First-Principles Study of Stability and N₂ Activation on the Octahedron RuRh Clusters

Abstract

The geometric and electronic structures of different octahedron RuRh clusters are studied using density functional theory calculations. The binding energy, electronic structure, and energy gap of the clusters have been obtained to determine the possible stable structures. The results show that the Ru₄Rh₂ cluster is the most stable structure which has D_4h symmetry with the largest ionization potential, smallest affinity energy and larger energy gap. Furthermore, the information on adsorption and dissociation of multiple nitrogen molecules and the density of state for the octahedral Ru₄Rh₂ cluster is analyzed. The dissociation barrier of three nitrogen molecules further decreases to 1.18 eV with an increase in the number of N₂ molecules. The co-adsorption of multiple N₂ molecules facilitates the dissociation of N₂ on the Ru₄Rh₂ cluster. The strong interaction between the antibonding orbital of N₂ and the d orbital of the Ru₄Rh₂ cluster is illustrated by calculating and analyzing the results of PDOS, which stretches the N−N bond length and reduces the activation energy to dissociation. The antibonding orbital of the nitrogen molecule shows distinct and unique catalytic activity for the dissociation of the adsorbed nitrogen molecule on the octahedral Ru₄Rh₂ cluster.

1. Introduction

Ammonia is an important inorganic chemical product that sustains the global food supply chain. It is widely used in the manufacture of nitrogen-containing compounds, such as fertilizer and ammoniated feed, and plays an important role in the global economy [1,2,3]. However, the process of reducing N₂ to ammonia is slow and needs to be carried out under high temperatures (400–600 °C), high pressure (20–40 MPa) and catalytic conditions, resulting in high energy consumption and serious environmental pollution [4,5]. Accordingly, it is of great significance to devise an environmentally benign and low energy consumption synthetic method of ammonia for the sustainable development of the global economy. The dissociation of N₂ on the catalyst surface is the rate-limiting step for ammonia synthesis [6,7]. Therefore, it is necessary to further optimize the performance of the catalyst toward N₂ dissociation and understand the reaction mechanism. In past decades, great progress has been made in the development of new catalysts. The direct dissociation of N₂ is facilitated by the strong interaction of N₂ molecules on the surfaces of Ru(0001) [8], Ru($11\overline{2}1$) [9] and Ru($10\overline{1}9$) [10], indicating that the ruthenium catalyst has recently become an industrial catalyst after Pd(110) [11], Fe(111) [12] and W(110) [13]. Although some studies have shown that the catalytic activity of the Ru step surface is better than that of the terrace surface, the stability of the material has not been considered in the literature. Stability is an important parameter that determines the long-term performance of the material during the reaction process.

Recently, it has been shown that sub-nanoscale, atomically dispersed, and fully exposed metal cluster catalysts can not only provide adjacent metal atoms as catalytic sites, but also maintain full atomic utilization efficiency, providing a variety of structural possibilities and catalytic feasibility [14,15,16]. Studying the formation, structure, and properties of clusters not only plays a bridging role in the transition between atomic and molecular physics and condensed matter physics, but also plays an important role in developing the theory of interaction between atoms and molecules, material science and environmental science. Furthermore, various sub-nano to nano-scale clusters with different structures have been experimentally and theoretically studied [17,18]. Theoretical studies on the dissociation of N₂ on clusters of ruthenium, rhodium, aluminum, and tantalum have been reported [19,20,21,22,23]. Then, ruthenium-rhodium binary alloys are often used as catalysts in the industry. Early theoretical studies showed that Ru₆ and Rh₆ clusters with octahedral geometry had high catalytic activity in several reactions [24,25,26,27]. RuRh clusters further enhance the catalytic activity and also reduces the disadvantages that are present in pure Rh catalysts [25,28]. It is important to study the geometric structure and stability of ruthenium-rhodium clusters for their material reaction and catalytic mechanism.

