Investigation of Structures, Stabilities, and Electronic and Magnetic Properties of Niobium Carbon Clusters Nb₇C_n (n = 1–7)

Abstract

The geometrical structures, relative stabilities, and electronic and magnetic properties of niobium carbon clusters, Nb₇C_n (n = 1–7), are investigated in this study. Density functional theory (DFT) calculations, coupled with the Saunders Kick global search, are conducted to explore the structural properties of Nb₇C_n (n = 1–7). The results regarding the average binding energy, second-order difference energy, dissociation energy, HOMO-LUMO gap, and chemical hardness highlight the robust stability of Nb₇C₃. Analysis of the density of states suggests that the molecular orbitals of Nb₇C_n primarily consist of orbitals from the transition metal Nb, with minimal involvement of C atoms. Spin density and natural population analysis reveal that the total magnetic moment of Nb₇C_n predominantly resides on the Nb atoms. The contribution of Nb atoms to the total magnetic moment stems mainly from the 4d orbital, followed by the 5p, 5s, and 6s orbitals.

1. Introduction

The research on nanoclusters represents a significant branch within the fields of nanoscience and nanotechnology, providing a potent pathway for crafting tailored nanomaterials characterized by unique physical and chemical attributes. Consequently, significant advancements have been achieved in this field over the past few decades [1,2,3,4,5,6,7]. Transition metal-carbide clusters possess unique electronic and magnetic properties that make them promising candidates for various applications, ranging from nanoelectronics to energy storage. By delving into the structures, stabilities, and electronic and magnetic properties of transition metal–carbon clusters, researchers can unravel the underlying principles governing the behavior of transition metal-carbide systems at the nanoscale. Moreover, the investigation of transition metal–carbon clusters offers insights into the intricate interplay between metal and carbon atoms within the cluster framework. The coordination environment of metal atoms with carbon atoms influences not only the structural stability but also the electronic and magnetic properties of the clusters. Understanding how these interactions manifest at the atomic level is crucial for designing nanomaterials with tailored functionalities. Niobium carbon clusters, Nb_nC_m clusters, have been the subject of significant interest in recent years due to their potential applications in numerous areas including catalysis, optoelectronics, and materials science [8,9,10,11,12,13,14]. With niobium’s versatile oxidation states and carbide’s decisive role in determining cluster stability and reactivity, understanding the behavior of niobium carbide clusters is essential and could enable the design of advanced materials and catalysts with specific properties. Furthermore, the study of niobium carbon clusters can also contribute to the broader understanding of transition metal-carbide clusters.

Various experimental techniques, such as photoelectron spectroscopy [15,16,17,18], mass spectroscopy [19,20,21,22], X-ray diffraction spectroscopy [17,23], and infrared absorption spectroscopy [24], have been employed to explore the physical and chemical properties of niobium carbides and their ions. Wang and co-workers investigated a series of mono-niobium carbide clusters, NbC_n⁻ (n = 2–7), using anion photoelectron spectroscopy [16]. The linear NbC_n⁻ were observed to have high electron binding energies and exhibited an even–odd alternation, similar to that observed for pure linear carbon clusters in the same size range. NbC₂⁻ and NbC₃⁻ were shown to have C_2v cyclic structures. The mass spectroscopic studies conducted by Duncan and co-workers [19] as well as Castleman and co-workers [20] have revealed that the smaller Nb_mC_n⁺ clusters, especially for n = 2–4, are particularly stable. Jarrold and co-workers reported experimental studies of the structures of gas-phase NbC_n⁺ (n = 28–50) clusters. The experiments, which use injected-ion drift-tube techniques, indicate that for fullerenes containing an even number of carbon atoms, the niobium metal is endohedral, but for fullerenes with an odd number of carbon atoms, the niobium metal is bound as part of the carbon cage [25]. Mafuné and co-workers determined ionization energies for a series of Nb_nC_m (n = 3–10, m = 0–7) clusters [21].

