## 3 Results and Discussions
To understand the process clearly, the Nudged Elastic Band (NEB) method [38] was applied to investigate the energy profile for the whole reaction process of an incident H atom approaching the hydrocarbon chain and capturing a hydrogen atom, then forming a H\({}_{2}\) molecule and leaving the hydrocarbon chain. The incident H atom approaches the chain along the \(x\)-axis (Figure 3). It must pass over an energy barrier, about 1.4 eV, to reach the position where it can bond with the hydrogen atom in the hydrocarbon chain. Note, that even if the H\({}_{2}\) molecule reaches the bonding position, it cannot cleave the H atom from the chain, as indicated by the above analysis (Figure 2b). Therefore, the atomic hydrogen beam (AHB) [39; 40] with a proper incident energy could be used as a hammer to knock off the hydrogen atom from the hydrocarbon chain, which offers better performance than H\({}_{2}\) molecule proposed by [PERSON] et al. [27]. In the next step, we built a model of two infinitely long hydrocarbon chains in parallel along the \(z\)-direction to simulate the cross-linking effect. Each of the hydrocarbon chains had one hydrogen atom removed in the supercell, called R2 configuration. The initial and relaxed structures are shown in Figure 4a,b, respectively. After structural relaxation, the two carbon atoms with dangling bonds approached each other and form a C-C bond (Figure 4b). The bond length was 1.63 A, comparable with the typical C-C single bond length (1.52 A) in alkane chains. The positions of the adjacent carbon atoms and their bonded hydrogen atoms also deviated much from the ideal case, which made the cross-linking possible. The total energy difference for the configurations in Figure 4a,b was \(-\)89.87 eV (E\({}_{4(b)}\) \(-\) E\({}_{4(a)}\)), which shows that the cross-linking does help to stabilize the system. The charge-density difference is illustrated in Figure 4b, indicating that the electron accumulates in the cross-linking region, helping the formation of C-C direct bond. For comparison, we also calculated the two hydrocarbon chains, in which only one chain has a hydrogen atom removed (called the R1 configuration), and the two hydrocarbon chains with two hydrogen atoms per supercell removed from each chain individually (called R4). The calculations show that for the R1 case, no cross-linking was observed. Hence, the removal of at least one hydrogen atom per supercell from each chain (R2) was the minimum requirement  
Figure 2: (Color online) The difference of the charge densities between the hydrocarbon chain with H (**a**) or H\({}_{2}\) (**b**) adsorbed at site s1 and the pristine hydrocarbon chain. The blue and white spheres are the carbon and hydrogen atoms, respectively. for the cross-linking between two hydrocarbon chains. While for the R4 configuration, the cross-linking happens, however, owing to the higher energy needed to cleave four hydrogen atoms totally per supercell, we suppose that this case is difficult to achieve in experiments, compared with the R2 configuration. Figure 4: (Color online) The initial (**a**) and relaxed (**b**) structures of two infinitely long hydrocarbon chains in parallel along the \(z\)-axis. The two dotted lines in (**a**) show the borders of the supercell along the \(z\) direction. For each hydrocarbon chain, one hydrogen atom in the supercell is removed. Brown and pink spheres represent the carbon and hydrogen atoms, respectively. The charge-density difference is also given for the relaxed structure in (**b**). The green and yellow colors illustrate the electron accumulation and depletion regions, respectively. (**c,d**) are the initial and relaxed structures of two finitely long molecules (C\({}_{20}\)H\({}_{2}\)) in ’head-to-head’ cross-linking mode. The charge-density difference for the relaxed structure is also given in (**d**). For each molecule, one hydrogen atom is removed. Figure 3: (Color online) The calculated minimum energy path (MEP). Blue circles correspond to NEB images.