Structures and Stabilities of Carbon Chain Clusters Influenced by Atomic Antimony

The C-C bond lengths of the linear magnetic neutral CnSb, CnSb+ cations and CnSb− anions are within 1.255–1.336 Å, which is typical for cumulene structures with moderately strong double-bonds. In this report, we found that the adiabatic ionization energy (IE) of CnSb decreased with n. When comparing the IE~n relationship of CnSb with that of pure Cn, we found that the latter exhibited a stair-step pattern (n ≥ 6), but the IE~n relationship of CnSb chains took the shape of a flat curve. The IEs of CnSb were lower than those of corresponding pure carbon chains. Different from pure carbon chains, the adiabatic electron affinity of CnSb does not exhibit a parity effect. There is an even-odd alternation for the incremental binding energies of the open chain CnSb (for n = 1–16) and CnSb+ (n = 1–10, when n > 10, the incremental binding energies of odd (n) chain of CnSb+ are larger than adjacent clusters). The difference in the incremental binding energies between the even and odd chains of both CnSb and pure Cn diminishes with the increase in n. The incremental binding energies for CnSb− anions do not exhibit a parity effect. For carbon chain clusters, the most favorable binding site of atomic antimony is the terminal carbon of the carbon cluster because the terminal carbon with a large spin density bonds in an unsaturated way. The C-Sb bond is a double bond with Wiberg bond index (WBI) between 1.41 and 2.13, which is obviously stronger for a carbon chain cluster with odd-number carbon atoms. The WBI of all C-C bonds was determined to be between 1.63 and 2.01, indicating the cumulene character of the carbon chain. Generally, the alteration of WBI and, in particular, the carbon chain cluster is consistent with the bond length alteration. However, the shorter C-C distance did not indicate a larger WBI. Rather than relying on the empirical comparison of bond distance, the WBI is a meaningful quantitative indicator for predicting the bonding strength in the carbon chain.


Introduction
Small magnetic carbon clusters and related carbon-based materials have attracted much attention on the account of their important role in astrophysics, terrestrial processes, electronic spintronics, catalysis, and chemical engineering [1][2][3][4][5]. A deep understanding of the carbon nanostructure with adjustable bonding should facilitate the designing and synthesizing of size-and morphology-controlled carbon-based functional materials [6][7][8][9]. Previous theoretical and experimental studies have revealed that small carbon clusters               We mainly focused on the linear carbon chain clusters initially because many ported carbon clusters mainly adopt a linear shape as the ground state structure. For t antimony-doped larger carbon clusters (n > 10), we conducted density functional calc lations to obtain the energies of ring structures. The total energies of C11Sb and C14 with a ring structure are 0.35 and 0.11 eV higher in energy than linear C11Sb and C14 clusters, while C12Sb, C13Sb, C15Sb, and C16Sb clusters with a ring structure are −0. −0.57, −0.53, and −0.83 eV lower in energy. Although the linear carbon cluster is not ways the global minima in the potential surface, the linear carbon clusters possess practical significance that the reactivity of carbon clusters increases dramatically w the increasing number of consecutive cumulene-like double bonds, and the linear carb clusters can be formed in space-confined nanotube materials [33][34][35]. The linear carb chain cluster with different lengths and doping atoms is a special allotrope type of c bon material different from graphite, diamond, graphene, carbon nanotubes, and full enes.
We made comparisons between 6-31G(d) and def2-TZVP basis sets for select structural parameters, C-Sb stretching frequencies, and HOMO-LUMO gaps in Table  The C-C bonds adjacent antimony atoms of C4Sb, C5Sb, C6Sb, C10Sb, and C11Sb show on  We mainly focused on the linear carbon chain clusters initially because many reported carbon clusters mainly adopt a linear shape as the ground state structure. For the antimony-doped larger carbon clusters (n > 10), we conducted density functional calculations to obtain the energies of ring structures. The total energies of C 11 Sb and C 14 Sb with a ring structure are 0.35 and 0.11 eV higher in energy than linear C 11 Sb and C 14 Sb clusters, while C 12 Sb, C 13 Sb, C 15 Sb, and C 16 Sb clusters with a ring structure are −0.14, −0.57, −0.53, and −0.83 eV lower in energy. Although the linear carbon cluster is not always the global minima in the potential surface, the linear carbon clusters possess a practical significance that the reactivity of carbon clusters increases dramatically with the increasing number of consecutive cumulene-like double bonds, and the linear carbon clusters can be formed in space-confined nanotube materials [33][34][35]. The linear carbon chain cluster with different lengths and doping atoms is a special allotrope type of carbon material different from graphite, diamond, graphene, carbon nanotubes, and fullerenes.
