Introduction

Emissions from human activities modify the concentrations of essential gasses, which causes a considerable change in the survival conditions. Notably, the records show that the atmospheric concentrations of important greenhouse gases have increased over the last two centuries [1]. Environmental pollution is, nowadays, one of the most critical topics for humankind. Many efforts guide the way to fix the already damaged ecosystem, all of them trying to achieve more sustainable development or lowering the impact caused by human activities. Anthropogenic reactive nitrogen has been increasing over the past two decades, often associated with waste industries generating clouds of environmental pollution that originate in combustion processes at high temperatures and in-car engines. In this sense, the nitrogen oxides (\(NO_{x}\)) present in those clouds of pollution involve reactions that harm human health. For example, nitric oxide (NO) and nitrous oxide (\(N_{2}O\)) are some of the many toxic molecules that can destroy ozone, besides being precursors of acid rain. The challenge for chemistry, physics, or materials science is setting the formation mechanism constituents of this pollution and, in this way, reduce or eliminate their formation.

This elimination usually involves a selective catalytic reduction (SCR) process, whose basis resides on the chemical reduction of the \(NO_{x}\) molecules using a metal-based catalyst with activated sites to increase the reduction reaction. A nitrogen-based reducing agent (reagent), such as ammonia or urea, selectively reacts with \(NO_{x}\) within a specific temperature range and in the presence of the catalyst and oxygen to reduce the \(NO_{x}\) into molecular nitrogen (\(N_{2}\)) and water vapor (\(H_{2}O\)) [2]. These reactions that occur in a wide temperature range accompany a significant increase in costs. These costs are mainly due to the large volumes of catalyst required for the reduction reaction [3]. Therefore, it is desirable to have better materials to reduce toxic molecules such as nitrogen oxides.

The exploration of the adsorption of nitrogen oxides on metallic surfaces [4] can constitute the initial step of understanding the corrosion processes. For example, Piskorz et al. [4] analyzed the catalytic reactivity of the spinel \(Co_{3}O_{4}\) sample, where different stages are suggested in the decomposition of \(N_{2}O\). The first step refers to the reaction of the oxide on active metal sites, followed by an oxygen diffusion on the metal surface, and finally recombination of atoms to form \(N_{2}\) and \(O_{2}\). In the last decades, the study of metallic clusters has received considerable attention mainly because their information makes them a model of great utility on this topic. In this sense, it is essential to understand the behavior of its electronic structure and hence its geometry. Quantum confinement and low-dimensional effects of the clusters allow a wide variety of structural and electronic behaviors to be analyzed that are generally not scalable [5]. The clusters are pretty sensitive to the addition or subtraction of a single atom; besides, the size of these particles allows the theoretical study efficiently with a low computational cost [6]. In this way, it is possible to apply various properties that they exhibit and which make them of great interest in processes of adsorption and catalysis of other molecules [7]. For example, in the literature, several experimental and theoretical studies exist on different transition metals that show the dissociative chemisorption of NO on surfaces such as Rh, Cr, Fe, and Co [8, 9]. The Fourier-transform ion cyclotron resonance (FT-ICR) is a technique that is usually followed by mass spectroscopy, which produces cationic rhodium ions (\(Rh^{+}_{6}\)) and reacting with nitric oxide [10]. This reaction leads to the dissociative chemisorption of the oxide and the formation of rhodium dioxide and rhodium tetraoxide, releasing nitrogen molecules. In other reports, theoretical studies were performed on clusters with one to six Rh atoms to study \(N_{2}O\) adsorption and dissociation [11]. In this case, \(N_{2}O\) molecule is bonded to bridge sites; hence, the decomposition of this oxide on the metal surface is performed. It is interesting how \(N_{2}O\) is decomposed through \(N-O\) bond, forming \(N_{2}\) and O as the final products. On the other hand, theoretical studies on structural properties of \(Ag_{n}Rh\) clusters were performed to investigate the adsorption of NO on the surface of these systems [12]. In that work is observed a size-dependent reactivity, indicating that the Rh atom acts as a more active adsorption site for NO than Ag. In this sense, the \(Ag_{n}\) clusters doped with Rh increase the reactivity toward NO adsorption.

