Quantum Monte Carlo study on the structures and energetics of cyclic and linear carbon clusters ${\mathrm{C}}_n$ ($n$ = 1,...,10)

Using fixed-node diffusion quantum Monte Carlo (DMC) simulation we investigate the structural properties and energetics of the linear and cyclic carbon clusters ${\mathrm{C}}_n$ for $n\le 10$. We calculate the binding energy, the electron correlation energy, the dissociation energy, and the second difference in energy. We also present an analysis of the structural properties of the clusters. It is found that the bond lengths, binding energies, and dissociation energies obtained from the DMC calculations are in excellent agreement with the available experimental results. The electron-correlation contribution to the binding energy indicates that in the case of the linear isomers, the clusters of odd-number size are relatively more favored than their neighbors of even-number size, whereas for the cyclic isomers, we do not observe the oscillation pattern. In the range of cluster size under investigation, we find that the electron-correlation impact in the binding energy of the cyclic clusters is larger than that of the corresponding linear ones, varying from $30\%$ to $40\%$ of the binding energy values. The electron correlation is also essential to the stability of the clusters enhancing by up to $52\%$ their dissociation energy. A comparative analysis of the dissociation energy and second difference in energy indicates that the linear isomers ${\mathrm{C}}_3$ and ${\mathrm{C}}_5$ are the most stable ones.

Figure 1

The cyclic and linear structures of the ${\mathrm{C}}_n$ clusters obtained by DFT with B3LYP using the gaussian03 program. The main bond lengths are given in Table 1.

Figure 2

The weighted average bond length ($d_{\mathrm{av}}$) and effective coordination number (ECN) of the carbon clusters as a function of the cluster size. The blue triangle and red circles are calculation results for the cyclic and linear structures, respectively. The solid lines are only guides to the eyes.

Figure 3

The average correlation energy per electron in the clusters. The black diamond, blue triangles, and red circles are results for the carbon atom and cyclic and linear clusters, respectively.

Figure 4

Total magnetic moment per atom of the linear (red circles) and cyclic (blue triangles) clusters as a function of the cluster size. The black diamond shows the magnetic moment of a carbon atom.

Figure 5

The binding energy per atom of the linear and cyclic structures as a function of the cluster size $n$. The DMC and DFT-B3LYP results are from this work, the DFT-PBE binding energy obtained by Mauney et al. [38], and the experimental values from Refs. [27] and [37]. The inset shows the difference between the binding energies of the linear and cyclic clusters.

Figure 6

The electron-correlation contribution to the binding energy per atom as a function of the cluster size $n$. The blue triangles and red circles are results for cyclic and linear structures, respectively.

Figure 7

(a) The dissociation energy and (b) the second difference in energy of the more stable carbon clusters as a function of the cluster size $n$. The blue lozenges and green squares are the DMC and HF results, respectively, with the optimized ground-state structures. In (a) the black stars show the experimental values from Ref. [39] and the red crosses indicate the DMC results within the linear structures.