We made comparisons between 6-31G(d) and def2-TZVP basis sets for selected structural parameters, C-Sb stretching frequencies, and HOMO-LUMO gaps in Table 7. The C-C bonds adjacent antimony atoms of C 4 Sb, C 5 Sb, C 6 Sb, C 10 Sb, and C 11 Sb show only slight differences less than 0.011 Å, and the C-Sb bond distances are almost the same through using def2-TZVP and 6-31G(d) basis sets. C-Sb stretching frequencies tend to show negligible differences for the larger C 11 Sb and C 12 Sb. Although the C-Sb stretching frequency differences for C 4 Sb and C 5 Sb, it is slightly larger (6.2 cm −1 and 3.9 cm −1 ) under different basis sets, and we can clearly recognize that the theoretically predicted vibration modes are the same by comparing normal coordinates. The HOMO-LUMO gaps for C 4 Sb, C 5 Sb, C 6 Sb, C 10 Sb, and C 11 Sb are 1.27, 1.25, 1.17, 1.03, and 1.01 eV with 6-31G(d), while the HOMO-LUMO gaps are nearly identical with def2-TZVP basis set (1.26, 1.27, 1.16, 1.04, and 0.99 eV, respectively). Table 7. The selected structural parameters (the C-C bond adjacent to Sb and the C-Sb bond in Å), C-Sb stretching frequencies (in cm −1 ), and HOMO-LUMO gaps (in eV) for results obtained at the B3LYP/6-31G(d)/SDD and B3LYP/def2-TZVP/SDD level (denoted as normal and italic fonts, respectively) for C 4 Sb, C 5 Sb, C 6 Sb, C 10 Sb, and C 11 Sb.
Molecules 2021, 26, x FOR PEER REVIEW 9 of 22 The frontier molecular orbitals are depicted in Figure 7 with an isovalue of 0.02. Except for the CSb molecule, the LUMO of the other carbon chain clusters exhibits the same orbital shape as HOMO. The smallest CSb shows the σ-character of HOMO, while all the other CnSb chain clusters show π-character HOMO with overlapping p orbitals shoulder to shoulder. For HOMO, the p orbital of the terminal carbon of the even-numbered carbon cluster always presents a different sign from the p orbital of antimony. While the p orbital of the terminal carbon of odd-numbered carbon cluster exhibits the same sign with a p orbital of antimony, it can form π bonding orbital with antimony. Therefore, the C-Sb bond of CnSb with even-numbered carbon should be weaker than the C-Sb bond of Cn ± 1Sb with odd-numbered carbon. This result agrees well with the Wiberg bond index analysis that the C-Sb WBI of CnSb with an even n tends to be smaller than the C-Sb WBI of CnSb with odd n.

