Isomers of C(20): an energy profile II

Semi-empirical calculations, at the PM3 level provided within the Winmopac v2.0 software package, are used to geometrically optimize and determine the absolute energies (heats of formation) of a variety of C(20) isomers that are predicted to exist in and around the bowl and cage isomers. Using the optimized Cartesian coordinates for the bowl and the cage isomers, a saddle-point calculation was performed. The output file generated, containing energy, distance, and geometry information, is then organized into a graphical format. The resulting graph, which plots the energy of the 20-atom system as a function of the distance from the geometric midpoint, is a two-dimensional energy profile. This profile illustrates an estimation of the contours on the potential energy surface, showing energy minima and maxima that are encountered as the bowl evolves into the cage structure, or vice-versa. To expand the surface into three dimensions, geometry optimizations were performed on the sets of Cartesian coordinates that correspond to energy minima in the bowl-cage profile. Based on these optimizations, eight additional isomers of C(20) have been identified and are predicted to be energetically stable. These additional isomers were subsequently subjected to saddle-point calculations in order to identify those isomers that lie adjacent to one another on the three-dimensional surface. Two isomers that are adjacent to each other will exhibit an energy profile that progresses smoothly from the potential well of each isomer up to the saddle point separating them. Consequently, these adjacent pairs of isomers establish a step-wise transformation between the bowl and the cage. This process, which extends out over the three-dimensional surface, is predicted to require less energy than that of the direct, two-dimensional transformation predicted in the bowl-cage profile.     

Isomers of C(20): an energy profile

Semiempirical calucaltions, at the PM 3 level, are used to geometrically optimize and determine the absolute energies (heats of formation) of a variety of C(20) isomers. Based on the geometrically optimized Cartesian coordinates of the ring and the bowl isomers, and the subsequent saddle-point calculation, a two-dimensional energy profile between these two isomers is generated. Performing geometry optimization on the Cartesian coordinates that correspond to energy minima within the ring-bowl profile, we have been able to identify several more isomers of C(20) that are predicted to be energitically stable. With these additional stable structures, we have identified pairs of isomers that lie adjacent to one another on the potential energy surface, as is evidenced by the form of their respective energy profiles. These adjacent pairs of isomers establish a step-wise transformation between the ring and the bowl. This process, which extends out over the three-dimensional surface, is predicted to require less energy than that of the direct, two-dimensional transformation predicted in the ring-bowl profile.     

Ab initio calculation of bowl, cage, and ring isomers of C20 and C20-

High-level ab initio calculations have been carried out to reexamine relative stability of bowl, cage, and ring isomers of C(20) and C(20)(-). The total electronic energies of the three isomers show different energy orderings, strongly depending on the hybrid functionals selected. It is found that among three popular hybrid density-functional (DF) methods B3LYP, B3PW91, PBE1PBE, and a new hybrid-meta-DF method TPSSKCIS, only the PBE1PBE method (with cc-pVTZ basis set) gives qualitatively correct energy ordering as that predicted from ab initio CCSD(T)/cc-pVDZ [CCSD(T)-coupled-cluster method including singles, doubles, and noniterative perturbative triples; cc-pVDZ-correlation consistent polarized valence double zeta] as well as from MP4(SDQ)/cc-pVTZ [MP4-fourth-order Moller-Plesset; cc-pVTZ-correlation consistent polarized valence triple zeta] calculations. Both CCSD(T) and MP4 calculations indicate that the bowl is most likely the global minimum of neutral C(20) isomers, followed by the fullerene cage and ring. For the anionic counterparts, the PBE1PBE calculation also agrees with MP4/cc-pVTZ calculation, both predicting that the bowl is still the lowest-energy structure of C(20)(-) at T=0 K, followed by the ring and the cage. In contrast, both B3LYP/cc-pVTZ and B3PW91/cc-pVTZ calculations predict that the ring is the lowest-energy structure of C(20)(-). Apparently, this good reliability in predicting the energy ordering renders the hybrid PBE method a leading choice for predicting relative stability among large-sized carbon clusters and other carbon nanostructures (e.g., finite-size carbon nanotubes, nano-onions, or nanohorns). The relative stabilities derived from total energy with Gibbs free-energy corrections demonstrate a changing ordering in which ring becomes more favorable for both C(20) and C(20)(-) at high temperatures. Finally, photoelectron spectra (PES) for the anionic C(20)(-) isomers have been computed. With binding energies up to 7 eV, the simulated PES show ample spectral features to distinguish the three competitive C(20)(-) isomers.     

