Cohesive energy per carbon atom of carbon clusters: monocyclic ring, bicyclic rings and fullerenes. 

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... calculations on ring systems up to C 60 are shown in figure 1. The energies quoted here are cohesive energy per carbon atom. ...

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... fullerenes structures can be considered as finite two-dimensional analogues, in which each carbon is distorted (strained) from its preferred planar configuration. Since the strain should be proportional to the square of the planar distortion angle, δψ, we expect that the strain energy should scale as 1/n We have performed the DFT(Becke/LYP) calculations on C n fullerenes with n = 20, 32 and 60. Figure 1 shows the cohesive energies per carbon atom. ...

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... energies of the structures were computed via the following procedures that combine the DFT with MD as illustrated in figure 3. each of I, II and III, the system is partitioned into two parts: part A involving bond lengths changes, and part B involving continuous deformation. (2) The energy of I is determined directly from figure 1. (3) The energy difference between I and III is a strain energy which can be calculated using the MSX FF. ...

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... of I, II and III, the system is partitioned into two parts: part A involving bond lengths changes, and part B involving continuous deformation. (2) The energy of I is determined directly from figure 1. (3) The energy difference between I and III is a strain energy which can be calculated using the MSX FF. (4) The energy difference between III and II is calculated in two parts. (a) part A: DFT calculations are carried out on the reaction of two C 6 H 2 molecules to form to the 4-membered ring, C 12 H 2 . (b) part B: we calculate the corresponding change in going from III to II using MSX FF. Then we combine A and B to get the energy difference between III and II. (5) Thus the energy of C 60 bicyclic ring (II) can be calculated by II–III–I. We extend the MSX FF to include terms capable of describing the different bonding schemes. The key components are the additive energy terms for the dangling bond and the energy cost for bending a triple bond to form a 1,2-benzyne. Our FF are defined as follows: E tot (n 2 ) = E bond + E radical + E strain = n 2 ( 1 − 2 ) + d 1 n R + d 2 n σ π + E str (n 2 ). We have chosen E 0 = 60 1 , as zero point. Here, n 2 is the sp 2 bonded carbons, n 2 ( 1 − 2 ) gives the energy gained by converting sp 1 bonded carbon into sp 2 bonded carbon, with 1 = − 6 . 56 eV and 2 = − 7 . 71 eV. d 1 is the energy of a dangling bond relative to the σ -bonded state, n R number of such dangling bond(radicals); d 2 is the energy of an atom participating bended planar π -bond relative to the σ -bonded state and n σ π is the number of such atoms. We use the Benson-like scheme to evaluate d 1 and d 2 (Guo 1992) and found d 1 = 2 . 32 eV and d 2 = 1 . 64 eV. E str (n 2 ) is the strain energy and it is evaluated at the minimum energy structure. We use the fine model for the initial steps in the C 60 formation. As the reaction takes off and begins to release more and more energy, we switch to the coarse one. At the beginning, atomic carbons combine themselves to form dimers and trimer, C 2 , C 3 . These would then grow into linear chain of carbons C n , etc, for n < 10 (Hutter et al 1994). When n > 10 the carbon clusters prefer ring structure (Hutter et al 1994) because beyond n > 10 the energy gain in killing the dangling bonds at the two ends overcompensates for the strain energy incurred by folding up the chain. At around n > 30 the ring structures give way to fullerene structures (von Helden et al 1993a, b) because replacing more π -bonds by σ -bonds overcompensates for the strain of folding the 2D net. One process of C 60 formation, as suggested by Jarrold’s experiments (von Helden et al 1993a, b, Hunter et al 1994) is to combine two C 30 rings to form a bicyclic C 60 ring, which in turn is isomerized into a C 60 fullerene. This unimolecular reaction will be the focus of our study. As a mnemonic for referring to the various structures, we will simply denote the ring sizes of a structure. Thus the simple C 60 ring is denoted as 60, while the double ring system, 1, is 30 + 4 + 30. This notation does not uniquely describe a structure, but is for the species we will consider. We take the reference energy to be E o = 60 1 , where 1 = − 6 . 56 eV. Following Jarrold, the first few steps in the reaction are as follows (see figure ...