Electronic Properties
As is known, the ion signal intensity in a mass spectrum is related to the electron affinity (EA) or ionization energies (IE, also called electron detachment energy) of the

Electronic Properties
As is known, the ion signal intensity in a mass spectrum is related to the electron affinity (EA) or ionization energies (IE, also called electron detachment energy) of the molecule. Thus, adiabatic ionization energies, defined as the energy required to remove an electron from the neutral clusters with a geometric change, are important parameters to understand the relative stability of the antimony-doped clusters with different sizes. Usually, there are three types of IE: Koopmans IE, vertical IE, and adiabatic IE. Koopmans IE is the HOMO energy, vertical IE is the energy difference between the neutral and ionic clusters at the neutral equilibrium geometry, and adiabatic IE is the energy difference between the neutral and ionic clusters at their respective equilibrium geometry (i.e., IE = E(optimized cation) − E(optimized neutral)). In this work, the adiabatic IE of C n Sb and C n clusters for their optimized structures were calculated and shown in Table 4 and Figure 8. As a whole, the ionization energies decrease with the size of the clusters, suggesting that larger C n Sb chains become less stable, e.g., when exposed to a strong electrical field or high temperature. As shown in Figure 8, the adiabatic IE of pure carbon chains C n has a stair-step shape (n ≥ 6), whereas C n Sb chains take the shape of a flat curve with a smaller gradient than pure carbon chains. The adiabatic IEs of the even and odd pure carbon chains (n ≥ 6) follow some nonlinear relationship; the IEs of carbon chains are larger than that of corresponding C n Sb, but the energy difference between them (IE Cn -IE CnSb ) decreases when n increases. The electron affinity (EA) of C n Sb is defined as the energy released when an electron is attached to neutral C n Sb:  This property is also related to the stability of the molecule in the TOF experiment and molecule. A higher EA means that more energy is released when an electron is added to a neutral molecule, and the generation of the corresponding anion is more readily performed. The calculated adiabatic EA data are presented in Table 4. In Figure 9, we compared the adiabatic EAs of C n Sb with that of pure carbon chains: the EAs of carbon chains have a clear alternation parity effect, the EAs of carbon chains with an even n are larger than the EAs of adjacent n-odd members but carbon chains doped with the antimony atom do not exhibit a parity effect. The computed EAs of the even and odd carbon chains followed a non-linear relationship with the number of carbons, respectively. The energy difference between EA Cn and EA CnSb was much smaller than (IE Cn -IE CnSb ), except when n = 1, and EA Cn was slightly larger than EA CnSb when n ≥ 10.    (Table 9). Without doping antimony, the two anomeric carbons at the right and the left end of the pure carbon chain show an identical charge value. With the antimony doping, the anomeric carbon shows significant charge differences (Table 9 and Figure 10). The charge differences between anomeric carbon atoms are calculated to be 0. 43    The Wiberg bond indices of CnSb are listed in Figure 11. The C-Sb bond is a double bond with WBI between 1.41 and 2.13, which is obviously stronger for a carbon chain The Wiberg bond indices of C n Sb are listed in Figure 11. The C-Sb bond is a double bond with WBI between 1.41 and 2.13, which is obviously stronger for a carbon chain cluster with odd-number carbon atoms (compared with neighboring carbon chain clusters with even-number carbon atoms). The WBI of all C-C bonds was determined to be between 1.63 and 2.01, indicating the cumulene character of the carbon chain. Generally, the alteration of WBI and, in particular, the carbon chain cluster is consistent with the bond length alteration. However, the shorter C-C distance did not indicate a larger WBI. For example, the largest WBI for C n Sb (n > 1) was calculated to be the terminal C-C bond index, while this terminal C-C bond was not the shortest C-C bond. Therefore, rather than relying on the empirical comparison of bond distance, the WBI is a meaningful quantitative indicator for predicting the bonding strength in the carbon chain. than relying on the empirical comparison of bond distance, the WBI is a meaningful quantitative indicator for predicting the bonding strength in the carbon chain.