In this paper, the geometric structure and stability of octahedral Ru_mRh_n (m + n = 6) clusters are systematically studied by density functional theory (DFT). It is predicted that the properties of Ru_mRh_n clusters are different from those of pure clusters, and provide some theoretical references for the study of larger RuRh clusters and their related chemical reactions. Overall Ru₄Rh₂ has the best energy and electronic stability and better chemical stability. The structure of nitrogen adsorbed on the Ru₄Rh₂ cluster was further systematically studied, and the potential energy surface of nitrogen bond breaking on the Ru₄Rh₂ cluster was calculated. Our work shows that the Ru₄Rh₂ cluster is an N₂ dissociation active catalyst, which revises the previous practice of Ru₂Rh₄ [25] for catalytic reactions.

2. Results and Discussion

2.1. Geometric Structure of Clusters

The original pure Ru₆ and Rh₆ structures are the globally lowest energy structures obtained by the Birmingham cluster genetic algorithm [29]. The resulting lowest global energy structure is an octahedron. Bimetallic RuRh clusters were obtained using Rh atoms instead of Ru atoms in Ru₆ clusters. The global optimization of the octahedron structure of six-atom bimetallic Ru_mRh_n clusters was carried out using the BFGS structure optimization algorithm [30] of the DFT method. The ground state structures of these Ru_mRh_n clusters are obtained according to the principle of minimum energy. Figure 1 shows the geometric structure of Ru_mRh_n clusters.

Table 1. Optimized symmetry PG, binding energy E_b (eV), bond order σ, and average bond length d (Å) of octahedral Ru_mRh_n clusters, including the bond length d (Å) of the Ru-Ru, Ru-Rh, and Rh-Rh. When the atomic number ratio of Ru and Rh is 2:1, 1:1 and 1:2, each proportion corresponds to two structures, represented as Ru₄Rh₂ and Ru₄Rh₂-I, Ru₃Rh₃ and Ru₃Rh₃-I, and Ru₂Rh₄ and Ru₂Rh₄-I.

	PG	Eb	σ	d(Ru−Ru)	d(Ru−Rh)	d(Rh−Rh)
Ru6	Oh	−3.40	1	2.508	-	-
Ru5Rh1	C4v	−3.34	0.33	2.488	2.602	-
Ru4Rh2	D4h	−3.28	−0.33	2.421	2.607	-
Ru4Rh2-I	C2v	−3.25	0	2.482	2.567	2.614
Ru3Rh3	C2v	−3.20	−0.33	2.407	2.605	2.618
Ru3Rh3-I	C3v	−3.18	0	2.517	2.582	2.589
Ru2Rh4	D4h	−3.12	−0.33	-	2.536	2.625
Ru2Rh4-I	C2v	−3.12	0	2.377	2.571	2.602
Ru1Rh5	C4v	−3.05	0.33	-	2.542	2.602
Rh6	Oh	−2.99	1	-	-	2.583

lists the symmetry, binding energy, bond order and average bond length of neutral Ru_mRh_n clusters. As shown in Figure 1 and Table 1, the binding energy E_b of Ru_mRh_n clusters increases with an increasing Ru atom number and tends to converge to a certain point. This is because the cohesive energy of Ru is 6.74 eV/atom, which is greater than that of Rh is 5.75 eV/atom [31], suggesting that the constructed structure model is reasonable. The symmetry distribution of Ru_mRh_n clusters is symmetrical. The symmetry of Ru₆ is the same as that of Rh₆, which are both O_h. The symmetry of Ru₅Rh₁ is the same as that of Ru₁Rh₅, both of which belong to C_4v. When the Ru/Rh atomic ratio of clusters is 1:2 or 2:1, each ratio has two different structures. In the first structure, two Rh or two Ru atoms are located in the axial direction, and the symmetry of Ru₄Rh₂ and Ru₂Rh₄ clusters is D_4h. In the second structure, two Rh or two Ru atoms are adjacent, and the symmetry of Ru₄Rh₂-I and Ru₂Rh₄-I clusters is C_2v. Since the binding energy of Ru₄Rh₂ and Ru₂Rh₄ clusters is greater than that of Ru₄Rh₂-I and Ru₂Rh₄-I clusters, Ru₄Rh₂ and Ru₂Rh₄ clusters with higher stability will be studied below. The Ru₃Rh₃ has two possible geometries with the symmetries C_2v and C_3v.