Energetic analysis of 24 C20 isomers

Twenty-four individual geometric structures of the 20-carbon system are investigated using hybrid Hartree–Fock/density functional theory (DFT-B3LYP) in combination with the 6-31G and 6-311G* basis sets. These computations are carried out in order to evaluate the relative energies of the various C₂₀ isomers utilizing both geometry optimization and single-point energy calculations. The relative energies of the three most widely studied C₂₀ isomers (ring, bowl, and cage) are compared to the energies of other cyclic, bowl-like, and miscellaneous isomers. Although the ring and the bowl isomers are predicted to be the two most stable species, we have identified three to four additional cyclic structures that are predicted to be energetically competitive with the fullerene structure.     

A quantum chemical study of three isomers of N20

Ab initio molecular orbital (MP2) and density functional theory (B3LYP) calculations using different basis sets have been employed to study the structures, energetics and vibrational frequencies of the large homonuclear polynitrogen compound, N₂₀. In the present study, three distinct forms were found to represent local minima on the potential energy surface. They are the fullerene-type cage form of I_h symmetry, a corannulene-like bowl form of C_5v symmetry, and a ring isomer with D₅ symmetry of which the cage form turns out to be the highest energy form. Both the bowl and ring forms are calculated to be more stable than the cage form by about 200 kcal/mol. The molecular properties calculated for these isomers may serve as valuable predictions for future experimental searches for new high energy density materials (HEDM).     

Super-valence phenomenon of carbon atoms in C20 molecule

Fourteen structures of C₂₀ are studied at DFT/B3LYP/6-31G* theoretical level. Except ring, bowl, cage and isomer 1 which have been studied before, other isomers have not been reported so far. Calculated results show that the ring has the lowest energy at this level and isomers 1, 2, 3 and 4 have lower energies than that of cage. Analyses of optimized bond lengths, electronic structure indicate that some carbon atoms express super-valence property. In addition, NICS value is consistent with molecular orbital character in denoting aromaticity of C₂₀ molecule. Delocalization character averts influence of curvature strain, which can well explain the stability of the cage.     

Isomers of C36 and free energy criteria for cluster growth

A molecular dynamics procedure is developed to search for cluster isomers and is used to study the isomer spectrum of C36 with the Brenner potential. Beginning with isolated carbon atom, the procedure quickly arrives at the D6h cage with the lowest potential and produces other 410 isomers. Among these isomers, we selected ones of typical cage, bowl, and sheet structures to calculate their free energies at 2300 K and performed molecular dynamics simulations starting either from 36 free carbon atoms diluted in He buffer gas kept at 2300 K or from the D6h cage under the same conditions, which show that the microsystem reaches a kinetic equilibrium within about 100 ns and that the isomer of the lowest free energy rather than the D6h cage of the lowest potential energy dominates in the resultant cluster.     

富勒烯衍生物(C36OH)+异构体间的相互转化机理──羟基(OH)的分子内迁移

用从头算HF/STO-3G方法对(C36OH)+的3种异构体之间的重排机理进行了理论研究.计算结果表明,异构体2的热力学稳定性最强;从动力学角度考虑,由异构体1和3转化为2比反方向转化容易得多.结合电荷在碳笼表面的分布,预计动力学上最终在C7和C26位置上发生羟基加成的几率最大,该结论与热力学上预计的跨赤道六元环内羟基1,4-加成所得到的C36(OH)2异构体最稳定的结果一致.     

Aromaticity of Heterofullerenes C18BxNy (x+y=2) and Their Molecular Ions

The aromaticity of all possible substituted fullerene isomers of C18N2, C18B2, C18BN, and their molecularions which originate from the C20 (Ih) cage were studied by the topological resonance energy (TRE) and the percentage topological resonance energy methods. The relationship between the aromaticity of C18BxNy isomers and the sites where the heteroatoms dope at the C20 (Ih) cage is discussed. Calculation results show that at the neutral and cationic states all the isomers are predicted to be antiaromatic with negative TREs, but their polyvalent anions are predicted to be aromatic with positive TREs. The most stable isomer is formed by heteroatom doping at the 1,11-sites in C18N2, C18B2, and C18BN. Heterofullerenes are more aromatic than C20. The stability order in the neutral states is C18N2>C18BN>C18B2>C20. The stability order in closed-shell is C18B2 8->C20 6->C18BN6->C18N2 4-. This predicts theoretically that their polyvalent anions have high aromaticity.     