Incremental Energies, Fragmental Energies, and Binding Sites
The relative stability of clusters can be also analyzed in terms of the energy differences between the neighboring size of the clusters, which is correlated with the "magic number" in cluster science [36]. Energy differences between CnSb and Cn−1Sb, CnSb + and Cn−1Sb + , CnSb − and Cn−1Sb − (differential energies △En, defined as △En = En − En−1) are listed in Tables 5-7, respectively. For the clusters with different sizes, the concept of the incremental binding energy, labeled as △E I , was introduced to compare their relative stabilities. As suggested by Pascoli and Lavendy [37,38], △E I is just the reaction energy of the following processes: They can be computed as the consecutive binding energy (BE, atomization energy, listed in Tables 5-7) differences between the adjacent clusters, ΔE I (CnSb/CnSb + / CnSb − ) = BE(CnSb/CnSb + /CnSb − ) − BE(Cn−1Sb/Cn−1Sb + /Cn−1Sb − ) Where BE can be defined as the energy difference between a molecule and its component atoms: The results for the incremental binding energy as a function of the number of carbon atoms for the different open-chain CnSb/CnSb + /CnSbclusters and pure carbon chains are shown in Figures 12-14. It can be observed that there is an even-odd alternation for

Incremental Energies, Fragmental Energies, and Binding Sites
The relative stability of clusters can be also analyzed in terms of the energy differences between the neighboring size of the clusters, which is correlated with the "magic number" in cluster science [36]. Energy differences between C n Sb and C n−1 Sb, C n Sb + and C n−1 Sb + , C n Sb − and C n−1 Sb − (differential energies ∆E n , defined as ∆E n = E n − E n−1 ) are listed in Tables 5-7, respectively. For the clusters with different sizes, the concept of the incremental binding energy, labeled as ∆E I , was introduced to compare their relative stabilities. As suggested by Pascoli and Lavendy [37,38], ∆E I is just the reaction energy of the following processes: They can be computed as the consecutive binding energy (BE, atomization energy, listed in Tables 5-7) differences between the adjacent clusters, ∆E I (C n Sb/C n Sb + / C n Sb − ) = BE(C n Sb/C n Sb + /C n Sb − ) − BE(C n−1 Sb/C n−1 Sb + /C n−1 Sb − ) where BE can be defined as the energy difference between a molecule and its component atoms: BE(C n Sb/ C n Sb + / C n Sb − ) = nE(C) + E(Sb) − E(C n Sb/ C n Sb + / C n Sb − ) The results for the incremental binding energy as a function of the number of carbon atoms for the different open-chain C n Sb/C n Sb + /C n Sb − clusters and pure carbon chains are shown in Figures 12-14. It can be observed that there is an even-odd alternation for the open chain C n Sb and C n Sb + (n = 1-10, when n > 10, the parity effect is less obvious and n-odd members of C n Sb + are slightly larger), with even species being comparatively more stable than odd ones; the parity variation tendency of neutral C n Sb is opposite to that of pure C n (i.e., n-even carbon chains are less stable than adjacent n-odd ones) and the variation amplitude is much smaller than pure C n . The difference in ∆E I between the even and odd species of both C n Sb and pure C n chains diminishes with the increase in carbon atoms; different from C n anions, the ∆E I for C n Sb − anions did not exhibit a parity effect. However, these patterns of ∆E I for C n Sb + and C n Sb − cannot be simply explained by the "valence π-electron number" rule. the open chain CnSb and CnSb + (n = 1-10, when n > 10, the parity effect is less obvious and n-odd members of CnSb + are slightly larger), with even species being comparatively more stable than odd ones; the parity variation tendency of neutral CnSb is opposite to that of pure Cn (i.e., n-even carbon chains are less stable than adjacent n-odd ones) and the variation amplitude is much smaller than pure Cn. The difference in ΔE I between the even and odd species of both CnSb and pure Cn chains diminishes with the increase in carbon atoms; different from Cnanions, the ΔE I for CnSbanions did not exhibit a parity effect. However, these patterns of ΔE I for CnSb + and CnSbcannot be simply explained by the "valence π-electron number" rule.    the open chain CnSb and CnSb + (n = 1-10, when n > 10, the parity effect is less obvious and n-odd members of CnSb + are slightly larger), with even species being comparatively more stable than odd ones; the parity variation tendency of neutral CnSb is opposite to that of pure Cn (i.e., n-even carbon chains are less stable than adjacent n-odd ones) and the variation amplitude is much smaller than pure Cn. The difference in ΔE I between the even and odd species of both CnSb and pure Cn chains diminishes with the increase in carbon atoms; different from Cnanions, the ΔE I for CnSbanions did not exhibit a parity effect. However, these patterns of ΔE I for CnSb + and CnSbcannot be simply explained by the "valence π-electron number" rule.   