The bond order parameter σ of octahedral Ru_mRh_n clusters varies with composition as shown in Table 1 and Figure 2. In Ru₆, Ru₅Rh₁, Ru₁Rh₅, and Ru₆ clusters, the bond order σ > 0. In particular, the positive σ values of Ru₅Rh₁ and Ru₁Rh₅ indicate different degrees of separation between Ru and Rh atoms. The bond order σ = 0 of Ru₄Rh₂-I, Ru₃Rh₃-I and Ru₂Rh₄-I indicate a distorted mixing of clusters. The bond order σ < 0 of Ru₄Rh₂, Ru₃Rh_3, and Ru₂Rh₄ indicate that the clusters are mixed. Therefore, we prove that ruthenium and rhodium can be mixed at the nanoscale. The bond order σ = 0 and σ < 0 correspond to Figure 2c–e. For all the optimized structures, the bond length of Ru-Ru is shorter than the theoretical value of 2.70 [32]. All cluster model bond lengths show small contraction, as with some metal cluster models, which can be ascribed to surface stress [33,34]. The bond length between metal and metal in the cluster model is shorter than that between bulk metals [35]. The binding energies E_b per atom for Ru_mRh_n clusters reflect the degree of deviation from bulk metals in cluster stability. It can be seen from Table 1 that E_b decreases with the increase in the number of Rh atoms, indicating that the stability of metal clusters increases with an increase in the number of Ru atoms. For clusters with the same number of Ru and Rh atoms, the number of Ru-Ru and Ru-Rh bonds is positively correlated with the binding energy of clusters. Therefore, the stability of octahedral Ru_mRh_n clusters is mainly determined by the number of Ru−Ru bonds and Ru−Rh bonds.

2.2. Stability of Clusters

The electronic states of small metal clusters possess the characteristics of discrete energy levels instead of bands. The reciprocal of the highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) gap can reflect the activity of electrons and the degree of deformation of their distribution [36,37,38]. That is, the HOMO−LUMO gap (E_g) is inversely correlated with the ability to participate in chemical reactions, suggesting that E_g determines the chemical stability of the cluster. Therefore, it is necessary to study the energy level distribution of clusters. Figure 3 shows the HOMO, LUMO and corresponding orbital energy levels of the RuRh cluster. Figure 4 shows the calculated E_g, i.e., the difference between the calculated HOMO and LUMO energy levels. The E_g of Ru_mRh_n clusters becomes smaller than that of the pure Ru or Rh clusters, which means that the chemical activity of Ru_mRh_n clusters is higher. According to Figure 4, Ru₆ has the highest chemical stability among the studied Ru_mRh_n clusters, with the largest energy gap of 0.47 eV. Such a large energy gap indicates that electrons find it difficult to transition from an occupied orbital to a vacant orbital. This means that clusters of large energy gaps have high chemical stability [39]. We find that the E_g value of the Ru₃Rh₃ and Ru₄Rh₂ cluster is slightly lower than that of the Ru₆ cluster, indicating that their ability to participate in chemical reactions is higher than Ru₆, that is, the stability of the Ru₃Rh₃ and Ru₄Rh₂ cluster is relatively higher than that of other Ru_mRh_n clusters.

