Exploring the Structural and Electronic Properties of Niobium Carbide Clusters: A Density Functional Theory Study

This paper systematically investigates the structure, stability, and electronic properties of niobium carbide clusters, NbmCn (m = 5, 6; n = 1–7), using density functional theory. Nb5C2 and Nb5C6 possess higher dissociation energies and second-order difference energies, indicating that they have higher thermodynamic stability. Moreover, ab initio molecular dynamics (AIMD) simulations are used to demonstrate the thermal stability of these structures. The analysis of the density of states indicates that the molecular orbitals of NbmCn (m = 5, 6; n = 1–7) are primarily contributed by niobium atoms, with carbon atoms having a smaller contribution. The composition of the frontier molecular orbitals reveals that niobium atoms contribute approximately 73.1% to 99.8% to NbmCn clusters, while carbon atoms contribute about 0.2% to 26.9%.


Introduction
Previous studies have indicated that detailed experimental and theoretical investigations of transition metal clusters often reveal remarkable relationships between their physical properties and chemical reactivity [1][2][3][4].Niobium clusters are among the most interesting transition metal clusters that have been extensively studied.They possess a long and rich history of experimental and theoretical investigations, attributed to several special properties, including the relative tendency to form clusters, the existence of a single naturally occurring isotope, high melting point, high temperature resistance, and superconductivity [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].The relative propensity to form clusters is experimentally advantageous, and the presence of only one naturally occurring isotope simplifies the interpretation of niobium spectra by avoiding complications from overlapping features caused by different isotopomers.On the experimental side, the ionization potential (IP) [5], dissociation energy (DE) [6][7][8], reactivity [9][10][11], and electron affinity (EA) [12][13][14] have been measured on niobium clusters and their ions.The results have revealed dramatic size-dependent fluctuations of the electronic and chemical properties of niobium clusters.For example, both the reactivity rates of Nb n (n ≤ 20) clusters with D 2 and the ionization potential show that Nb 8 , Nb 10 , and Nb 16 are relatively unreactive [15,16].The strong size dependence of their chemical reactivity further makes them attractive for study, as certain species may possess the required balance of properties for nanotechnology applications.On the theoretical side, niobium has garnered significant attention, with a focus on studying the structural and electronic properties of a series of pure niobium or niobium-related clusters through the application of density functional theory (DFT) [19][20][21].Many studies on small Nb clusters confirm that Nb 8 , Nb 10 , and Nb 16 clusters have higher stability than other clusters.Pansini and co-workers studied the electronic structure and electrical properties of Al-doped niobium clusters [21].By using all-electron density functional theory with Douglas-Kroll-Hess correction, the spin state, geometry, hardness, and mean static dipole polarizability of Nb x Al y clusters were examined.
In this study, we investigate the geometry, stability, and electronic structures of Nb m C n (m = 5, 6; n = 1-7) clusters using DFT calculations with the BPW91 functional.Firstly, a global search is performed to obtain the geometric structures of Nb 5 C n and Nb 6 C n (n = 1-7) clusters.Secondly, the relative stability of the optimized structure is determined by analyzing the average binding energy, dissociation energy, and second-order difference energy, and the stability reasons are explained.Finally, the density of states and compositions of the frontier molecular orbitals are discussed.

Geometry
The structures of different isomers for Nb 5 C n and Nb 6 C n (n = 1-7) are obtained using the BPW91 functional with the 6-311G+(2d) basis set for C and SDD basis set for Nb. Figure 1 displays the lowest-energy geometries and some low-lying isomers of Nb 5 C n and Nb 6 C n (n = 1-7).According to the total energy from low to high, the isomers of Nb 5 C n (n = 1-7) are designated by 5-na, 5-nb, and 5-nc, while the isomers of Nb 6 C n (n = 1-7) are designated by 6-na, 6-nb, and 6-nc, where n is the number of C atoms.The Cartesian coordinates of the lowest-energy structures of Nb 5 C n and Nb 6 C n (n = 1-7) are summarized in Table S1 of the Supplementary Materials.The optimized results show that the most stable structures of Nb 5 C n (n = 1-7) exhibit a doublet spin state, whereas those of Nb 6 C n (n = 1-7) exhibit a singlet spin state.In the following sections, we discuss these results in detail.