The investigation of the reaction of NO on cobalt clusters reports the competition of dissociative chemisorption against molecular chemisorption of NO using ionic techniques in the gas phase. It shows the nature of the NO structure chemisorbed on Co clusters by examining reactions of displacement with molecular oxygen [13, 14]. Other studies show the reactions of cobalt cluster cations with nitrous oxide and nitric oxide by FT-ICR mass spectrometry [15], where there is evidence of NO decomposition suggested by the loss of molecular nitrogen, therefore forming \(Co^{+}_{n}(O)\). This fact indicates that clusters of the formula \(Co_{n}\) showed new properties not present in bulk or compact structures. The latter leads to interesting chemistry on the cluster’s surface owing to the formation of stable oxide clusters and the decomposition of nitric oxide with the loss of molecular nitrogen. In an attempt to rationalize results, theoretical studies have been carried out by density functional theory (DFT) of the interaction of nitric oxide with Co clusters in a wide range of both structures and multiplicities, showing changes in reactivity as a function of size [16,17,18]. For example, Hanmura et al. clearly showed the dissociative chemisorption of two NO molecules on \(Co^{+}_{n}\) (n = 3-10) [17] using a tandem mass spectrometer and DFT calculation.

Accordingly, full geometry optimizations of these systems are desirable in order to make a comparison between theoretical and experimental results. There is much to this chemistry that remains to be understood, in particular, how these reactions are in function of particle size and structure [16], since experimental results show that these clusters present an exciting change in reactivity as a function of the cluster size [15, 18]. Although the experiments carried out show the formation of these systems, as well as dissociation of the nitric and nitrous oxide with the interaction with the cobalt clusters, there are no reports about the calculations that show either reactivity and geometry of these small size clusters, which is the aim of the present work. Specifically, clusters between four and nine cobalt atoms are naked and interacting with a single NO and \(N_{2}O\) molecule. Additionally, both neutral and charged systems are studied. There were several spin configurations considered for each case. Therefore, to explain the behavior of the \(Co_{n}\) clusters (n = 4–9), a theoretical density functional study has been developed. The goal is to understand the reactivity and geometry of \(Co_{n}\) interacting with both NO and \(N_{2}O\) to offer a possible explanation for the dissociation of nitrogen oxides in these systems. In this work, the global and local reactivity indexes explain dissociative chemisorption and predict the subsequent addition of nitrogen oxides.

The remainder of this article is organized as follows: In the next section, the methodology used for calculations is presented. Later, molecular structural arrangements and global and local reactivity results are shown, in addition to the binding energies. Finally, a summary and conclusions are given.

Methodology

The geometry optimizations were performed in neutral and cation clusters for both naked and capped structures with NO or \(N_{2}O\). Each structure was evaluated with several multiplicities (see Table 1). In the first stage, the minimum energy search was performed through Gradient Embedded Genetic Algorithm (GEGA) [19] in order to assure that the global minimum structure was found. This search of minimum energy structures was obtained through VWN functional [20] and LANL2DZ basis set for cobalt atoms [21] as well as D95 basis set for nitrogen and oxygen atoms [22]. This level of theory is supported in previous work [23].

Table 1 Multiplicities (\(2S+1\)) utilized in the search for the minimum energy structure of the potential energy surface using GEGA
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The second stage of the calculations involved an improvement in geometry and electronic structure. In this sense, the calculations were developed using the GAUSSIAN 09 [24] and GAUSSIAN 16 [25] computational packages. In this way, a calibration was made through several basis sets and functionals testing. Such functionals were BP86 [26, 27], BPW91 [26, 28], PW91 [28, 29], M05 [30], M05-2X [31], M06 [32], M06-2X [32], and M06-L [33]. These functionals were combined with four basis sets 6-311G [34], DZVP [35, 36], LANL2MB [21], and SDD [37]. This calibration, developed for Co cluster dimers, was based on the ionization potential results since it is possible to compare with the experimental values reported in the literature [38, 39].