The IP and EA can also be used to determine the electronic stability of the clusters. The IP of a cluster is defined as the energy needed for ionizing an electron away from the cluster. The energy required for electron ionization increases with the increase in the IP value, resulting in a more difficult expression of adiabatic electron ionization affinity, indicating that clusters tend to generate negative ions [40,41]. The stability of clusters decreases with the increase in the tendency of a cluster to generate negative ions. As shown in Figure 4, with the increase in the number of Ru atoms, the IP of Ru_mRh_n clusters first decreases, then increases, and finally tends to decrease. Among them, the IP of the Ru₄Rh₂ cluster is the largest. The low IP indicates that cations can easily form clusters by losing electrons. The ability of other Ru_mRh_n clusters to lose electrons increases with the increase in the number of Ru atoms, and the stability of Ru_mRh_n clusters are decreased, except Ru₄Rh₂. Moreover, the electron affinity EA of the clusters increases first to Ru₁Rh₅, then decreases to Ru₄Rh₂, and finally it tends to increase with the increase in the number of Ru atoms in the clusters. This phenomenon indicates that the cluster has a small tendency to acquire electrons to form anions, which corresponds to high stability. From the above-mentioned observations, we can conclude that Ru₄Rh₂ clusters have the largest IP (5.37 eV) and smallest EA (1.89 eV), which is more stable in terms of reactivity.

In order to better characterize the structural stability of clusters, two parameters, namely the mixing energy and quadratic difference energy, are used for comparison. The mixing energy E_mix is used to characterize the degree of mixing, or the segregation stability. A negative value of E_mix means that mixing is favorable. E_mix is the analogue of the formation energy of bulk alloys, used in the case of nanoclusters. Another index of energetic stability is the quadratic difference in energy Δ₂, whers the Δ₂ describes the relative stability of the cluster as compared to the neighboring compositions. The value of Δ₂ is positively correlated with the closeness of bonding between atoms, and the high closeness of bonding represents the high stability of clusters. In addition, the magic number structure is often the maximum value in the Δ₂ image. Figure 5 compares the E_mix and Δ₂ values of all Ru_mRh_n clusters. It is found that when the number of Ru atoms is four, the E_mix value becomes the smallest and the value of Δ₂ become the largest. Therefore, the Ru₄Rh₂ cluster has the highest energy stability among all Ru_mRh_n clusters. Hence we will take Ru₄Rh₂ as the main research object in the following study to analyze its catalytic activity for N₂ decomposition.

2.3. Adsorption and Dissociation of N₂ on Ru₄Rh₂

Figure 6 gives the configurations of N₂ adsorbed on the octahedral Ru₄Rh₂ cluster. The effective adsorption sites of Ru₄Rh₂ metal clusters vary with atoms as shown in Figure 6. T1 and T2 are the top sites, at the top position of the Ru and Rh atoms in the cluster, respectively. B1 and B2 are two bridge sites, which refer to the bridges between the Ru and Ru, and Ru and Rh atoms, respectively. Two Ru atoms and one Rh atom make up the hollow adsorption site (H). As shown in Figure 6, the adsorption configurations of N₂ molecules on clusters can be divided into two types: End-on (standing) and side-on (lying) conformations. The optimal adsorption site was found by placing a single N₂ molecule at each location of each configuration and minimizing the local DFT. By placing a single N₂ molecule at every position of each configuration, we can find the best adsorption site based on adsorption energy.

For the end-on conformations, as shown in Figure 6a and

Table 2. Adsorption energy E_a (eV), bond length d (Å), vibrational wavenumbers ν (cm⁻¹) and total charge number Δq (|e|) of N₂ adsorbed on Ru₄Rh₂ clusters.

Configurations	Sites	Ea	d(N−N)	d(Ru-N)	Δq	ν(N−N)
End-on	T1	−1.07	1.135	1.958	−0.403	1996
	T2	−1.02	1.131	1.950	−0.403	2058
	B1 1	−1.05	1.134	1.973	−0.403	1951
	B2	−0.69	1.155	2.021	−0.472	1601
	H1	−1.04	1.132	1.965	−0.404	2062
Side-on	T1	−0.65	1.142	2.125	−0.504	1896
	T2	−0.47	1.142	2.126	−0.465	1899
	B1 2	−1.03	1.135	1.981	−0.405	1989
	B2	−0.77	1.169	2.135	−0.711	1556
	H2	−0.77	1.169	2.134	−0.711	1557

¹ In the end-on configuration, the adsorption of N₂ at the B1 site is unstable and shifts to the T1 site. The adsorption of N₂ at the H site is also unstable and shifts to the T2 site.² In the side-on configuration, the adsorption of N₂ at the B1 site is unstable and shifts to the T1 site in a vertical configuration. The adsorption of N₂ at the H site is also unstable and shifts to the B2 site.