Nb 5 C n (n = 1-7)
The lowest-energy structure of Nb 5 C is found to be a distorted prism with a doublet spin state, which can be viewed as the one obtained from replacing one Nb atom by a C atom in the Nb 6 structure.The corresponding quartet states are higher in energy than the doublet state by 0.26 eV.Another two similar structures, 5-1b and 5-1c, are obtained, and their relative energies to the ground state are 0.53 and 0.82 eV, respectively.
Nb 5 C 2 is an irregular pentagonal bipyramid with similar features as the structure of Nb 7 .The structure of Nb 5 C 2 can be described as the one obtained from replacing two Nb atoms by C atoms sitting on the pentagonal ring.The point group symmetry is C 2v and the calculated equatorial bond lengths of the pentagonal ring are 2.04, 2.05, and 2.84 Å, whereas the polar-equatorial bond lengths are 2.19, 2.63, and 2.55 Å.The pole-to-pole bond length is 3.13 Å. Structure 5-2b is also a pentagonal bipyramid with C 2v point group symmetry.Comparing structures 5-2a and 5-2b, the two C atoms occupy opposite positions in the pentagonal ring in structure 5-2a and adjacent positions in structure 5-2b.Structure 5-2b lies 0.69 eV above structure 5-2a.Structure 5-2c possesses low C s symmetry and is less stable than 5-2a by 0.93 eV.
Structure 5-2b lies 0.69 eV above structure 5-2a.Structure 5-2c possesses low Cs symmetry and is less stable than 5-2a by 0.93 eV.A previous study predicted that Nb5C3 has a similar cubic structure as Nb4C4.The BPW91 calculation determined that Nb5C3 is a distorted cubic structure and has low C1 symmetry.Structure 5-3a consists of two quadrangles.The upper quadrangle contains two C atoms in opposite positions and the lower quadrangle has one C atom.Isomer 5-3b presents a similar geometry as structure 5-3a, with a slight distortion, but with two C atoms occupying adjacent positions.Isomers 5-3b and 5-3c are less stable than 5-3a by 0.30 and 0.53 eV, respectively.
The three low-lying isomers of Nb5C4 listed in this paper are all capped structures, and the main difference between the three isomers is the position of the four C atoms in the cluster.It should be pointed out that the energies of structures 4a and 4b are very close, and the energy of structure 5-4b is only 0.02 eV higher than that of 5-4a.In view of such a small relative energy and in order to ensure the accuracy of calculations, we conducted DFT computations with the Def2-DZVPP basis set instead of the SDD basis set.It was found that the energy difference between 5-4a and 5-4b was very small, but the energy level order was reversed.The energy of structure 5-4b was 0.01 eV lower than that of the 5-4a structure (see Table S2).Additionally, DFT calculations were also performed using the B3LYP functional.The B3LYP functional calculations showed that isomer 5-4b is the lowest energy structure, in which each of the two C atoms occupies the opposite position A previous study predicted that Nb 5 C 3 has a similar cubic structure as Nb 4 C 4 .The BPW91 calculation determined that Nb 5 C 3 is a distorted cubic structure and has low C 1 symmetry.Structure 5-3a consists of two quadrangles.The upper quadrangle contains two C atoms in opposite positions and the lower quadrangle has one C atom.Isomer 5-3b presents a similar geometry as structure 5-3a, with a slight distortion, but with two C atoms occupying adjacent positions.Isomers 5-3b and 5-3c are less stable than 5-3a by 0.30 and 0.53 eV, respectively.
The three low-lying isomers of Nb 5 C 4 listed in this paper are all capped structures, and the main difference between the three isomers is the position of the four C atoms in the cluster.It should be pointed out that the energies of structures 4a and 4b are very close, and the energy of structure 5-4b is only 0.02 eV higher than that of 5-4a.In view of such a small relative energy and in order to ensure the accuracy of calculations, we conducted DFT computations with the Def2-DZVPP basis set instead of the SDD basis set.It was found that the energy difference between 5-4a and 5-4b was very small, but the energy level order was reversed.The energy of structure 5-4b was 0.01 eV lower than that of the 5-4a structure (see Table S2).Additionally, DFT calculations were also performed using the B3LYP functional.The B3LYP functional calculations showed that isomer 5-4b is the lowest energy structure, in which each of the two C atoms occupies the opposite position of the upper and lower quadrangle.In this case, we calculated the vertical ionization potentials (VIPs) of both isomers 5-4a and 5-4b.The results found that the calculated VIP of isomer 5-4b is 5.15 eV (B3LYP) or 5.13 eV (BPW91), and the calculated VIP of isomer 5-4a is 4.85 eV (B3LYP) or 4.90 eV (BPW91).No matter the B3LYP or BPW91 functional, the calculated VIP value for structure 5-4b is in better agreement with the experimental result (5.12 ± 0.12 eV) than for structure 5-4a.Therefore, isomer 5-4b should be the ground state structure.Isomer 5-4c is less stable than 5-4a by 0.15 eV.
From the point of view of growth, the lowest-energy structure of Nb 5 C 5 can be viewed as the one obtained from capping of the C atom to the lowest-energy structure of Nb 5 C 4 with a little local distortion.Another isomer, 5-5b, which possesses similar shape but a different C atom position with isomer 5-5a, is less stable than 5-5a by 0.45 eV.Structure 5-5c, which has relatively high C 2v point group symmetry and can be obtained by adding one C atom to the bottom of structure 5-4c, is 1.03 eV higher in energy than structure 5-5a.
The lowest-energy structure for Nb 5 C 6 is found to be a ring, consisting of four C atoms and three Nb atoms capped with one C atom and one Nb atom on each side of the ring.The point group symmetry is C 2v .Another two structures, 5-6b and 5-6c, with lower C 1 symmetry, are less stable than structure 5-6a by 1.27 and 2.22 eV, respectively.For Nb 5 C 7 , when a C atom is added to the lowest-energy structure of 5-6a at different positions, isomers 5-7a, 5-7b, and 5-7c are formed.Isomer 5-7a is more stable than 5-7b and 5-7c by 0.07 and 0.25 eV, respectively.
For Nb 6 C, the lowest-energy structure is a pentagonal bipyramid with four Nb atoms and one C atom in a plane, and the other two Nb atoms are located on each side of the pentagonal ring.The energies of the other two structures, 6-1b and 6-1c, are higher than those of structure 6-1a by 0.66 eV and 0.99 eV, respectively.We found the lowest-energy structure of Nb 6 C 2 is approximately C 2 point group symmetry.Three Nb atoms and two C atoms make up a ring that is not in the plane, and one Nb atom and a dimer Nb 2 occupy each side of the ring, respectively.Another structure, 6-2b, with C s point group symmetry, is less stable than 6-2a by 0.67 eV.The lowest-energy structure of Nb 6 C 3 can be obtained by capping of an atom to Nb 5 C 3 with different positions of C atoms.Structures 6-3b and 6-3c are less stable than 6-3a by 0.50 and 0.64 eV, respectively.The lowest-energy structure 6-4a has relatively high D 2d point group symmetry.The structure can be viewed as the one obtained from capping of a Nb atom to the bottom of structure 5-4c of Nb 5 C 4 , with small structure distortion.Structures 6-4b and 6-4c are 0.75 and 0.95 eV higher in energy than structure 6-4a.As for Nb 6 C 5 , Nb 6 C 6 , and Nb 6 C 7 clusters, three low-lying isomers of each system are listed in this paper; isomer 6-5a is the structure with the lowest energy, and the energies of the other two isomers are 0.24 and 0.43 eV higher than that of structure 6-5a, respectively.Isomer 6-6a is the lowest-energy structure for the Nb 6 C 6 cluster, and isomers 6-6b and 6-6c are less stable than isomer 6-6a by 0.73 and 0.84 eV, respectively.Isomers 6-7b and 6-7c are less stable than the lowest-energy isomer 6-7a by 1.75 and 2.31 eV, respectively.