Theoretical studies primarily focus on employing density functional theory (DFT) calculations to explore the geometric and electronic structures of pure niobium clusters and niobium carbon clusters. This provides crucial insights into their stability, reactivity, and electronic properties, which plays a pivotal role in customizing these clusters to meet specific application requirements. A theoretical investigation of structures, spectra, and energies of niobium clusters from Nb₁₃ to Nb₂₀ is performed by Pham and Minh [26]. The Nb₁₅ system is observed to be stable and it can form a highly symmetric structure in all charged states with both open and closed electron shells. The geometric and electronic structures of niobium carbon clusters, ranging in size from [Nb₂C₂] to [Nb₁₂C₂₀], were discussed by Dance and Harris [27]. Vertical ionization energies and normalized binding energies are provided for all isomers. The species [Nb₈C₁₂] and [Nb₁₄C₁₃] have the same structures as the Ti analogs. Balasubramanian and co-workers determined the geometric and electronic structures of NbC_n (n = 3–8), and equilibrium geometries, Gibbs free energies, and heat capacity functions as a function of temperatures were computed [28].

In this paper, we present the theoretical investigation of a series of niobium carbon clusters, Nb₇C_n (n = 1–7). The geometric structures are predicted by employing the Saunders ‘Kick’ (SK) global search method combined with DFT calculation. To better understand how the Nb and C atoms affect the molecular orbitals, we plot the total density of states (TDOS) and the partial density of states (PDOS) of Nb₇ and C_n. Meanwhile, the magnetic properties are discussed in detail.

2. Results and Discussion

2.1. Structure of Nb₇C_n (n = 1–7)

The lowest-energy structure and low-lying isomers of Nb₇C_n (n = 1–7) at the B3LYP/Nb/SDD//C/6−311+G(2d) level are listed in Figure 1. The Cartesian coordinates of the lowest-energy structures of Nb₇C_n (n = 1–7) are summarized in Table S1 of the Supplementary Materials. The two criteria for selecting the ground-state structure of clusters in this study are (1) the principle of lowest energy and (2) the agreement between theoretical predictions and experimental results of vertical ionization potential (VIP). The ground-state structures of the Nb₇C_n series exhibit an outer body structure. It is predicted that as the number of C atoms increases, the structures of Nb₇C_n clusters gradually tend towards hollow spherical structures. In addition, C atom aggregation areas will appear on the surface of the structure in the subsequent structural evolution process.

The lowest-energy structure 1a of the Nb₇C cluster is a distorted tetragonal prism in which carbon atoms occupy one corner of the quadrangle. The point group symmetry is C_s and the spin state is a doublet state. Structure 1b exhibits a twisted hexagonal bipyramid-shaped structure higher in energy than the ground-state structure by 0.22 eV. The quartet-state structure 1c is higher in energy than doublet state 1a by 0.52 eV. The low-energy isomers of the Nb₇C₂ cluster all exhibit C₁ symmetry. Among them, 2a adopts a twisted bipyramidal shape, while 2b and 2c are three-dimensional structures formed by splicing three twisted pyramids, with relative energies of 0.16 eV and 0.22 eV, respectively. Moving on to the Nb₇C₃ cluster, the three C atoms in 3a attach to the outer side of the C₁ symmetric isomer, resulting in an irregular three-dimensional shape. Isomers 3b and 3c have relative energies 0.42 eV and 0.85 eV higher than that of 3a, respectively. The lowest-energy isomer 4a of Nb₇C₄ can be viewed as that obtained from the capping of a C atom to the lowest-energy structure of Nb₇C₃. Isomers 4b and 4c are less stable than 4a by 0.38 and 0.57 eV, respectively. The Nb₇C₅ system exhibits three isomers with 5a being the ground-state structure. Isomer 5b is only 0.08 eV higher than that of isomer 5a. Within the expected accuracy of the methods, isomers 5a and 5b can be regarded as nearly degenerate. In this case, we calculate the VIPs of both isomers 5a and 5b. The calculated VIPs of 5a and 5b are 4.86 and 5.14 eV, respectively. Compared with the results of isomers 5a and 5b, the calculated VIP value for structure 5a obviously agrees better with the experimental result of 4.7 ± 0.1 eV. These results indicate that isomer 5a is the most stable isomer in current DFT calculations of the Nb₇C₅ cluster. Isomer 5c is less stable than 5a by 0.14 eV. As for the neutral Nb₇C₆ cluster, the energy gap of isomers 6a and 6b is 0.06 eV. Given the small energy difference between structures 6a and 6b, we also performed VIPs calculations for both the isomers. The results found that the calculated VIP for isomer 6a (5.08 eV) is in better agreement with the experimental values (4.91 eV) than that for isomer 6b (5.16 eV). Therefore, isomer 6a should be the ground-state structure. In the doped Nb₇C₇ cluster, an equal number of Nb and C atoms does not result in a uniform distribution but instead leads to the aggregation of carbon atoms. This phenomenon results in shorter C-C bond lengths and greater bond energies, further enhancing the stability of the structure. Compared to the 7b isomer with a more uniform atomic distribution, the energy is 0.14 eV lower. The energy of 7c is 0.50eV higher than that of the ground-state structure.