, the energy of N₂ adsorption on T1 and T2 sites is −1.07 eV and −1.02 eV, and the bond length of N−N is 1.135 Å and 1.131 Å, respectively. Besides, the energy of N₂ adsorption on the B2 site (−0.69 eV) is clearly smaller than that on the T1 and T2 sites. The adsorption of N₂ at the B1 site is unstable and shifts to the T1 site. Similarly, the adsorption of N₂ at the H site is unstable and shifts to the T2 site. For side-on conformations, as shown in Figure 6b and Table 2, B2 has adsorption energy of −0.77 eV, and the bond length of N−N is 1.169 Å, which is in accordance with the values from previous results ~1.130 Å. [42]. The N–Ru bond distance of 2.135 Å is also in line with the available investigations [43]. The adsorption of N₂ at B1 and H sites is unstable and shifts to the end-on T1 and B2 sites, respectively. Finally, the adsorption energies on the T1 and T2 sites of the end-on configuration are −0.65 eV and −0.47 eV, respectively, with the same bond length of 1.142 Å. In contrast to B2 sites, the bond lengths of N−N on T1 and T2 sites are shorter, so the dissociation at these two sites is not studied for the time being.

For these adsorption configurations, only the molecules with side-on configurations can dissociate. Therefore, we have considered the dissociation path at the B2 adsorption site and draw the calculated dissociation potential energy surface as shown in Figure 7. The adsorption energy of the nitrogen molecule at the B2 site on the Ru₄Rh₂ cluster is −0.77 eV. The N₂ molecule dissociates from the B2 site to two H sites with an exothermicity of −1.42 eV; the dissociation barrier is 1.38 eV, which is less than 1.59 eV of the pure Ru₆ cluster in this work and 1.80 eV of the terrace Ru(0001) of the previous study [44]. This indicates that the catalytic performance of the alloyed clusters is better than that of the pure Ru. We explain this phenomenon by comparing the d-band centers of the two clusters and the Mulliken charge transfer on the alloyed Ru₄Rh₂ cluster. In comparison with the pure Ru cluster, the charge transfer from the Ru atom (0.037) to the Rh atom (−0.073) in the Ru₄Rh₂ cluster results in the up-shift of the d-band center from −1.76 eV to −1.36 eV. In addition, the atomic size caused by the length of the Ru−Rh bond (2.607 Å) in Ru₄Rh₂ cluster is larger than that of the Ru−Ru bond (2.508 Å) in the Ru₆ cluster (Table 1), which also causes the center of the d-band to move up, thus enhancing the catalytic activity.

In order to further verify the reliability of the results, the bond lengths, electronic configurations, and vibrational wavenumbers of N₂ on the Ru₄Rh₂ clusters were studied. Table 2 shows that the bond length of the N₂ molecule at B2 is elongated to 1.169 Å compared with 1.11 Å of the gas−phase nitrogen molecule [20]. It will undergo more deformation when nitrogen molecules attach to the cluster, which makes the dissociation barrier smaller. We also calculated the Mulliken charge of the adsorbed N₂ molecule. A positive value represents the transfer of charge from the metal to the nitrogen molecule. As shown in Table 2, the number of electronic configurations of a nitrogen molecule in B2 is 0.711 |e|, which is larger than that of other sites, indicating a strong correlation with the breakage of the nitrogen bond in this adsorption configuration. We calculated the vibrational wave number of the nitrogen molecule as listed in Table 2. Compared with the vibration wave number of the gas−phase nitrogen molecule (2359 cm⁻¹) [20] and that of others listed in Table 2, the calculated vibration wave number of the nitrogen molecule at B2 is 1556 cm⁻¹ indicating the adsorption potential of the nitrogen molecule at B2 is the weakest among all adsorption sites, and the dissociation barrier is smallest when the nitrogen molecule dissociates.