The fragmentation energies accompanying channels DN1, DC1, and DA1, i.e., △E I , have been discussed. In addition, the fragmentation energies for many other dissociation reactions are calculated and exhibited in Figures 15-17, including the following seven channels for neutral CnSb clusters: The fragmentation energies accompanying channels DN1, DC1, and DA1, i.e., E I , have been discussed. In addition, the fragmentation energies for many other dissociation reactions are calculated and exhibited in Figures 15-17, including the following seven channels for neutral C n Sb clusters: channel is the loss of the Sb + ion (channel DC8). However, when n ≥ 10, losing an antimony atom (DC4) becomes the dominant channel. The most favorable dissociation pathway for CSband CnSb -(n = 2-16) is the loss of the Sbion and the loss of the antimony atom, respectively. The most favorable dissociation channels for CnSb/CnSb + /CnSbare illustrated in Figure 18, from which we can draw some conclusions: the fragmentation energies for DN4 exhibit a parity effect, i.e., the even CnSb clusters are more stable while the odd CnSb clusters are easy to dissociate; the fragmentation energies for DC8 (n = 1-9) also showed an alternation effect with the same alternation trend as DN4. the fragmentation energies for DC4 did not show fluctuation or a decrease monotonically as the n number rose; when n > 4 CnSb + , is more stable than CnSbanions and neutral CnSb. The subtle alternation for DA7 exists when n ≤ 10; when n > 10, the CnSbanion is in its least stable form compared with CnSb and CnSb + .  channel is the loss of the Sb + ion (channel DC8). However, when n ≥ 10, losing an antimony atom (DC4) becomes the dominant channel. The most favorable dissociation pathway for CSband CnSb -(n = 2-16) is the loss of the Sbion and the loss of the antimony atom, respectively. The most favorable dissociation channels for CnSb/CnSb + /CnSbare illustrated in Figure 18, from which we can draw some conclusions: the fragmentation energies for DN4 exhibit a parity effect, i.e., the even CnSb clusters are more stable while the odd CnSb clusters are easy to dissociate; the fragmentation energies for DC8 (n = 1-9) also showed an alternation effect with the same alternation trend as DN4. the fragmentation energies for DC4 did not show fluctuation or a decrease monotonically as the n number rose; when n > 4 CnSb + , is more stable than CnSbanions and neutral CnSb. The subtle alternation for DA7 exists when n ≤ 10; when n > 10, the CnSbanion is in its least stable form compared with CnSb and CnSb + .    C n Sb → C n−2 Sb + C 2 (DN2) C n Sb → C n−3 Sb + C 3 (DN3) C n Sb → C n + Sb (DN4) C n Sb → C n−1 + CSb (DN5) C n Sb → C n−2 + C 2 Sb (DN6) C n Sb → C n−3 + C 3 Sb (DN7) C n Sb → C n -+ Sb + (DN8) The following ten channels for cationic CnSb+ cations: C n Sb + → C n−2 Sb + + C 2 (DC2) C n Sb + → C n−3 Sb + + C 3 (DC3) C n Sb + → C n + + Sb (DC4) C n Sb + → C n−1 + + CSb (DC5) C n Sb + → C n−2 + + C 2 Sb (DC6) C n Sb + → C n−3 + + C 3 Sb (DC7) C n Sb + → C n + Sb + (DC8) C n Sb + → C n−1 + CSb + (DC9) C n Sb + → C n−2 + C 2 Sb + (DC10) C n Sb + → C n−3 + C 3 Sb + (DC11) and the following thirteen channels for anionic C n Sb − anions: These channels can be divided into four types: (1) losing neutral small carbon particles such as C, C 2 , or C 3 ; (2) losing neutral antimony-contained small fragments such as Sb, CSb, C 2 Sb, or C 3 Sb; (3) losing ionic defects, such as Sb + /Sb − , CSb + /CSb − , C 2 Sb + /C 2 Sb − , or C 3 Sb + /C 3 Sb − fragments (for C n Sb + /C n Sb − ); and (4) losing anionic carbons, such as C − , C 2 − , and C 3 − fragments (only for C n Sb − ). Comparing the fragmentation energies can help us to find some dominant channels for each kind of cluster in the discussion.