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... A of difficulty the most puzzling in current aspects experiments of fullerenes (Hunter (C 60 et , C al 70 1993, , etc) 1994, is how von such Helden complicated et al 1993a, symmetric b) is molecules that the products are formed can only from be a gas detected of atomic on time carbons, scales namely, of μ s, long the atomistic after many or of chemical the important mechanism. formation Are the steps atoms have added been one completed. by one Consequently, or as molecules it is (C necessary 2 , C 3 )? to Is use the first C 60 principles fullerene quantum formed mechanical by adding C theory 1 , C 2 , to or determine C 3 to some these smaller initial states; fullerene however, or is the C 60 experiments formed by isomerization serve to provide of boundary some type conditions of precursor that severely molecule limit C 60 ? the Is there possibilities, a critical making nucleus the beyond use of which first principles formation proceeds theory practicable. at gas kinetic rates? What determines the balance bwtween forming buckyballs, buckytubes, graphite, and soot? The answer to these questions might lead to means of manipulating the systems to achieve particular products. A difficulty in current experiments (Hunter et al 1993, 1994, von Helden et al 1993a, b) is that the products can only be detected on time scales of μ s, long after many of the important formation steps have been completed. Consequently, it is necessary to use first principles quantum mechanical theory to determine these initial states; however, the experiments serve to provide boundary conditions that severely limit the possibilities, making the use of first principles theory practicable. We selected density functional theory (DFT) as the best compromise between accuracy and speed for studying these systems. We use the Becke gradient corrected exchange and the gradient corrected correlation functional of Lee, Yang, and Parr (Johnson et al 1993). The calculations were carried out using Jaguar program (PS-GVB) with the 6-31G* basis set (Rignalda et al 1995). Because the carbon rings play a central role, we studied how the structures and energetics of such rings changed with size and extracted a force field (denoted as the MSX FF) that would reproduce the DFT energetics and structures. This MSX FF would be used later in conjunction with the DFT calculations on various multi-ring systems to estimate the energetics of the full 60-atom systems without the necessity of DFT on the complete system. The calculations on ring systems up to C 60 are shown in figure 1. The energies quoted here are cohesive energy per carbon atom. In calculating these energies we used as our reference the triplet C atom, calculated by LSDA. We found that • For n = 4 m , the minimum energy structure has a polyacytelene geometry of alternating single and triple bonds. The bond length difference is from 0.5 to 0.9 Å. We find that inclusion of correlation reduces the dimerization amplitude, similar to the case ...

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... q r 1 (l) = R i (l) − R i 0 (l) is the bond strain term, where for n = 4 m , i = 1 is the triple bond and i = 2 is the single bonds; for n = 4 m + 2 their bonds are equivalent. The angle strain term is q θ (l) = θ (l) − θ 0 (l) We use the periodic boundary condition so that, with θ = π , n/ 2 + 1 = 1 where n is the total number of atoms in the system and n/ 2 is the number of unit cells. E 0 is a reference energy corresponding to zero strain energy structure (infinite linear chain). Comparing E MSX with E DFT for several structures, we can derive the force field parameters. In a similar fashion we can derive the force field for the sp 2 bonded carbons. The optimum structure for bulk carbon is graphite, which has each carbon bonded to three others (sp 2 bonding) to form hexagonal sheets stacked on each other. The fullerenes structures can be considered as finite two-dimensional analogues, in which each carbon is distorted (strained) from its preferred planar configuration. Since the strain should be proportional to the square of the planar distortion angle, δψ , we expect that the strain energy should scale as 1 /n We have performed the DFT(Becke/LYP) calculations on C n fullerenes with n = 20, 32 and 60. Figure 1 shows the cohesive energies per carbon atom. Extrapolating the calculated cohesive energy to n → ∞ leads to a cohesive energy per sp 2 carbon of E coh ( sp 2 ) = 7 . 71 eV. This can be compared to the experimental cohesive energy of a single graphitic sheet of E coh sheet = 7 . 74 eV. This is derived from the experimental cohesive energy (CRC Handbook) of graphite of E graphite = 7 . 8 eV plus total Van der Waals attraction of E vdw = 0 . 056 eV between sheets calculated using the graphite force field (Guo 1992). Now that we have the energy and force field of both sp 1 and sp 2 hybridized carbon we can get the energetics of any carbon clusters. Adding the entropic contribution within the harmonic approximation using FF, we get the free energy of various species at different temperature, which dictates the thermal equilibrium distribution of these species. Our population analysis is shown in figure 2. For studying formation reaction sequence we adopt two level of models, a fine one and a coarse one, as explained below. The energies of the structures were computed via the following procedures that combine the DFT with MD as illustrated in figure 3. (1) The reaction from two C 30 ring (I) to C 60 bicyclic ring molecules (II) is achieved via an intermediate III. ...