2.4. Adsorption and Dissociation of Multiple N₂ on Ru₄Rh₂

Because of the highly symmetrical structure of the Ru₄Rh₂ cluster, it is possible for several nitrogen molecules to adsorb together on the Ru₄Rh₂ cluster. According to the above calculations, nitrogen molecules have the largest adsorption energy at the end-on configuration of T2, which makes it unlikely to decompose into nitrogen atoms. When two nitrogen molecules co-adsorb on Ru₄Rh₂, one at T2 and the other at T1, the adsorption energy obtained is −1.84 eV. The dissociation path structure and relative potential energy are shown in Figure 7. Compared with the dissociation barrier of one nitrogen molecule of 1.38 eV, the dissociation barrier of two N₂ molecules at the most active B2 site is slightly reduced to 1.27 eV. When three nitrogen molecules are adsorbed to Ru₄Rh₂, one nitrogen molecule at T1, one at top T2, and the other at bottom T2, the adsorption energy obtained is −2.93 eV and the energy barrier of nitrogen bond breaking is further reduced to 1.18 eV. This result implies that for multiple nitrogen molecules adsorbed on Ru₄Rh₂ clusters together, the dissociation of adsorbed nitrogen molecules will be accelerated according to the real conditions that may occur. To explain this situation, we compare the N−N bond lengths in the transition state (TS) structures for three cases. In a single N₂ adsorption system, the N−N bond length for the reactant and TS is 1.169 and 1.680 Å, respectively, indicating the bond length elongation of 0.511 Å. For the two N₂ adsorption systems the N−N bond length elongation of 0.630 Å between the reactant state (1.130 Å) and the TS (1.760 Å) is obtained. On the other hand, for the three N₂ adsorption systems, the N−N bond length elongation between the reactants (1.132 Å) and TS (1.895 Å) is 0.763 Å. To further illustrate that multiple nitrogen molecules can accelerate the dissociation of adsorbed nitrogen molecules, we added four nitrogen molecule dissociation path structures and relative potential energy as shown in Figure S1. When the four nitrogen molecules were adsorbed on the T1, T1, T2, and T2 positions of Ru₄Rh₂, the adsorption energy was −3.28 eV and the nitrogen dissociation energy barrier was further reduced to 1.05 eV. The elongation of the N bond length between the reactants N−N bond length (1.135 Å) and TS (1.994 Å) is 0.859 Å, which is larger than the tensile length of the three nitrogen atoms adsorbed. From the observed results, we can conclude that deformation occurs in N₂ after the formation of TS indicating the adsorption of multiple N₂ can increase the degree of N₂ structural deformation, which is conducive to accelerating the reaction process.