The fragmentation energies of channels DN1, DC1, and DA1 are also included for comparison. It is clear that losing an antimony atom is the dominant channel for neutral C n Sb (channel DN4). For small C n Sb + cations (n = 1-9), the most favorable fragmentation channel is the loss of the Sb + ion (channel DC8). However, when n ≥ 10, losing an antimony atom (DC4) becomes the dominant channel. The most favorable dissociation pathway for CSb − and C n Sb − (n = 2-16) is the loss of the Sb − ion and the loss of the antimony atom, respectively. The most favorable dissociation channels for C n Sb/C n Sb + /C n Sb − are illustrated in Figure 18, from which we can draw some conclusions: the fragmentation energies for DN4 exhibit a parity effect, i.e., the even C n Sb clusters are more stable while the odd C n Sb clusters are easy to dissociate; the fragmentation energies for DC8 (n = 1-9) also showed an alternation effect with the same alternation trend as DN4. the fragmentation energies for DC4 did not show fluctuation or a decrease monotonically as the n number rose; when n > 4 C n Sb + , is more stable than C n Sb − anions and neutral C n Sb. The subtle alternation for DA7 exists when n ≤ 10; when n > 10, the C n Sb − anion is in its least stable form compared with C n Sb and C n Sb + .  For carbon chain clusters, the most favorable binding site of atomic antimony is the terminal carbon of the carbon cluster because the terminal carbon with a large spin density bonds in an unsaturated way. We have constructed initial structures with antimony binding on nonterminal carbon and found that the optimization of these structures often encounters a convergence problem or loop back to the linear structure with antimony binding to the terminal carbon. For the structures which encounter convergence problems or converge to geometry with antimony binding on nonterminal carbon, we further optimized them and obtained the local minima ( Figure 19). For antimony binding on C2, the 2a nonlinear structure showed a C-C distance of 1.315 Å , longer than the linear structure. The results indicate that the construction of the nonlinear shape CnSb can further activate the C-C bond. Nonlinear C3Sb and C4Sb show C2V symmetric structures. The C-C bonds of C3Sb are all 1.335 Å , while the C-C bond adjacent to antimony shows a substantially longer distance of 1.380 Å . Nonlinear C5Sb with antimony binding on the For carbon chain clusters, the most favorable binding site of atomic antimony is the terminal carbon of the carbon cluster because the terminal carbon with a large spin density bonds in an unsaturated way. We have constructed initial structures with antimony binding on nonterminal carbon and found that the optimization of these structures often encounters a convergence problem or loop back to the linear structure with antimony binding to the terminal carbon. For the structures which encounter convergence problems or converge to geometry with antimony binding on nonterminal carbon, we further optimized them and obtained the local minima ( Figure 19). For antimony binding on C 2 , the 2a nonlinear structure showed a C-C distance of 1.315 Å, longer than the linear structure. The results indicate that the construction of the nonlinear shape C n Sb can further activate the C-C bond. Nonlinear C 3 Sb and C 4 Sb show C 2V symmetric structures. The C-C bonds of C 3 Sb are all 1.335 Å, while the C-C bond adjacent to antimony shows a substantially longer distance of 1.380 Å. Nonlinear C 5 Sb with antimony binding on the second or third carbon does not exhibit bond length alteration. Nonlinear C 6 Sb and C 7 Sb with antimony binding on the third carbon or binding on the bridge site between the third and fourth carbon do not exhibit bond length alteration effects. The C-C bond distances adjacent to antimony are calculated to be 1.346 and 1.360 Å for 5a and 5b, respectively, which are much longer than 1.288 Å for linear C 5 Sb. The C-C bond distances adjacent to the antimony of 6a and 6b are calculated to be 1.355 and 1.440 Å, which is much longer than 1.277 Å for the linear carbon chain C 6 Sb. The bond length alteration does not exist for nonlinear C 7 Sb. The antimony binding on the second or third carbon of C 7 leads to the reconstruction and formation of the linear carbon chain. The bond length alteration effect exists in unstable nonlinear C 8 Sb (8c) with s high RBE of 3.93 eV. Structures 8a and 8b do not show the bond length alteration effect.
For the low energy minia of nonlinear C 2 Sb to C 8 Sb, the adiabatic ionization energies were calculated to be 8.54, 7.19, 9.09, 6.14, 4.82, 8.06, and 8.26 eV, respectively, indicating that the nonlinear C 5 Sb and C 6 Sb could be more easily ionized. Comparatively, the linear carbon chains C n Sb show more constant adiabatic ionization energy with the increase in the carbon number. The relative binding energies of nonlinear C 2 Sb to C 8 Sb with antimony binding on the sides of carbon clusters (relative to the binding energy of the linear chain cluster) are calculated to be 0.4, 1.23, 1.53, 1.8, 2.86, 2.54, and 2.82 eV, respectively. The results indicate that the relative binding energy is significantly large except for the much smaller carbon cluster C 2 Sb. Quantitatively, the binding energy of nonlinear C 2 Sb to C 8 Sb with antimony binding on the sides of carbon clusters were calculated to be −3.33, −3.24, −1.85, −2.64, −0.34, −1.52, and −0.28 eV, respectively. Antimony is a metallic element exhibiting low electron affinity. If the antimony atom is changed to nitrogen with strong electronegativity, we can test the structural geometry, adiabatic ionization energy, and natural charge population of C 6 N and C 7 N. The bond length alteration effect of C 6 N and C 7 N is much different from that in C 6 Sb and C 7 Sb. The bond length difference between the adjacent C-C bonds is larger than that in the antimony-doped carbon chain cluster. The C-C bond adjacent to nitrogen is substantially longer than that adjacent to antimony. The adiabatic ionization energies of C 6 N and C 7 N were calculated to be 8.82 and 9.17 eV, which are larger than the C 6 Sb and C 7 Sb, indicating the higher stability of the nitrogen-doped carbon chain. Different from the charging state of antimony, the natural charge population analysis of C 6 N and C 7 N indicates that nitrogen atoms are both negatively charged with −0.42 |e|.