Based on the lowest-energy structures, VIPs are calculated and compared with the experimental values, and the results are shown in

Table 1. The vertical ionization potential (VIP), average binding energy (E_b), second-order difference energy (Δ₂E), dissociation energy (DE), HOMO-LUMO gap (E-gap), and chemical hardness (η) for Nb₇C_n (n = 1–7) clusters; all energies are in eV.

Isomers	VIP		Eb	Δ2E	DE	E-Gap	η
	Calc.	Expt. [21]
Nb7C	5.16	5.20 ± 0.08	4.01	−0.39	7.21	1.36	3.80
Nb7C2	4.72	4.7 ± 0.1	4.47	−0.12	6.60	1.15	3.51
Nb7C3	4.78	4.7 ± 0.08	4.79	1.27	7.72	1.24	3.57
Nb7C4	4.76	4.75 ± 0.07	4.94	−0.67	6.45	1.09	3.42
Nb7C5	4.86	4.7 ± 0.1	5.13	−0.60	7.12	1.11	3.48
Nb7C6	5.08	4.91 ± 0.07	5.33	−0.28	7.72	1.23	3.55
Nb7C7	5.19	5.1 ± 0.1	5.52		8.00	1.25	3.64

. The vertical ionization potential is defined as VIP = E (cation at optimized neutral geometry) − E (optimized neutral geometry). From Table 1, we can see that the calculated VIPs are all in good agreement with the experimental results. The calculated VIP values of the lowest-energy structures 1a (5.16 eV), 2a (4.72 eV), 3a (4.78 eV), 4a (4.76 eV), 5a (4.86 eV), 6a (5.08 eV), and 7a (5.19 eV) agree well with the experimental values of 5.20 ± 0.08, 4.7 ± 0.1, 4.7 ± 0.08, 4.75 ± 0.07, 4.7 ± 0.1, 4.91 ± 0.07, and 5.1 ± 0.1 eV [21], respectively.

The infrared (IR) spectra of the lowest-energy structures of Nb₇C_n (n = 1–7) clusters are computed based on DFT calculations. The calculated spectra are plotted in Figure S1 of the Supporting Information (ESI). Based on Figure S1, the IR spectra of Nb₇C_n (n = 1–7) clusters cover the range of 0 to 2000 cm⁻¹. We applied a frequency scaling factor of 0.9692 [29], determined at the B3LYP/6−311G(2d) level, to adjust the IR spectra during plotting. From Figure S1, it is evident that the IR spectra of Nb₇C_n (n = 1–4) clusters span from 0 to 800 cm⁻¹. However, for Nb₇C_n (n = 5–7) clusters, a noticeable peak emerges within the range of 1200–1400 cm⁻¹. In the case of Nb₇C clusters, two distinct and strong spectral peaks are found around 600 and 700 cm⁻¹, alongside some weaker peaks below 250 cm⁻¹. For Nb₇C₂, the highest peak is centered at around 600 cm⁻¹, and there are some minor peaks within the range of 100–400 cm⁻¹. The highest-intensity peak of Nb₇C₃ is observed at around 670 cm⁻¹, with several lower-intensity peaks falling between 100 and 350 cm⁻¹. For Nb₇C₄, we mainly distinguish one broad band ranging from 450 to 750 cm⁻¹, and the broadness of the peak suggests that it originates from multiple vibrational modes. The spectra of Nb₇C₅ and Nb₇C₆ clusters exhibit an intense peak at around 1300 cm⁻¹, with some lower-intensity peaks below 800 cm⁻¹. The spectrum for Nb₇C₇ contains two distinguishable peaks around 1250 and 1350 cm⁻¹, as well as a wide band below 800 cm⁻¹ with a maximum around 600 cm⁻¹.