The best approximation was provided by M06-L functional in combination with DZVP and SDD basis sets (see Table 2). However, convergence problems were obtained with the four Co atoms cluster. According to Table 2, the next level of theory with the best approximation is M05/DZVP; however, convergence problems persisted, in this case, with four Co atoms cluster interacting with NO molecule (it was suspected that the problem was in the basis sets). Therefore, it was decided to use the third basis set that offered the best results for experimental ionization potential, the 6-311G basis set. On the other hand, in line with Gutsev et al. [40] and Guzmán-Ramirez et al. [23], BPW91 is one of the most widely used functionals in small clusters of transition metals, especially in Co clusters. Consequently, the level of theory selected in this work to re-optimize the geometries found in the first search stage is BPW91/6-311G. To make a comparison of this level of theory and the use of polarization functions, at that same stage, a reoptimization of the structures was also carried out using the basis set 6-31G* [41, 42] with the same functional for the calculation of exchange and correlation energy.

Table 2 Calibration results obtained for Co neutral dimer with various DFT levels of theory
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Concerning the local reactivity indexes, Fukui functions were obtained as proposed by Yang and Parr [43] in a finite difference scheme through

$$\begin{aligned} f^{+}(\mathbf {r})= & {} \rho _{N+1}(\mathbf {r})- \rho _{N}(\mathbf {r}),\end{aligned}$$
(1)
$$\begin{aligned} f^{-}(\mathbf {r})= & {} \rho _{N}(\mathbf {r})- \rho _{N-1}(\mathbf {r}) \end{aligned}$$
(2)

and

$$\begin{aligned} f^{0}(\mathbf {r})=\frac{\rho _{N+1}(\mathbf {r})- \rho _{N-1}(\mathbf {r})}{2}, \end{aligned}$$
(3)

where \(\rho _{N+1}(\mathbf {r})\), \(\rho _{N}(\mathbf {r})\) and \(\rho _{N-1}(\mathbf {r})\) are the electron densities of the system with N+1, N, and N-1 electrons, respectively. The vertical ionization energies (IE) and electron affinities (EA) were calculated as \(IE= E_{cation}-E_{neutral}\), and \(EA= E_{neutral}-E_{anion}\). The electron chemical potential (\(\mu\)) for a N-electron system with an external potential \(V(\mathbf {r}\)) and total energy E is the partial derivative of the energy to the number of electrons at a constant external potential. \(\mu\) is the negative of Mulliken’s electronegativity definition, the negative mean of IE and EA, obtained through a finite difference approximation of \(\left[ \frac{\partial E}{\partial N}\right] _{V}\) [44, 45]

$$\begin{aligned} \mu =-{\frac{(IE+EA)}{2}} \end{aligned}$$
(4)

Besides, the response of the system to charge donation (\(\mu ^{-}\)) and charge acceptance (\(\mu ^{+}\)) fundamentally drives the direction of flow of charge [46] and can be obtained through

$$\begin{aligned} \mu ^{-}=-{\frac{(3IE+EA)}{4}} \end{aligned}$$
(5)

and

$$\begin{aligned} \mu ^{+}=-{\frac{(IE+3EA)}{4}} \end{aligned}$$
(6)

This approach makes it possible to differentiate the charge addition from the charge subtraction in a system.