Comparison between Calculated and Experimental Vertical Ionization Potential
The VIPs of the stable structures of Nb 5 C n and Nb 6 C n (n = 1-7), determined with BPW91 calculations, are shown in Table 1 and Figure 2, along with the experimental results [28].The calculated VIP values are in excellent agreement with the experiments for Nb 5 C n (n = 1-6); however, for Nb 5 C 7 , the calculated result likely overestimates this parameter, and the relative error is 8.6%.We tried to obtain more of the various structures and calculate the VIPs of different isomers, but the results were not satisfactory.For Nb 6 C n (n = 1-7) clusters, the employed DFT method predicted VIPs with acceptable agreement with the experiments.The calculation results showed that the relative error of Nb 6 C n (n = 1-7) clusters is within 3.5%.The good agreement between the calculated VIPs and the experimental values gives confidence in the assigned the ground state for the complexes considered in the present paper.clusters is within 3.5%.The good agreement between the calculated VIPs and the experimental values gives confidence in the assigned the ground state for the complexes considered in the present paper.5.13 (0.2%) 5.12 ± 0.12 6-4a 4.94 (0.8%) 4.9 ± 0.1 5-5a 5.02 (0.6%) 5.05 ± 0.10 6-5a 5.18 (1.6%) 5.1 ± 0.1 5-6a 5.00 (0.4%) 5.02 ± 0.08 6-6a 5.28 (2.5%) 5.15 ± 0.12 5-7a 5.54 (8.6%) 5.10 ± 0.12 6-7a 5.28 (3.5%) 5.1 ± 0.12