C20三个异构体稳定性的从头计算研究

采用HF,B3LYP,MP2和QCISD等理论方法在6-31G和6-31G^*基组水平上,对C20的3个主要异构体碟型(Bowl)、笼型(Cage)和环型(Ring)的结构进行理论研究,得出这3个异构体的相对稳定性的次序为:Ring>Bowl>Cage.计算结果显示,分子轨道具有离域特性,电子的这种离域特征有利于原子间更好的成键,可避免出现大的张力,这是Cage型异构体虽然违反“五元环隔离规则”(Isolatedpentagonrule)却能稳定存在的原因.     

The use of internal cartesian coordinates for describing molecular vibrations

An analysis of the influence of isotope substitution on the system of electronic-nuclear equations for an arbitrary molecular system was used as a basis for formulating invariance conditions with respect to isotope substitution of the potential energy surface written in the Cartesian coordinates rigidly bound with the center of mass of the molecule (internal Cartesian coordinates). This property of the potential function obviates the necessity of using curvilinear natural coordinates, which can be replaced by Cartesian coordinates, in theoretical studies of the vibrational spectra of molecules and their isotopomers and in solving the direct and inverse anharmonic problems. An equation for the quantum-mechanical Hamiltonian of a normal molecule in internal Cartesian coordinates was obtained.     

Role of four-membered rings in C32 fullerene stability and mechanisms of generalized Stone-Wales transformation: a density functional theory investigation

Density functional theory (DFT) methods have been applied to study C(32) fullerenes built from four-, five-, and six-membered rings. The relative energies of pure C(32) fullerenes have been evaluated to locate three most stable structures, 32:D(4d) with two squares, 1:D(3) without square and 5:C(s) with one square. Structural analysis reveals that there is a rearrangement pathway between the lowest energy classical isomer 1:D(3) and the lowest energy non-classical isomer 32:D(4d), and 5:C(s) behaves just as an intermediate between them. The kinetic processes of generalized Stone-Wales transformation (GSWT) with four-membered rings have been explored and two distinct reaction mechanisms are determined by all the transition states and intrinsic reaction coordinates with PBE1PBE/6-31G(d) approach for the first time. One mechanism is the concerted reaction with a rotating dimer closed to the cage surface and another is the stepwise reaction with a carbene-like sp(3) structure, whereas the latter is sorted into two paths based on four-membered ring vanishing before or after the formation of the carbene-like structure. It is indicated that there is no absolute preference for any mechanism, which depends on the adaptability of different reactants on the diverse mechanisms. Furthermore, it's found that the interconversion process with the participation of squares is more reactive than the rearrangement between C(60)_I(h) and C(60)_C(2v), implying some potential importance of non-classical small fullerenes in the fullerene isomerization.     

Predicting Potential Stable Isomers on the Singlet Surface of the [H,P,C,S] System by the MP2 and QCISD(T) Methods

A detailed singlet potential energy surface of [H,P, C,S] system is investigated by means of the MP2 and QCISD(T) methods. Eight isomers are located on the potential energy surface, and at the final QCISD(T)/6-311++G (3df,2p)//MP2/6-311++G(d,p) level with zero-point energy correction, the chainlike isomer HPCS is found to be kinetically and thermodynamically the most stable species followed by the chainlike HSCP, planar three-membered ring HC(S)P, chainlike HCPS, and stereo three-membered ring HP(C)S, which are predicted to be also kinetically stable isomers and should be experimentally observable provided that accurate experimental conditions are available. The dissociation processes from the kinetically and thermodynamically most stable species HPCS to the low-lying molecular dissociation fragments are not more favorable in energy than the isomerization process from HPCS to HSCP. Therefore, the experimental observation for potential isomer HSCP with C ≡ P triple bond is possible by means of photoisomerization technology using HPCS as precursor.     

Direct dynamics simulations using Hessian-based predictor-corrector integration algorithms

In previous research [J. Chem. Phys. 111, 3800 (1999)] a Hessian-based integration algorithm was derived for performing direct dynamics simulations. In the work presented here, improvements to this algorithm are described. The algorithm has a predictor step based on a local second-order Taylor expansion of the potential in Cartesian coordinates, within a trust radius, and a fifth-order correction to this predicted trajectory. The current algorithm determines the predicted trajectory in Cartesian coordinates, instead of the instantaneous normal mode coordinates used previously, to ensure angular momentum conservation. For the previous algorithm the corrected step was evaluated in rotated Cartesian coordinates. Since the local potential expanded in Cartesian coordinates is not invariant to rotation, the constants of motion are not necessarily conserved during the corrector step. An approximate correction to this shortcoming was made by projecting translation and rotation out of the rotated coordinates. For the current algorithm unrotated Cartesian coordinates are used for the corrected step to assure the constants of motion are conserved. An algorithm is proposed for updating the trust radius to enhance the accuracy and efficiency of the numerical integration. This modified Hessian-based integration algorithm, with its new components, has been implemented into the VENUS/NWChem software package and compared with the velocity-Verlet algorithm for the H(2)CO-->H(2)+CO, O(3)+C(3)H(6), and F(-)+CH(3)OOH chemical reactions.     

