerenes was used as an index to establish a relationship between the structures of ferrocenyl derivatives (1–15), ferrocenyl fullerene complexes 1–15 (the most well-known redox systems) and fullerenes Cn (n = 60, 70, 76, 82 and 86) A-1 to O-5, which are commonly denoted as [Fc-x].Cn. The Rehm-Weller equation was applied to build the relationship between the structures and the free energies for electron transfer (ΔGet(1) to ΔGet(4) ) of the complexes [Fc-x].Cn (A-1 to O-5). The calculations are presented for the four reduction potentials ( Red.E1 to Red.E4 ) of the fullerenes Cn . The reduction potentials of the fullerenes Cn have been applied to calculate the four free energies of electron transfer (ΔGet(1) to ΔGet(4) ) of supramolecular complexes [Fc-x].Cn (A-1 to O-5). Based on Marcus theory and Planck's equation, the first four free activation energies of electron transfer and the wavelengths (ΔGet(1) # to ΔGet(4) # and λ et , n = 1–4, respectively) for the photoelectron transfer process were also calculated for [Fc-x].Cn (A-1 to O-5).

Exploring magnetic properties inside full equatorial BN-substituted fullerenes Cn (n=20, 24, 30, 36, 60, 80): A computational NICS characterization

A density functional study has been performed at B3LYP/6-31G(d) level of theory to investigate the magnetic properties along the principal axes of tubular belts of full equatorial substituted BCN fullerenes in comparison with their parent fullerenes Cn (n=20, 24, 30, 36, 50, 80) based on NICS index. Independent of type and size of tubular BN belt, BCN fullerenes, unlike their parent fullerenes, represent short range of negative NICS values (aromatic character) calculated at the cage center. Full equatorial BN substitution noticeably affects the aromaticity character of antiaromatic fullerenes, like C24, and converts them to aromatic systems. The magnetic field inside two specific hybrid BCN fullerenes C10B5N5 and C10B10N10 has completely different intensities in each of their hemispheres which might lead to nanoparticles with unusual magnetic properties as promising candidates for electronic devices. Finally the influence of the carbon caps on such properties is also analyzed and it is concluded that the presence of the caps significantly affects electron delocalization and consequently aromaticity depending on the properties characterizing the caps and the remaining BN-tubular parts.

Study of Electrochemical Properties, Free Energies of Electron Transfer and Reduction Potentials of Supramolecular [X-UT-Y]@Cn Complexes and Fullerenes C60 to C300

Supramolecular complexes of unsaturated thiocrown ethers with cis geometry [X-UT-Y] 1–9 and fullerenes Cn have been reported to demonstrate a host-guest interaction for electron transfer processes. To establish a good relationship between the structure of molecules 1–9 and fullerenes, which create [X-UT-Y]@Cn supramolecular complexes (C60: A-1 to A-9 , C70: B-1 to B-9 , C76: C-1 to C-9 , C82: D-1 to D-9 , C86: E-1 to E-9 , F-6 to R-6 ), the number of carbon atoms within the fullerenes was utilized as an index. The relationship between the number of carbon atoms and free energies of electron transfer (ΔGet(n) , n = 1–4) are assessed using the Rehm-Weller equation for A-1 to A-9 , B1 to B-9 , C-1 to C-9 , D-1 to D-9 and E-1 to E-9 complexes. Calculations are presented for the four reduction potentials ( Red.En , n = 1–4) of fullerenes. The results were extended to calculate ΔGet (n) (n = 1–4) of F-6 to R-6 and Red.E1 to Red.E4 for fullerenes C60 to C300.