Relative Stability
To determine the structural stabilities of these Nb5Cn and Nb6Cn (n = 1-7) clusters, the average binding energy per atom (Eb) can be calculated using the following formulas: Here, E(Nb), E(C), E(Nb5Cn), and E(Nb6Cn) represent the total energy of Nb, C, Nb5Cn, and Nb6Cn, respectively, where "n" denotes the number of C atoms.The average binding energy is a measurement of the thermodynamic stability of the clusters, and the results are listed in Table 2 and plotted in Figure 3 as a function of the number of C atoms.As shown in Figure 3, the Eb values of these Nb5Cn and Nb6Cn (n = 1-7) clusters are quite large, ranging from approximately 4 to 6 eV, and are close to each other.For Nb5Cn (n = 1-7), the Eb becomes a growing function of the number of C atoms and has two inflection points, corresponding to n = 2 and 6, indicating that Nb5C2 and Nb5C6 clusters are more stable

Relative Stability
To determine the structural stabilities of these Nb 5 C n and Nb 6 C n (n = 1-7) clusters, the average binding energy per atom (E b ) can be calculated using the following formulas: Here, E(Nb), E(C), E(Nb 5 C n ), and E(Nb 6 C n ) represent the total energy of Nb, C, Nb 5 C n , and Nb 6 C n , respectively, where "n" denotes the number of C atoms.The average binding energy is a measurement of the thermodynamic stability of the clusters, and the results are listed in Table 2 and plotted in Figure 3 as a function of the number of C atoms.As shown in Figure 3, the E b values of these Nb 5 C n and Nb 6 C n (n = 1-7) clusters are quite large, ranging from approximately 4 to 6 eV, and are close to each other.For Nb 5 C n (n = 1-7), the E b becomes a growing function of the number of C atoms and has two inflection points, corresponding to n = 2 and 6, indicating that Nb 5 C 2 and Nb 5 C 6 clusters are more stable than others.For the Nb 6 C n cluster, the E b shows a monotonic increasing trend with increasing cluster size.Starting from n = 3, the rate of increase in average binding energy slightly decelerates, indicating that Nb 6 C 3 exhibits slightly higher but not significantly pronounced stability.The stability of Nb5Cn and Nb6Cn (n = 1-7) clusters was further investigated by calculating the dissociation energy (DE), defined as follows: The dissociation energies to remove a C atom from the clusters are illustrated in Table 2 and Figure     The stability of Nb 5 C n and Nb 6 C n (n = 1-7) clusters was further investigated by calculating the dissociation energy (DE), defined as follows: The dissociation energies to remove a C atom from the clusters are illustrated in Table 2 and Figure 3.For the Nb 5 C n (n = 1-7) cluster, the dissociation energies of Nb 5 C 2 and Nb 5 C 6 corresponding to dissociation paths Nb 5 C 2 → Nb 5 C + C and Nb 5 C 6 → Nb 5 C 5 + C yield the local maxima of all calculated clusters, indicating that Nb 5 C 2 and Nb 5 C 6 clusters are more stable than their neighbors.The dissociation energy of Nb 5 C 3 corresponding to the dissociation path Nb 5 C 3 → Nb 5 C 2 + C yields the local minima of all calculated clusters, indicating that Nb 5 C 3 has relatively weak stability.For Nb 6 C n , the dissociation energy of Nb 6 C 3 corresponding to the dissociation path Nb 6 C 3 → Nb 6 C 2 + C yields a small local maximum, indicating slightly higher stability compared to neighboring clusters.This result is consistent with the results of the average binding energy.
The second-order difference (∆ 2 E) refers to taking the difference again based on the first-order difference of energy.∆ 2 E, a more sensitive quantity, reflects the relative stability of the clusters.Positive peaks of ∆ 2 E indicate greater stability of the cluster, while a smaller ∆ 2 E suggests weaker stability of the cluster.The ∆ 2 E values of Nb 5 C n and Nb 6 C n (n = 1-7) clusters were determined using the formulas: where E(Nb 5/6 C n−1 ), E(Nb 5/6 C n ), and E(Nb 5/6 C n+1 ) represent the total energy of Nb 5/6 C n−1 , Nb 5/6 C n , and Nb 5/6 C n+1 , respectively.The ∆ 2 E is presented in Table 2 and plotted in Figure 3 as a function of the number of C atoms.The minima of Nb 5 C n (n = 1-7) are found at Nb 5 C 3 and Nb 5 C 5 , indicating that these clusters show obviously weak stability.The maxima of Nb 5 C n are found at n = 2 and 6, indicating that Nb 5 C 2 and Nb 5 C 6 possess higher stability than their neighbors.The curve for DE shows a similar behavior in Figure 3, confirming the thermodynamic stability of these clusters.For Nb 6 C n (n = 1-7), ∆ 2 E initially decreases, then increases, and gradually decreases again from n = 3 to 6.The curve of ∆ 2 E reaches a small local maximum at n = 3, showing a trend similar to the dissociation energy, implying that the stability of Nb 6 C 3 clusters is slightly higher than that of neighboring clusters.
Comparing the analysis results of average binding energy, dissociation energy, and second-order difference energy, it can be observed that Nb 5 C 2 and Nb 5 C 6 exhibit significantly higher stability.The stability of Nb 6 C 3 is slightly higher among Nb 6 C n clusters but not notably so.Next, we use ab initio molecular dynamics (AIMD) analysis to explain the reasons for the higher stability of Nb 5 C 2 and Nb 5 C 6 .Furthermore, when n = 1 and n = 7, due to the limitations imposed by the cluster sizes computed in this study, Nb 5 C and Nb 5 C 7 as well as Nb 6 C and Nb 6 C 7 are positioned at the endpoints of the line.Their stability will also be further discussed in conjunction with AIMD analysis.