Ab initio calculation of carbon clusters. II. Relative stabilities of fullerene and nonfullerene C24

Chemical stabilities of six low-energy isomers of C24 derived from global-minimum search are investigated. The six isomers include one classical fullerene (isomer 1) whose cage is composed of only five- and six-membered rings (56-MRs), three nonclassical fullerene structures whose cages contain at least one four-membered ring (4-MR), one plate, and one monocyclic ring. Chemical and electronic properties of the six C24 isomers are calculated based on a density-functional theory method (hybrid PBE1PBE functional and cc-pVTZ basis set). The properties include the nucleus-independent chemical shifts (NICS), singlet-triplet splitting, electron affinity, ionization potential, and gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital (HOMO-LUMO) gap. The calculation suggests that the neutral isomer 2, a nonclassical fullerene with two 4-MRs, may be more chemically stable than the classical fullerene (isomer 1). Analyses of molecular orbital NICS show that the incorporations of 4-MRs into the cage considerably reduce paratropic contributions from HOMO, HOMO-1, and HOMO-2, which are mainly responsible for the sign change in NICS from positive for isomer 1 (42) to negative (-19) for isomer 2, although C24 clusters satisfy neither 4N+2 nor 2(N+1)2 aromaticity rule. Anion photoelectron spectra of four cage isomers, one plate, one monocyclic ring, and one tadpole isomer, as well as three bicyclic ring isomers are calculated. The simulated photoelectron spectra of mono- and bicyclic rings (with C1 symmetry) appear to match the measured HOMO-LUMO gap (between the first and second band in the experimental spectra) [S. Yang et al., Chem. Phys. Lett. 144, 431 (1988)]. Nevertheless, the nonclassical fullerene isomers 3 and 4 apparently also match the measured vertical detachment energy (2.90 eV) reasonably well. These results suggest possible coexistence of nonclassical fullerene isomers with the mono- and bicyclic ring isomers of C24(-) under the experimental conditions.     

Structure, stability, and cluster-cage interactions in nitride clusterfullerenes M3N@C2n (M = Sc, Y; 2n = 68-98): a density functional theory study

Extensive semiempirical calculations of the hexaanions of IPR (isolated pentagon rule) and non-IPR isomers of C(68)-C(88) and IPR isomers of C(90)-C(98) followed by DFT calculations of the lowest energy structures were performed to find the carbon cages that can provide the most stable isomers of M(3)N@C(2n) clusterfullerenes (M = Sc, Y) with Y as a model for rare earth ions. DFT calculations of isomers of M(3)N@C(2n) (M = Sc, Y; 2n = 68-98) based on the most stable C(2n)(6-) cages were also performed. The lowest energy isomers found by this methodology for Sc(3)N@C(68), Sc(3)N@C(78), Sc(3)N@C(80), Y(3)N@C(78), Y(3)N@C(80), Y(3)N@C(84), Y(3)N@C(86), and Y(3)N@C(88) are those that have been shown to exist by single-crystal X-ray studies as Sc(3)N@C(2n) (2n = 68, 78, 80), Dy(3)N@C(80), and Tb(3)N@C(2n) (2n = 80, 84, 86, 88) clusterfullerenes. Reassignment of the carbon cage of Sc(2)@C(76) to the non-IPR Cs: 17490 isomer is also proposed. The stability of nitride clusterfullerenes was found to correlate well with the stability of the empty 6-fold charged cages. However, the dimensions of the cage in terms of its ability to encapsulate M(3)N clusters were also found to be an important factor, especially for the medium size cages and the large Y(3)N cluster. In some cases the most stable structures are based on the different cage isomers for Sc(3)N and Y(3)N clusters. Up to the cage size of C(84), non-IPR isomers of C(2n)(6-) and M(3)N@C(2n) were found to compete with or to be even more stable than IPR isomers. However, the number of adjacent pentagon pairs in the most stable non-IPR isomers decreases as cage size increases: the most stable M(3)N@C(2n) isomers have three such pairs for 2n = 68-72, two pairs for n = 74-80, and only one pair for n = 82, 84. For C(86) and C(88) the lowest energy IPR isomers are much more stable than any non-IPR isomer. The trends in the stability of the fullerene isomers and the cluster-cage binding energies are discussed, and general rules for stability of clusterfullerenes are established. Finally, the high yield of M(3)N@C(80) (Ih) clusterfullerenes for any metal is explained by the exceptional stability of the C(80)(6-) (Ih: 31924) cage, rationalized by the optimum distribution of the pentagons leading to the minimization of the steric strain, and structural similarities of C(80) (Ih: 31924) with the lowest energy non-IPR isomers of C(760(6-), C(78)(6-), C(82)(6-), and C(84)(6-) pointed out.     