increase in the carbon number. The relative binding energies of nonlinear C2Sb to C8Sb with antimony binding on the sides of carbon clusters (relative to the binding energy of the linear chain cluster) are calculated to be 0.4, 1.23, 1.53, 1.8, 2.86, 2.54, and 2.82 eV, respectively. The results indicate that the relative binding energy is significantly large except for the much smaller carbon cluster C2Sb. Quantitatively, the binding energy of nonlinear C2Sb to C8Sb with antimony binding on the sides of carbon clusters were calculated to be −3.33, −3.24, −1.85, −2.64, −0.34, −1.52, and −0.28 eV, respectively. Antimony is a metallic element exhibiting low electron affinity. If the antimony atom is changed to nitrogen with strong electronegativity, we can test the structural geometry, adiabatic ionization energy, and natural charge population of C6N and C7N. The bond length alteration effect of C6N and C7N is much different from that in C6Sb and C7Sb. The bond length difference between the adjacent C-C bonds is larger than that in the antimony-doped carbon chain cluster. The C-C bond adjacent to nitrogen is substantially longer than that adjacent to antimony. The adiabatic ionization energies of C6N and C7N were calculated to be 8.82 and 9.17 eV, which are larger than the C6Sb and C7Sb, indicating the higher stability of the nitrogen-doped carbon chain. Different from the charging state of antimony, the natural charge population analysis of C6N and C7N indicates that nitrogen atoms are both negatively charged with −0.42 |e|. Figure 19. Structural geometry, adiabatic ionization energy (AIE), relative binding energy (RBE), and charge population influenced by binding site of antimony atom. Relative binding energy is calculated by equation: , where E b (nt) and E b (t) stand for structure with antimony binding on nonterminal carbon and linear structure with antimony binding on terminal carbon, respectively. Notations na, nb, and nc represent the structures with n carbon atoms.

Computational Methods
To explore the structure and energetics in the linear antimony-doped carbon clusters, full geometry optimizations were performed using density-functional theory methods implemented in the Gaussian 03 program [39]. The Molecular mechanic's algorithm is frequently employed to investigate very large carbon-based materials because of the efficiency in predicting the binding and delivery mechanism. Density-functional calculations are verified to be effective and have an accurate strategy to reveal the geometric structures [40,41] and electronic properties of various nanomaterials [42][43][44][45]. The B3LYP exchange-correlation function consists of Fock's exact exchange and Beck's three-parameter nonlocal exchange function, along with the nonlocal correlation function developed by Lee et al. [46]. B3LYP was chosen here because the previous research suggests that hybrid-functional is reliable and highly efficient for molecules and clusters [47,48], while the post-HF Ab initio method is accurate but time-consuming [49,50]. A medium-size basis set 6-31G(d) was used for a carbon atom, and the Stuttgart/Bonn relativistic effective core potential (SDD) basis set was used for antimony. The geometries and relative energies of heteroatom-doped carbon clusters obtained with the B3LYP method were very close to those with the coupled cluster single and double (triple) (CCSD(T)) and QCISD(T) method [51,52]. Vibrational frequencies were computed at the same level using a harmonic approximation to assess the nature of the optimized structures. Zero-point energies (ZPE) were evaluated as well using the same methodology. The optimized structures were then used for single-point calculations at the B3LYP/6-311++G(3df,3pd) level. In the computation, bond lengths of the magnetic neutral CnSb, CnSb + cations, and CnSb − anions (n = 1-16) clusters have been optimized through the use of B3LYP methods with a 6-31G(d) basis set. Subsequently, the corresponding harmonic vibrational frequencies are evaluated at the same level. To further verify the reliability of the optimized geometries, we have carefully checked every computation step that might cause possible numerical calculation errors. We have adopted the default convergence criteria for self-consistent-field (SCF) calculation, i.e., 10 −8 for the root mean square density and 10 −6 for the maximum density, and for geometry optimization, 0.00030 This includes the Hartree/Bohr radius for the root mean square force and 0.00045 Hartree/Bohr radius for the maximum force. In addition, the convergence criteria for the energy change in the final step of geometry optimization was set to be 10 −7 Hartree.