Stability Analysis via AIMD Simulations
In the above discussion, through a comprehensive comparison of the results of average binding energy, dissociation energy, and second-order difference energy, we identified that Nb 5 C 2 and Nb 5 C 6 clusters exhibit higher stability.To analyze the reasons for the stability of these clusters, AIMD simulations, implemented in Born-Oppenheimer MD (BOMD) format, were conducted to analyze their stability.Additionally, AIMD calculations were performed on clusters positioned at the endpoints, namely Nb 5 C, Nb 5 C 7 , Nb 6 C, and Nb 6 C 7 .Each system was set at three different temperatures (100 K, 200 K, and 300 K) under vacuum conditions and maintained a certain temperature stability every 15 fs.The simulations allowed us to observe the thermal motion of the atoms and the geometric fluctuations in these systems.Figure 4 presents the root mean square deviation (RMSD) curves for the three AIMD trajectories at these temperatures.Here, RMSD is defined as the square root of the mean square difference between the position where each atom moves and its initial position.The trajectory from the beginning to the end is represented by three colors: red, white, and blue.The more overlapping of the atoms represented by these three colors, the smaller the range of atomic fluctuations, corresponding to smaller RMSD values and better thermal stability.
As can be seen in Figure 4, by comparing the RMSD of these systems, it is evident that the order of fluctuation amplitude is Nb 5 C 2 (Nb 5 C 6 ) < Nb 6 C < Nb 5 C 7 < Nb 6 C 7 < Nb 5 C. Neither isomerization nor dissociation of these systems was observed during 2000 fs simulations at the three temperatures, which implies that the thermal stability should never be underestimated, at least in vacuum and not too high temperature.As the temperature increases, the differences in the ability of different systems to maintain equilibrium gradually manifest.For Nb 5 C, Nb 5 C 7 , and Nb 6 C 7 , the fluctuation amplitude significantly increases with the increase in temperature, and no regular vibration period forms.Especially, the RMSD values of Nb 5 C do not show a convergence trend, indicating that it is likely to undergo dissociation.By contrast, for Nb 5 C 2 , Nb 5 C 6 , and Nb 6 C, the variation amplitude with temperature change is not significant (usually less than 0.1 Å), and it has a relatively obvious vibration period (about 250 fs) with a clear convergence trend, indicating that these structures have a certain degree of flexibility and can also maintain good stability.As can be seen in Figure 4, by comparing the RMSD of these systems, it is evident that the order of fluctuation amplitude is Nb5C2(Nb5C6) < Nb6C < Nb5C7 < Nb6C7 < Nb5C.Neither isomerization nor dissociation of these systems was observed during 2000 fs simulations at the three temperatures, which implies that the thermal stability should never be underestimated, at least in vacuum and not too high temperature.As the temperature increases, the differences in the ability of different systems to maintain equilibrium gradually manifest.For Nb5C, Nb5C7, and Nb6C7, the fluctuation amplitude significantly increases with the increase in temperature, and no regular vibration period forms.Especially, the RMSD values of Nb5C do not show a convergence trend, indicating that it is likely to undergo dissociation.By contrast, for Nb5C2, Nb5C6, and Nb6C, the variation amplitude with temperature change is not significant (usually less than 0.1 Å), and it has a relatively obvious vibration period (about 250 fs) with a clear convergence trend, indicating that these structures have a certain degree of flexibility and can also maintain good stability.
The reasons for the differences in stability among the different systems can also be analyzed based on the atom colors (red-white-blue) in Figure 5.For Nb5C2 and Nb5C6, they both exhibit a certain degree of symmetry and balanced bond lengths.The horizontal movements have relatively small amplitudes, and the movements perpendicular to the paper surface are almost stationary.This results in these structures having a better ability The reasons for the differences in stability among the different systems can also be analyzed based on the atom colors (red-white-blue) in Figure 5.For Nb 5 C 2 and Nb 5 C 6 , they both exhibit a certain degree of symmetry and balanced bond lengths.The horizontal movements have relatively small amplitudes, and the movements perpendicular to the paper surface are almost stationary.This results in these structures having a better ability to resist interference.On the other hand, the other candidates show a reduction or loss of symmetry, leading to a decrease in their ability to resist interference.One example is the loss of partial symmetry, as seen in Nb 6 C.Although the positioning of the doped atoms on the pentagonal ring distorts the structure of the pentagonal bipyramid, the overall structure remains largely unchanged, allowing it to maintain a certain degree of stability.The second example is the loss of symmetry through the addition of atoms, as observed in Nb 5 C 7 and Nb 6 C 7 .The additional atoms have a greater impact on their neighboring atoms, while their impact on other atoms is relatively small, resulting in fluctuations occurring only between a portion of the atoms and minimal overall decreases in stability.The worst case is Nb 5 C, where the uneven bond lengths and fewer atoms result in weak binding forces between the atoms.Consequently, this leads to the highest volatility at high temperatures.
The worst case is Nb5C, where the uneven bond lengths and fewer atoms result in weak binding forces between the atoms.Consequently, this leads to the highest volatility at high temperatures.
When comparing Figures 3-5, we can observe that Nb5C2, Nb5C6, and Nb6C clusters, which exhibit high thermal stability, also perform well in terms of dissociation energy (Ed) and second-order difference energy (Δ2E).Additionally, Nb5C2 and Nb5C6 correspond to points where the slope of the average binding energy increases rapidly, indicating that these structures are possible magic number clusters.(f) Nb6C7.The left and right sides of each small image represent the trajectories at 100 K and 300 K, respectively.The structures are extracted every 100 fs, the color corresponds to the time step and varies as red-white-blue.