The calculation of chemical hardness [47], differentiating the chemical potential to the number of electrons at constant energy potential, equals to:

$$\begin{aligned} \eta =\left[ \frac{\partial \mu }{\partial N} \right] _{V} \end{aligned}$$
(7)

The electrophilicity index calculation was as [48]:

$$\begin{aligned} \omega =\frac{\mu ^{2}}{2\eta } \end{aligned}$$
(8)

Finally, in order to confirm the interaction between the different sizes of cobalt clusters and the NO and \(N_{2}O\) molecules, the calculation of the binding energies (\(E_{b}\)) was performed, understood as the energy difference between the compounds \(Co^{q}_{n}(NO)\) and \(Co^{q}_{n}(N_{2}O)\), and the energies of each species separately (\(Co^{q}\), N, and O) [49], as shown in the following equations:

$$\begin{aligned} E_{b}\left[ Co^{q}_{n}(NO) \right]= & {} E\left[ Co^{q}_{n}(NO) \right] - nE(Co^{q})- E(N)-E(O) \end{aligned}$$
(9)
$$\begin{aligned} E_{b}\left[ Co^{q}_{n}(N_{2}O) \right]= & {} E\left[ Co^{q}_{n}(N_{2}O) \right] - nE(Co^{q})- 2E(N)-E(O) \end{aligned}$$
(10)

where n=4-9 and q=0-1.

Results

As mentioned before, in the first stage to search for structures with lower energy using the GEGA computational package [19], we considered different spin multiplicities (\(M=2S+1\)) in all calculations in order to find the most stable spin state for each system. The range of multiplicities taken into account was based on each predecessor system’s minimum energy spin state. These multiplicities are indicated in Table 1.

This way, from the relative energies for each cluster and with each respective multiplicity, the minimum energy geometries were found for the different systems. The following section describes the geometrical structure, one of the most critical features of the clusters studied here.

\(Co_{n}\) and related structures

As mentioned in the Methodology section, the lowest energy structures found in the first stage were reoptimized under the BPW91/6-311G and BPW91/6-31G* theory levels. Geometries obtained for neutral and cation structures naked and interacting with NO or \(N_{2}O\) are shown in Figs. 1, 2, 3, 4, 5, and 6, respectively; bond distances are indicated in angstroms (Å). These figures indicate the bond distances for the two levels of theory used in the second stage, BPW91/6-311G and BPW91/6-31G*. Numbers without parentheses are bond distances for the first level of theory, while parentheses are used for the second. q indicates the electron charge, M defines the multiplicity of the lowest energy cluster, and \(E_{r}\) is the relative energy in eV for the 6-31G* basis set.

Fig. 1
figure 1

Geometries obtained for \(Co^{q}_{4}\), \(Co^{q}_{4}(NO)\), and \(Co^{q}_{4}(N_{2}O)\) (q = 0,1). Cobalt atoms are shown in blue green, nitrogen atoms in blue, and oxygen atoms in red. The distances are given in angstroms, BPW91/6-311G without parentheses and BPW91/6-31G* with parentheses. \(M=2S+1\), q is the electron charge, and \(E_{r}\) the relative energy in eV using 6-31G* basis set

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For n=4, at both levels of theory, a tetrahedral structure is obtained with a spin multiplicity of 11 for neutral (see Fig. 1a). For the cation, the structures shown in Fig. 1b and c were obtained. With the 6-31G* basis set, the lowest energy geometry shown in part b was obtained with a multiplicity of 10; the second structure is shown in part c with a multiplicity of 8 (\(E_{r }\)= 0.18 eV). This last geometry also corresponds to the minimum energy obtained with the 6-311G basis set. It is essential to mention that these geometries are in good agreement with other reports obtained for neutral clusters [17, 50]; those references used a tandem mass spectrometer and DFT calculation besides multicanonical basin hopping method, respectively (see Fig. 1). It is interesting to see that geometry is according with the structure of other metal clusters since in a previous study based on density functional theory [51], for this size of a cluster of Ni atoms, the same atomic arrangement is presented. For \(Co_{4}(NO)\), \(Co^{+}_{4}(NO)\), and \(Co_{4}(N_{2}O)\), both basis sets (6-31G* and 6-311G) coincide in the multiplicities as well as in the lowest energy structures (see Fig. 1d, e, and f). The lowest energy structure when interacting with both NO and \(N_{2}O\) shows no bonding between N and O atoms. This latter confirms dissociative chemisorption. Privileged positions for nitrogen and oxygen atoms are between cobalt atoms. Spin multiplicity decreases compared with naked cobalt clusters, suggesting that oxygen and nitrogen fill some electron shells. For NO interaction, this effect lowers M to 8 and 7 for neutral and charged clusters, respectively. It is more evident for the neutral cluster with \(N_{2}O\) since it lowers multiplicity to 5 for the neutral one. The lowest energy geometry for \(Co^{+}_{4}(N_{2}O)\) via the basis set 6-31G*, shown in Fig. 1g, has a spin multiplicity of 10. With this same basis set, a similar structure is obtained with an energy difference of 0.04 eV and M=8. This system coincides geometrically with the one obtained by 6-311G (M=8), as shown in Fig. 1h. It is clear that using both basis sets, \(Co^{+}_{4}(N_{2}O)\) structure shows an arrangement of \(Co^{+}_{4}(O)\) bonded to a pair of N atoms, which are lightly symmetric.