2.2. Stability

To explore the stability of the ground-state structure of Nb₇C_n (n = 1–7) clusters, the average binding energy (E_b), second-order difference energy (Δ₂E), dissociation energy (DE), HOMO-LUMO gap (E-gap), and chemical hardness (η) of the lowest-energy isomers are calculated in Table 1 and plotted in Figure 2. They are defined as follows:

E_b(Nb₇C_n) = [7E(Nb) + nE(C) − E(Nb₇C_n)]/(7 + n)

(1)

Δ₂E(Nb₇C_n) = E(Nb₇C_n+1) + E (Nb₇C_n−1) − 2E(Nb₇C_n)

(2)

DE(Nb₇C_n) =E(Nb₇C_n−1) + E(C) − E (Nb₇C_n)

(3)

E-gap = ε(LUMO) − ε(HOMO)

(4)

η = VIP − VEA

(5)

where E(Nb) represents the energy of a single Nb atom, E(C) represents the energy of a single C atom, and ε(HOMO) and ε(LUMO) represent the energy of the highest occupied orbit and the energy of the lowest empty orbit of the ground-state structure of Nb₇C_n (n = 1–7) clusters, respectively.

As shown in Figure 2, the average binding energies of the ground-state structure of Nb₇C_n (n = 1–7) clusters generally show an increasing trend, with an obvious slope change at n = 3, which indicates that isomer Nb₇C₃ has excellent thermal stability among Nb₇C_n (n = 1–7) clusters. This corresponds to the high second-order differential energy and dissociation energy of the isomer Nb₇C₃. Chemical hardness and HOMO-LUMO gap represent the degree of response of the molecule to the outside world, especially the addition or removal of electrons under the action of external potential fields. The overall change trend of E-gap and η, which reflect the chemical stability of the ground-state structure of the Nb₇C_n cluster, is the same. The E-gap and η values of isomer Nb₇C₁-a are highest among all ground-state isomers, indicating that its chemical stability is higher than those of other isomers. In addition, when the number of carbon atoms n > 4, as the number of incorporated carbon atoms increases, the energy indicators of the cluster increase monotonically. This is because Nb atoms can form Nb-C bonds by sharing electrons with C atoms. Moreover, owing to the lower energy levels of Nb’s d orbitals, they can donate electrons to the p orbitals of the C atoms, thereby reinforcing the Nb-C bonds. Additionally, the distance between Nb and C atoms is shorter compared to that between Nb atoms, indicating a stronger van der Waals interaction between Nb and C atoms, thereby enhancing the stability of the present clusters.

Among all ground-state isomers of Nb₇C_n (n = 1–7) clusters, isomers Nb₇C₃ and Nb₇C₄ show obvious differences in stability. Isomer Nb₇C₃ shows a local maximum value among the ground-state isomers of all sizes of Nb₇C_n (n = 1–7) clusters in the curves of Δ₂E, DE, E-gap, and η shown in Figure 2. This observation suggests that Nb₇C₃ possesses higher thermodynamic and chemical stability, making it a potential candidate for a magic cluster. In contrast, minima appeared at n = 4, suggesting that isomer 4a has poor thermal and chemical stability and can be used as a potential detector material and chemical reaction indicator.