Density of States
To better understand the molecular orbitals of Nb5Cn and Nb6Cn (n = 1-7) clusters, the total density of states (TDOS) and the partial densities of states (PDOS) of Nb5/Nb6 and Cn atoms are plotted in Figures 6-9 using the BPW91 functional.TDOS refers to the density of electronic states or the number of electronic energy levels per unit energy within a given energy range.It describes the distribution of electronic energy levels within the cluster and can be used to analyze the electronic structure of the material.The calculation of PDOS involves decomposing DOS to obtain contributions from different atomic orbitals or interatomic interactions.The calculation of Nb5Cn (n = 1-7) is based on an unrestricted open-shell treatment, and both alpha-TDOS and beta-TDOS are obtained.For Nb6Cn (n = 1-7), the calculation predicts it to be a closed shell and the spin state is singlet.From these graphs, it can be observed that for both Nb5Cn and Nb6Cn (n = 1-7) series, the PDOS profiles of the C atoms are commonly lower, compared to the PDOS curves of Nb.With the increase in the number of C atoms, the contribution of C atoms to the molecular orbitals increases slightly.That is to say, Nb atoms play an important role during the formation of molecular orbitals compared with the usually smaller contribution of C atoms.For low energy regions (the first peak), C atoms are active and contribute significantly to the molecular orbitals.However, as the energy increases, the contribution of C atoms decreases.In particular, the contribution of C atoms to the frontier molecular orbitals is very When comparing Figures 3-5, we can observe that Nb 5 C 2 , Nb 5 C 6 , and Nb 6 C clusters, which exhibit high thermal stability, also perform well in terms of dissociation energy (E d ) and second-order difference energy (∆ 2 E).Additionally, Nb 5 C 2 and Nb 5 C 6 correspond to points where the slope of the average binding energy increases rapidly, indicating that these structures are possible magic number clusters.