In the case of n=5 the lowest energy structure obtained using the 6-31G* basis set for the neutral case has a multiplicity of 14 (Fig. 2a), which is the same geometry as that obtained with a multiplicity of 12 (Fig. 2b), except that they have an energetic difference of 0.65 eV respectively. This second structure of \(Co_{5}\), in addition to geometrically matching the energy minimum obtained using the 6-311G basis set, also matches the size of the multiplicity, which is shown in Fig. 2b. For \(Co^{+}_{5}\), the re-optimization calculations using the basis sets mentioned above agree that the energy minimum has a multiplicity of 11, both with a triangular bipyramid shape (see Fig. 2c). Previous reports for naked structures agree with that presented here [50]. For NO interaction, with both 6-31G* and 6-311G, a geometry with M=7 is obtained, where the dissociative chemisorption of NO on the cluster surface is evident (see Fig. 2d). Regarding \(Co^{+}_{5}(NO)\), Figs. 2e and 2f present minima with a multiplicity of 10 and 8 with 6-31G*. Geometry with the multiplicity of 8 for that basis set turns out to be the same as that obtained when using 6-311G. In both cases, neutral and cation, the lowest energy isomers always show dissociative chemisorption of NO, where also, nitrogen and oxygen atoms occupy intermediate positions between cobalt atoms. Concerning their spin multiplicity, there is a sensitive lowering, understood in terms of filling electron shells through N and O atoms. For \(Co_{5}(N_{2}O)\), the lowest energy structure found with 6-31G* has a multiplicity of 8, as shown in Fig. 2g. A second geometry for this system, with an energy of 0.08 eV higher than the previous one (M=6), corresponds to the minimum energy found with 6-311G (see Fig. 2h). The calculations made for \(Co^{+}_{5}(N_{2}O)\) with 6-31G* yield two possible energy minima with multiplicities of 9 and 11 with an energy difference of 0.54 eV. The geometry with a multiplicity of 11 is the most energetically favored with 6-311G (see Fig. 2j). In addition to the \(N_{2}O\) dissociative chemisorption on the cluster surface, a possible \(N_{2}\) formation is observed for the last structure.

Fig. 2
figure 2

Geometries obtained for \(Co^{q}_{5}\), \(Co^{q}_{5}(NO)\), and \(Co^{q}_{5}(N_{2}O)\) (q = 0,1). Cobalt atoms are shown in blue green, nitrogen atoms in blue, and oxygen atoms in red. The distances are given in angstroms, BPW91/6-311G without parentheses and BPW91/6-31G* with parentheses. \(M=2S+1\), q is the electron charge, and \(E_{r}\) the relative energy in eV using 6-31G* basis set

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For n=6, Fig. 3 shows that the neutral naked cluster (M =15) prefers a square bipyramidal structure [50]. A minimum distortion is due to the effect of the charge (M =14, see Fig. 3). Interaction with NO lowers (M =12 for neutral and 9 for cation) multiplicity. It is worth mentioning that the dissociative chemisorption of said nitrogen oxides is evident in the interaction of this cluster size with both NO and \(N_{2}O\). According to Fig. 3, minima found with both basis sets coincide in structural arrangements and multiplicity for the minimum energy structures of these systems.