2.3. Density of States

To further explore how Nb and C atoms affect the molecular orbital and HOMO-LUMO energy gap of the Nb₇C_n (n = 1–7) clusters, the density of states diagram of the ground-state structures of the Nb₇C_n (n = 1–7) clusters is shown in Figure 3 and Figure 4. Analysis of the total density of states (TDOS) plot for the lowest-energy structures of the Nb₇C_n (n = 1–7) clusters reveal a negligible disparity in profile between the alpha-TDOS and beta-TDOS, thereby signifying a minimal degree of spin polarization. To offer a more nuanced evaluation of the contribution of Nb and C atoms to the total density of states, respectively, we plotted the partial density of states (PDOS) of Nb₇ and C_n, respectively. Compared to the PDOS curves of Nb, the PDOS profiles of the C atoms generally exhibit lower magnitudes. At the HOMO and LUMO positions, the PDOS of the Nd fragment is very close to the TDOS. As the number of C atoms increases, a distinct aggregation phenomenon of carbon atoms is observed within the structure of the doped cluster. Upon comparing the density of states plots of Nb₇C_n clusters (Figure 3 and Figure 4), it becomes evident that the contribution of carbon atoms to the frontier orbitals demonstrates a trend of localized enhancement to some degree. We provide the compositions of the frontier molecular orbitals (alpha-HOMO, alpha-LUMO, beta-HOMO, beta-LUMO) for Nb₇C_n (n = 1–7) in

Table 2. The compositions of the frontier molecular orbitals for Nb₇C_n (n = 1–7).

Isomers	Atoms	α-HOMO	α-LUMO	β-HOMO	β-LUMO
Nb7C	Nb	98.72%	98.80%	98.68%	99.87%
	C	1.28%	1.20%	1.32%	0.13%
Nb7C2	Nb	99.45%	97.73%	98.88%	99.47%
	C	0.55%	2.27%	1.12%	0.53%
Nb7C3	Nb	94.32%	96.29%	92.69%	96.63%
	C	5.68%	3.71%	7.31%	3.37%
Nb7C4	Nb	91.37%	95.49%	96.44%	92.42%
	C	8.63%	4.51%	3.56%	7.58%
Nb7C5	Nb	88.95%	92.15%	88.60%	92.54%
	C	11.05%	7.85%	11.40%	7.46%
Nb7C6	Nb	93.80%	94.57%	84.35%	94.29%
	C	6.20%	5.43%	15.65%	5.71%
Nb7C7	Nb	94.49%	91.86%	84.66%	92.29%
	C	5.51%	8.14%	15.34%	7.71%

.

It can be seen in Table 2 that the percentage contribution of Nb in the front molecular orbitals is very high and far exceeds that of the C atom. For α-HOMO, Nb atoms contribute approximately from 88.95% to 99.45% to the Nb₇C_n clusters. For α-LUMO, Nb atoms contribute approximately from 91.86% to 98.8%. For β-HOMO, Nb atoms contribute approximately from 84.35% to 98.88%. For β-LUMO, Nb atoms contribute approximately from 92.29% to 99.87%. This indicates that HOMO and LUMO are mainly composed of orbitals of the transition metal Nb, with a weak involvement of C atoms. Such an observation suggests the pivotal role played by Nb atoms in modulating the electronic structure within these isomers of Nb₇C_n (n = 1–7) clusters.

2.4. Magnetic Properties

The magnetic moment is an important concept in physics, chemistry, and materials science, as it plays a significant role in understanding and controlling magnetic properties and phenomena. The spin density (${\rho }_{alpha}-{\rho }_{beta}$) refers to the distribution of unpaired electrons with different spin orientations within a molecule or material. It provides information about the electronic structure and magnetic properties of the system. We utilized Multiwfn in conjunction with the VMD program to visualize the spin-density isosurfaces of the lowest-energy structures of Nb₇C_n (n = 1–7), as shown in Figure 5. The total magnetic moments of Nb₇C_n (n = 1–7) as well as the local magnetic moments of Nb atoms and C atoms in Nb₇C_n clusters are presented in

Table 3. Total magnetic moments of Nb₇C_n clusters and local magnetic moments for the C atom and magnetic moments of 4d, 5s, 5p, 6s, and 6p orbitals for the Nb atoms; all units in μ_B.

Isomers	Nb					C	Total
	5s	4d	5p	6s	6p
Nb7C	0.15	0.74	0.12	0.00	−0.01	−0.04	1
Nb7C2	0.12	0.63	0.20	0.00	0.00	−0.02	1
Nb7C3	0.01	0.75	0.14	0.12	0.00	−0.01	1
Nb7C4	0.03	0.47	0.24	0.08	0.00	0.12	1
Nb7C5	0.02	0.36	0.18	0.14	0.15	0.16	1
Nb7C6	0.05	0.74	0.08	0.07	0.02	0.02	1
Nb7C7	0.00	0.66	0.13	0.14	0.05	0.02	1

.