Density of States
To better understand the molecular orbitals of Nb 5 C n and Nb 6 C n (n = 1-7) clusters, the total density of states (TDOS) and the partial densities of states (PDOS) of Nb 5 /Nb 6 and C n atoms are plotted in Figures 6-9 using the BPW91 functional.TDOS refers to the density of electronic states or the number of electronic energy levels per unit energy within a given energy range.It describes the distribution of electronic energy levels within the cluster and can be used to analyze the electronic structure of the material.The calculation of PDOS involves decomposing DOS to obtain contributions from different atomic orbitals or interatomic interactions.The calculation of Nb 5 C n (n = 1-7) is based on an unrestricted open-shell treatment, and both alpha-TDOS and beta-TDOS are obtained.For Nb 6 C n (n = 1-7), the calculation predicts it to be a closed shell and the spin state is singlet.From these graphs, it can be observed that for both Nb 5 C n and Nb 6 C n (n = 1-7) series, the PDOS profiles of the C atoms are commonly lower, compared to the PDOS curves of Nb.With the increase in the number of C atoms, the contribution of C atoms to the molecular orbitals increases slightly.That is to say, Nb atoms play an important role during the formation of molecular orbitals compared with the usually smaller contribution of C atoms.For low energy regions (the first peak), C atoms are active and contribute significantly to the molecular orbitals.However, as the energy increases, the contribution of C atoms decreases.In particular, the contribution of C atoms to the frontier molecular orbitals is very small.We provide the compositions of the frontier molecular orbitals (alpha-HOMO, alpha-LUMO, beta-HOMO, beta-LUMO) for the open-shell structures Nb 5 C n (n = 1-7) and closed-shell structures Nb 6 C n (n = 1-7) in Table 3 and Figures 10 and 11.The results indicated that the HOMOs and LUMOs are primarily derived from the contribution of Nb atoms, and the contribution from C atoms is commonly small.For example, for α-HOMO, 79.7-99.7% is contributed by Nb atoms, while 0.3-20.3% is contributed by C atoms.For α-LUMO, 87.4-99.6% is contributed by Nb atoms, while 0.4-12.6% is contributed by C atoms.For β-HOMO, 73.1-98.2% is contributed by Nb atoms, while 1.8-26.9% is contributed by C atoms.For β-LUMO, 82.1-99.5% is contributed by Nb atoms, while 0.5-17.9% is contributed by C atoms.For the Nb 6 C n (n = 1-7) cluster, 83.4-99.8% is contributed to HOMO by Nb atoms, while 0.2-16.6% is contributed to HOMO by C atoms.Additionally, 88.1-99.2% is contributed to LUMO by Nb atoms, while 0.8-11.9% is contributed to LUMO by C atoms.
pha-LUMO, beta-HOMO, beta-LUMO) for the open-shell structures Nb5Cn (n = 1-7) and closed-shell structures Nb6Cn (n = 1-7) in Table 3 and Figures 10 and 11.The results indicated that the HOMOs and LUMOs are primarily derived from the contribution of Nb atoms, and the contribution from C atoms is commonly small.For example, for α-HOMO, 79.7-99.7% is contributed by Nb atoms, while 0.3-20.3% is contributed by C atoms.For α-LUMO, 87.4-99.6% is contributed by Nb atoms, while 0.4-12.6% is contributed by C atoms.For β-HOMO, 73.1-98.2% is contributed by Nb atoms, while 1.8-26.9% is contributed by C atoms.For β-LUMO, 82.1-99.5% is contributed by Nb atoms, while 0.5-17.9% is contributed by C atoms.For the Nb6Cn (n = 1-7) cluster, 83.4-99.8% is contributed to HOMO by Nb atoms, while 0.2-16.6% is contributed to HOMO by C atoms.Additionally, 88.1-99.2% is contributed to LUMO by Nb atoms, while 0.8-11.9% is contributed to LUMO by C atoms.

Computational Methods
The ground states of Nb5Cn and Nb6Cn (n = 1-7) clusters were determined using DFT with the generalized gradient approximation by employing GAUSSIAN programs [36].In recent years, various algorithms and strategies have played an important role in exploring the potential energy surface (PES) of clusters [37][38][39].We first performed the

Computational Methods
The ground states of Nb5Cn and Nb6Cn (n = 1-7) clusters were determined using DFT with the generalized gradient approximation by employing GAUSSIAN programs [36].In recent years, various algorithms and strategies have played an important role in exploring the potential energy surface (PES) of clusters [37][38][39].We first performed the

Figure 1 .
Figure 1.Geometric structures, point group symmetries, as well as relative energies of Nb5Cn and Nb6Cn (n = 1-7) clusters.The relative energies are in eV.

Figure 1 .
Figure 1.Geometric structures, point group symmetries, as well as relative energies of Nb 5 C n and Nb 6 C n (n = 1-7) clusters.The relative energies are in eV.

Figure 2 .
Figure 2. The experimental and calculated VIPs of Nb 5 C n and Nb 6 C n (n = 1-7) clusters.