Conclusions
The linear carbon chain cluster with different lengths and doping atoms is a special allotrope type of carbon material different from graphite, diamond, graphene, carbon nanotubes, and fullerenes. We have conducted a systematic DFT study on linear C n Sb/C n Sb + /C n Sb − clusters with sizes of n = 1-16 and compared these with pure C n clusters. C-C bond lengths of the linear neutral C n Sb, C n Sb + cations and C n Sb − anions are within 0.1255−0.1336 nm, which is typical of cumulene structures with moderately strong double bonds. However, the alternation in C-C distances suggests that there is a substantial contribution of polyacetylenic valence-bond structures with an alternating triple and single C-C bond. The C-C BLA of neutral CnSb is obvious for n-even clusters, and for n-odd clusters, the BLA tends to be irregular in the vicinity of Sb. We can deduct from C-C BLA and C-Sb BLA that the antimony atom of n-odd C n Sb − anions and n-even C n Sb + (n = 1-10) cations are combined more firmly than adjacent clusters. This is roughly proved to be right by our calculation of their dissociation channels. When comparing the BLA of neutral and charged antimony-doped carbon chains to that of pure carbon chains, it is easy to deduce that through doping the antimony element, the properties of carbon clusters are changed. The adiabatic IE of C n Sb decreased with the rise in the n number, suggesting that larger C n Sb chains become less stable. When comparing the IE of C n Sb with that of pure C n , we found that the latter had a stair-step pattern (n ≥ 6), but C n Sb chains took the shape of a flat curve. The IEs of carbon chains are larger than that of corresponding C n Sb, but the energy difference (IE Cn -IE CnSb ) decreases with increasing n. Different from pure carbon chains, the adiabatic electron affinity of C n Sb do not exhibit a parity effect. There is an even-odd alternation for E I of the open chain C n Sb (n = 1-16) and C n Sb + (n = 1-10, when n > 10, E I of n-odd members of C n Sb + are larger), with the n-even species being comparatively more stable than n-odd ones. The E I for C n Sb − anions does not exhibit a parity effect. For carbon chain clusters, the most favorable binding site of atomic antimony is the terminal carbon of the carbon cluster because the terminal carbon with a large spin density bonds in an unsaturated way. The C-Sb bond is a double bond with WBI between 1.41 and 2.13, which is obviously stronger for a carbon chain cluster with odd-number carbon atoms (compared with neighboring carbon chain clusters with even-number carbon atoms). The WBI of all C-C bonds was determined to be between 1.63 and 2.01, indicating the cumulene character of the carbon chain. Generally, the alteration of WBI and, in particular, the carbon chain cluster is consistent with the bond length alteration. However, the shorter C-C distance did not indicate a larger WBI. For example, the largest WBI for C n Sb (n > 1) was calculated to be the terminal C-C bond index, while this terminal C-C bond was not the shortest C-C bond. Therefore, rather than relying on the empirical comparison of bond distance, the WBI is a meaningful quantitative indicator for predicting the bonding strength in the carbon chain. For HOMO, the p orbital of the terminal carbon of the even-numbered carbon cluster always presents a different sign from the p orbital of antimony. While the p orbital of the terminal carbon of odd-numbered carbon cluster exhibits the same sign with a p orbital of antimony, it can form π bonding orbital with antimony. Therefore, the C-Sb bond of C n Sb with even-numbered carbon should be weaker than the C-Sb bond of C n±1 Sb with odd-numbered carbon. This result agrees well with the Wiberg bond index analysis.