Geometric constraints in semiclassical initial value representation calculations in Cartesian coordinates: accurate reduction in zero-point energy

An approach for the inclusion of geometric constraints in semiclassical initial value representation calculations is introduced. An important aspect of the approach is that Cartesian coordinates are used throughout. We devised an algorithm for the constrained sampling of initial conditions through the use of multivariate Gaussian distribution based on a projected Hessian. We also propose an approach for the constrained evaluation of the so-called Herman-Kluk prefactor in its exact log-derivative form. Sample calculations are performed for free and constrained rare-gas trimers. The results show that the proposed approach provides an accurate evaluation of the reduction in zero-point energy. Exact basis set calculations are used to assess the accuracy of the semiclassical results. Since Cartesian coordinates are used, the approach is general and applicable to a variety of molecular and atomic systems.     

Giant motion of La atom inside C82 cage

An anomalous charge density distribution of La atom encapsulated in a C₈₂ cage has been revealed for La@C₈₂ by the maximum entropy method (MEM)/Rietveld analysis using synchrotron powder diffraction data. The obtained La atom charge density shows a feature almost like a bowl or a hemisphere, suggesting that the La atom has a giant motion (large amplitude motion) inside the C₈₂ cage at room temperature. From the obtained MEM charge density, the main results are (1) the cage structure of La@C₈₂ (I) has C_2V symmetry; (2) La atom locates at an off-centered position adjacent to a six-membered ring of the carbon cage; (3) the nearest La–C distance is 2.55(8) and (4) the amount of charge transfer from the La atom to the carbon cage is about 3.2 e, which corresponds to the nominal electronic structure, La³⁺@C₈₂³⁻.     

Isomerization study of C5H5NO molecules

The complex potential energy surface (PES) for the isomerization of C₅H₅NO species, including 18 isomers and 23 interconversion transition states, is probed theoretically at the B3LYP/6‐311++G(d,p) and MP2//B3LYP/6‐311++G(d,p) levels of theory. The geometries and relative energies for various stationary points were determined. The zero‐point vibrational energy (ZPVE) corrections have been made to calculate the reliable energy. We predicted a six‐membered ring structure as a global minima isomer I, which is 118.49 and 131.48 kcal · mol^?1 more stable than the least stable, four‐ and three‐membered ring isomer VIII at B3LYP and MP2//B3LYP levels of theory, respectively. The isomers and interconversion transition states have verified by frequency calculation. The intrinsic reaction coordinates (IRC) calculations have been performed to confirm that each transition state is linked by the desired reactants and products. The isomer stability has been studied using relative energies, chemical hardness, and chemical potential. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

A theoretical study of fullerene-ferrocene hybrids

Using density functional theory within the generalized gradient approximation, I analyzed the electronic structure of a C(60)-ferrocene hybrid [= C(60) (*) FeCp] around HOMO in comparison with that of ferrocene, where C(60) (*) and Cp denote C(60)(CH(3))(5) and a cyclopentadienyl ring. HOMO-LUMO gap is significantly smaller than that of ferrocene because of the intervention of pi(C(60) (*)) states below LUMO. In addition, geometrical and electronic structures of N@C(60) (*) FeCp are also investigated. I find that there are two isomers with the energy difference of 0.13 eV. In one of the two, the encased nitrogen atom is located at the center of the fullerene cage. The Fe atom is eta(5)-coordinated to both Cp and R*, where R* is a five-membered ring of C(60) (*) cage. On the other hand, the atom is coordinated to R* with eta(4)-hapticity, and the nitrogen atom is bonded to a carbon atom of the R* ring in the other isomer. Upon the isomerization between the two isomers, there occurs a partial transfer of spin density between the nitrogen and Fe atoms as well as the creation and breaking of a C-N bond.