The partial density of states (PDOS) of several nitrogen molecules co-adsorbed on the Ru₄Rh₂ cluster is shown in Figure 8, including the p orbital of a nitrogen molecule and the d orbital of the metal atom, and it is compared with that of the Ru₄Rh₂ cluster without adsorbed N₂ molecule as shown in Figure 8a. Figure 8b shows a single nitrogen molecule that has a maximum peak from −5.9 to −8.1 eV in the 1π and 5σ orbital at the B2 site, which is hybridized with the d orbital of the Ru₄Rh₂ cluster. In Figure 8c,d and Figure we find that the maximum value of the 5σ orbital decreases for the co-adsorption of two N₂ and three N₂ molecules, especially for three N₂ molecules (Figure 8d). This observation indicates that the strong interaction between the 5σ orbital of the nitrogen molecule and the d orbital of Ru₄Rh₂ with several nitrogen molecules co-adsorption can reduce the strength of the N−N bond. In Figure 8d, the antibonding orbital (2π*) of the nitrogen molecule has obvious peaks. In the region of +2.5 eV above the Fermi level, it shows that the unoccupied antibonding orbital 2π* from the d electron of Ru to the nitrogen molecule adsorbs more of the nitrogen molecule. This suggests that Ru, which has abundant d-orbital electron properties, can receive electrons from the σ bond of N₂ and give them to the antibonding orbital of N₂, thus effectively weakening the bond strength of N₂. This is consistent with previous studies [45]. As shown in Figure 8b-d the calculated p-band centers of a nitrogen molecule are −5.42, −5.16, and −5.81 eV; and that of the d-band centers of Ru atoms are −1.89, −1.68 and −1.56 eV. This result also supports the co-adsorption of N₂ on Ru₄Rh₂ and the stronger interaction between the nitrogen molecule and the Ru atom. This is consistent with the prediction of the d-band center theory [46]. The calculated wave numbers of single, two and three nitrogen molecules are 1540, 1468, and 1382 cm⁻¹, respectively. The weakest N−N bond of the three molecules provides clear evidence. With the larger co-adsorption of N₂, many more electrons are transferred to the antibonding orbital of N₂, which is testified by our calculations.Further, we cansuggest that the mechanism of ammonia formation is facilitated, after multiple N₂ co-adsorptions on the Ru₄Rh₂ cluster. The reaction mechanism of ammonia synthesis in the clusters is a complex and multi-step process. The dissociation of the N₂ molecule is only the first step toward the formation of NH₃. This study provides a catalyst that can facilitate the N₂ dissociation reaction, without high temperatures and pressures.

3. Materials and Methods

DFT calculations were carried out using the atomic orbital-based DMol³ package [47]. It was performed using the revised Perdew−Burke−Ernzerhof (RPBE) exchange and correlation functional within generalized gradient approximation (GGA) [48]. This function aptly describes the electronic structure of the transition metal and is widely used [49]. The DFT semi-core pseudo potentials (DSPPs) core treatment was implemented for relativistic effects [50]. Moreover, double numerical plus polarization (DNP) was chosen as the basis set. The spin-polarization method was used for all calculations. The convergence tolerance of energy is 1×10⁻⁶ Ha, the maximum force is 0.002 Ha Å⁻¹, and the maximum displacement is 0.005 Å. The range for the orbital cutoff was set to 5.0 Å. The transition states (TS) were obtained by LST/QST tools in Dmol³ code [51].

In order to study the electronic properties of metal clusters, and quantitatively study the separation of clusters, the ionization potential (IP), electron affinity (EA) and bond order parameter (σ) [52] are defined as follows:

$$IP=-E_{\mathrm{cluster}}+E_{{\mathrm{cluster}}^{-}},$$

(1)

$$EA=E_{{\mathrm{cluster}}^{-}}-E_{{\mathrm{cluster}}^{+}},$$

(2)

$$\sigma =\frac{N_{\mathrm{Ru}-\mathrm{Ru}}+N_{\mathrm{Rh}-\mathrm{Rh}}-N_{\mathrm{Ru}-\mathrm{Rh}}}{N_{\mathrm{Ru}-\mathrm{Ru}}+N_{\mathrm{Rh}-\mathrm{Rh}}+N_{\mathrm{Ru}-\mathrm{Rh}}},$$

(3)

where, $E_{\mathrm{cluster}}$ is the total energy of the cluster, $E_{{\mathrm{cluster}}^{+}}$ is the total energy of the cations with the same configuration of the cluster, and $E_{{\mathrm{cluster}}^{-}}$ is the total energy of the anions with the same configuration of the cluster. $N_{\mathrm{Ru}-\mathrm{Ru}}$, $N_{\mathrm{Ru}-\mathrm{Rh}}$, and $N_{\mathrm{Rh}-\mathrm{Rh}}$ are the numbers of Ru-Ru, Ru-Rh, and Rh-Rh bonds, respectively. The ionization potential reflects how easily a cluster can lose an electron, while the electron affinity potential measures the ability of a cluster to gain an electron [40,41]. Positive values of σ indicate segregation of the atoms in the cluster while σ would be zero for disorderly mixed clusters. Mixed and onion-like phases of clusters result in negative values of σ [52].