3 .
For the Nb5Cn (n = 1-7) cluster, the dissociation energies of Nb5C2 and Nb5C6 maxima of all calculated clusters, indicating that Nb5C2 and Nb5C6 clusters are more stable than their neighbors.The dissociation energy of Nb5C3 corresponding to the dissociation path minima of all calculated clusters, indicating that Nb5C3 has relatively weak stability.For Nb6Cn, the dissociation energy of Nb6C3 corresponding to the dissociation path local maximum, indicating slightly higher stability compared to neighboring clusters.This result is consistent with the results of the average binding energy.

Figure 3 .
Figure 3.The average binding energy per atom (E b ), dissociation energy (DE), and second-order difference energy (∆ 2 E) for Nb 5 C n and Nb 6 C n (n = 1-7) clusters.

Figure 4 .
Figure 4. RMSD of (a) Nb 5 C; (b) Nb 5 C 2 ; (c) Nb 5 C 6 ; (d) Nb 5 C 7 ; (e) Nb 6 C; and (f) Nb 6 C 7 at three different temperatures.The trajectories have been aligned to the first frame prior to the RMSD calculation.

Figure 5 .
Figure 5. AIMD simulation trajectories of (a) Nb5C; (b) Nb5C2; (c) Nb5C6; (d) Nb5C7; (e) Nb6C; and(f) Nb6C7.The left and right sides of each small image represent the trajectories at 100 K and 300 K, respectively.The structures are extracted every 100 fs, the color corresponds to the time step and varies as red-white-blue.

Figure 5 .
Figure 5. AIMD simulation trajectories of (a) Nb 5 C; (b) Nb 5 C 2 ; (c) Nb 5 C 6 ; (d) Nb 5 C 7 ; (e) Nb 6 C; and (f) Nb 6 C 7. The left and right sides of each small image represent the trajectories at 100 K and 300 K, respectively.The structures are extracted every 100 fs, the color corresponds to the time step and varies as red-white-blue.

Figure 6 .
Figure 6.The total density of state (TDOS) and partial density of state (PDOS) of Nb5Cn (n = 1-4) with a full width at half maximum (FWHM) of 0.5 eV.

Figure 6 . 16 Figure 7 .
Figure 6.The total density of state (TDOS) and partial density of state (PDOS) of Nb 5 C n (n = 1-4) with a full width at half maximum (FWHM) of 0.5 eV.Molecules 2024, 29, x FOR PEER REVIEW 11 of 16

Figure 7 .
Figure 7.The total density of state (TDOS) and partial density of state (PDOS) of Nb5Cn (n = 5-7) with a full width at half maximum (FWHM) of 0.5 eV.Figure 7. The total density of state (TDOS) and partial density of state (PDOS) of Nb 5 C n (n = 5-7) with a full at half maximum (FWHM) of 0.5 eV.

Figure 7 .
Figure 7.The total density of state (TDOS) and partial density of state (PDOS) of Nb5Cn (n = 5-7) with a full width at half maximum (FWHM) of 0.5 eV.

Figure 8 .
Figure 8.The total density of state (TDOS) and partial density of state (PDOS) of Nb6Cn (n = 1-4) with a full width at half maximum (FWHM) of 0.5 eV.

Figure 8 .
Figure 8.The total density of state (TDOS) and partial density of state (PDOS) of Nb 6 C n (n = 1-4) with a full width at half maximum (FWHM) of 0.5 eV.Molecules 2024, 29, x FOR PEER REVIEW 12 of 16

Figure 9 .
Figure 9.The total density of state (TDOS) and partial density of state (PDOS) of Nb6Cn (n = 5-7) with a full width at half maximum (FWHM) of 0.5 eV.

Figure 9 .
Figure 9.The total density of state (TDOS) and partial density of state (PDOS) of Nb 6 C n (n = 5-7) with a full width at half maximum (FWHM) of 0.5 eV.

Table 1 .
The vertical ionization potential (VIP) for Nb 5 C n and Nb 6 C n (n = 1-7) clusters, all energies are in eV, the values in parentheses are relative error (%).

Table 1 .
The vertical ionization potential (VIP) for Nb5Cn and Nb6Cn (n = 1-7) clusters, all energies are in eV, the values in parentheses are relative error (%).

Table 2 .
The average binding energy per atom (E b ), dissociation energy (DE), and second-order difference energy (∆ 2 E) for Nb 5 C n and Nb 6 C n (n = 1-7) clusters, all energies are in eV.

Table 3 .
The compositions of the frontier molecular orbitals for open-shell structures of Nb5Cn and closed-shell structures of Nb6Cn (n = 1-7).

Table 3 .
The compositions of the frontier molecular orbitals for open-shell structures of Nb 5 C n and closed-shell structures of Nb 6 C n (n = 1-7).