In Figure 5, the spin density diagram shows that the excess unpaired electrons are predominantly distributed around the Nb atoms. In Table 3, we can find that Nb₇C_n clusters have total magnetic moments of 1 μ_B and the total magnetic moment is mainly located on the Nb atoms, whereas the magnetic moment located on the C atoms is almost negligible. To further understand the magnetic properties of Nb₇C_n clusters, we performed a detailed analysis of the local magnetic moment of Nb atoms in Nb₇C_n clusters by natural population analysis calculations. Table 3 shows that for the Nb₇C and Nb₇C₂ clusters, the magnetic moment of Nb atoms is mainly from the 4d orbital, followed by the 5s and 5p orbitals with a small contribution to the magnetic moment of the Nb atom. For the other Nb₇C_n (n = 1–7), the magnetic moment of Nb atoms still comes mainly from the 4d orbital, followed by 5p and 6s orbitals. For the Nb₇C₅ cluster, the contribution of 5p, 6s, and 6p orbitals to the magnetic moment of Nb atom is nearly equal. The contribution of C atoms to the total moment is minimal and mainly originates from the 2p orbitals.

3. Computational Methods

Global minimum searches for low-lying structures of Nb₇C_n (n = 1–7) clusters were performed in three steps: Firstly, we employed the SK global search method [30,31,32] combined with DFT calculation to search and optimize the structure. In recent years, our research group has effectively predicted the ground-state structures and electronic properties of binary mixed clusters utilizing the SK-DFT method [33,34,35,36,37,38,39,40,41]. All the atoms are placed at the same point initially and then are “kicked” randomly with a sphere of some radius. The Kick method runs approximately 500 times at the B3LYP functional [42,43] using the 3–21 G [44] basis set until no new minima appear. Secondly, the isomers were ranked according to their total energy at the B3LYP/3-21G level. Thirdly, several pertinent lower-lying isomers were chosen for additional optimization employing the triply split basis set with polarization and diffuse functions. Specifically, the 6–311+G(2d) [45] basis set was employed for C. For the heavier atoms, previous studies [46,47,48,49] have confirmed the significance of relativistic effects in simulating the properties of systems involving heavy atoms. Therefore, using appropriate relativistic treatment methods is important when dealing with systems containing heavy atoms. The SDD pseudopotential basis set [50], which accounts for relativistic effects, was used for Nb. The interaction between valence electrons and the inert core is included in the pseudopotential. This allows us to reduce computational times and provide reliable calculations of geometry and electronic properties. Geometries are regarded as completing the optimization when the maximum force, the root-mean-square (RMS) force, the maximum displacement of atoms, and the RMS displacement of atoms have magnitudes less than 0.00045, 0.0003, 0.0018, and 0.0012 a.u., respectively. The self-consistent field convergence criterion was set as 10⁻⁸. Structural optimization with frequency analysis were considered at the same time. The image frequency would be eliminated until every structure optimized had no image frequency to confirm that the low-lying isomers were local minima. All calculations were performed with the GAUSSIAN09 program [51]. Density of states (DOS) and orbital composition analysis were performed by Multiwfn program [52] and visualized by Visual Molecular Dynamics (VMD 1.9.3) software [53].

4. Conclusions

We have investigated the structural, electronic, and magnetic properties of Nb₇C_n (n = 1–7) clusters by employing density functional theory calculations. The relative stability analysis indicates that Nb₇C₃ is more stable than the other clusters, as it exhibits a higher second-order difference energy, dissociation energy, HOMO-LUMO gap, and chemical hardness. In contrast, Nb₇C₄ is less stable, as it shows lower values of these stability indicators. The composition of the frontier molecular orbitals reveals that the predominant contribution originates from Nb atoms, accounting for 84.35% to 99.87% of the molecular orbital. The total magnetic moment of Nb₇C_n (n = 1–7) is mainly located on the Nb atoms, whereas the magnetic moment located on the C atoms is almost negligible. For the Nb atoms, the 4d orbit contributes most to the total magnetic moment, followed by the 5p, 5s, and 6s orbits.