To compare the relative stability of clusters, the binding energies (E_b) per atom for Ru_mRh_n clusters, mixing energy (E_mix) and quadratic difference energies (Δ₂) of Ru_mRh_n clusters are defined for six-atom nanoscale systems as follows: [53].

$$E_{\mathrm{b}}=\left[E\left({\mathrm{Ru}}_{\mathrm{m}}{\mathrm{Rh}}_{\mathrm{n}}\right)-m{E}_{\mathrm{Ru}}-n{E}_{\mathrm{Rh}}\right]/6,$$

(4)

$$E_{mix}=E\left({\mathrm{Ru}}_{\mathrm{m}}{\mathrm{Rh}}_{\mathrm{n}}\right)-\left[mE({\mathrm{Ru}}_6\right)+nE({\mathrm{Rh}}_6)]/6,$$

(5)

$${\Delta }_2=E\left({\mathrm{Ru}}_{\mathrm{m}+1}{\mathrm{Rh}}_{\mathrm{n}-1}\right)+E({\mathrm{Ru}}_{\mathrm{m}-1}{\mathrm{Rh}}_{\mathrm{n}+1})-2E({\mathrm{Ru}}_{\mathrm{m}}{\mathrm{Rh}}_{\mathrm{n}}),$$

(6)

where $E\left({\mathrm{Ru}}_{\mathrm{m}}{\mathrm{Rh}}_{\mathrm{n}}\right)$ (m + n = 6) is the total energy of each cluster. $E_{\mathrm{Ru}}$ and $E_{\mathrm{Rh}}$ are the energies of a single Ru atom and a single Rh atom, while $E({\mathrm{Ru}}_6)$ and $E({\mathrm{Rh}}_6)$ are the total energies of pure Ru and pure Rh clusters, respectively. Stable structures are identified by more negative $E_{\mathrm{b}}$ and $E_{\mathrm{mix}}$ values. Maxima of ${\mathsf{\Delta }}_2$ indicates structures of special relative stability which is compared to those of the same size and nearby compositions.

The formula for calculating the adsorption energy (E_a) is as follows:

$$E_a=E_{\left[\mathrm{cluster}+\mathrm{adsorbate}\right]}-\left(E_{\left[\mathrm{cluster}\right]}+E_{\left[\mathrm{adsorbate}\right]}\right),$$

(7)

where $E_{\left[\mathrm{cluster}+\mathrm{adsorbate}\right]}$ is the energy of substances adsorbed on the surface, $E_{\left[\mathrm{cluster}\right]}$ is the energy of a clean surface, and $E_{\left[\mathrm{adsorbate}\right]}$ is the energy of free molecules.

4. Conclusions

The geometric structure of different octahedron RuRh clusters was optimized by the DFT method. The possible structures were determined and their stability and electronic properties were studied. It is found that the average binding energy of clusters increases with an increase in the number of Ru atoms. The stability of six-atom Ru_mRh_n clusters is mainly determined by the number of Ru−Ru bonds and Ru−Rh bonds. The ability of Ru_mRh_n clusters to lose electrons increases with the increase in the number of Ru atoms. Overall, Ru₄Rh₂ has the best energy and electronic stability and better chemical activation stability. The adsorption and dissociation of the octahedral Ru₄Rh₂ cluster for several nitrogen molecules were studied. The dissociation barrier of nitrogen molecules further decreases to 1.18 eV with an increase in the number of N₂ molecules (for three nitrogen molecules). The strong interaction between the p orbital of N₂ and the d orbital of the Ru₄Rh₂ cluster in illustrated by calculating and analyzing the results of the PDOS. The antibonding orbital of the nitrogen molecule shows distinct and unique catalytic activity for the dissociation of adsorbed nitrogen molecules on the octahedral Ru₄Rh₂ cluster.