Fig. 3
figure 3

Geometries obtained for \(Co^{q}_{6}\), \(Co^{q}_{6}(NO)\), and \(Co^{q}_{6}(N_{2}O)\) (q = 0,1). Cobalt atoms are shown in blue green, nitrogen atoms in blue, and oxygen atoms in red. The distances are given in angstroms, BPW91/6-311G without parentheses and BPW91/6-31G* with parentheses. \(M=2S+1\), q is the electron charge, and \(E_{r}\) the relative energy in eV using 6-31G* basis set

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The results for \(Co_{7}\) suggest two different lower energy structures calculated by 6-31G*, which have an energy difference of 0.37 eV (see Fig. 4a and b) and the same spin multiplicity. The geometry shown in Fig. 4b is also obtained as the minimum energy for calculations performed by 6-311G. For that basis set, it is clear that the most stable structure for \(Co_{7}\) cluster corresponds to a pentagonal bipyramid having a spin multiplicity of 16, as presented in Fig. 4. Moreover, the number of unpaired electrons found in this work is the same as a result reported by Hong and coworkers [52]. It is important to note that this geometry also agrees with the reported by Zhan [50], Aguilera-Granja, and coworkers [52, 53]. In addition, as mentioned in the literature, the geometry of the small cluster is highly symmetric [53] such as that happening with this geometry. For the cation, the study with the polarized function suggests two structures as energy minima, which differ by 0.47 eV, as shown in Fig. 4c and d. Anew, the second isomer coincides with the minimum found when using the 6-311G basis set. For \(Co^{+}_{7}\) (with a spin multiplicity equal to 15), the results show a geometry quite similar to the neutral case; in other words, no distortion exists when an electron is removed (see Fig. 4). On the other hand, the most favored structure for \(Co_{7}(NO)\) system has a multiplicity of 11 (for both basis sets 6-31G* and 6-311G). This fact suggests that the molecule of NO decreases the multiplicity. For this geometry, Fig. 4 shows that the NO molecule has been dissociatively chemisorbed. Additionally, \(Co_{7}\) distorts due to its interaction with the NO. Cation \(Co^{+}_{7}(NO)\) has a multiplicity equal to 12, and there is no bond between N and O, geometrically coinciding in both basis sets used in this work. In this case, the multiplicity increases in one unit, suggesting the effect of losing an electron. In the same manner as in \(Co_{7}\), the geometry of the cation \(Co^{+}_{7}(NO)\) is similar to \(Co_{7}(NO)\) . For the interaction with \(N_{2}O\), the calculations made with 6-31G* suggest two lower energy isomers, although structurally quite similar, whose multiplicities are 12 and 10. The geometry with M=10 is the most favored (Fig. 4h) for the re-optimization performed with 6-311G. With either basis set, the lowest energy structure for \(Co_{7}(N_{2}O)\) showed the N and O atoms chemisorbed on the center of opposite faces of the geometry of \(Co_{7}\) cluster. For the cationic composition of \(Co^{+}_{7}(N_{2}O)\), Fig. 4 indicates that the most stable structure shows a multiplicity of 7 where dissociative chemisorption of \(N_{2}O\) (for both basis sets involved in this study) occurs; also, a significant similarity with the structure of \(Co_{7}(N_{2}O)\) is observed, suggesting a poor structural effect due to ionization of the neutral system.

Fig. 4
figure 4

Geometries obtained for \(Co^{q}_{7}\), \(Co^{q}_{7}(NO)\), and \(Co^{q}_{7}(N_{2}O)\) (q = 0,1). Cobalt atoms are shown in blue green, nitrogen atoms in blue, and oxygen atoms in red. The distances are given in angstroms, BPW91/6-311G without parentheses and BPW91/6-31G* with parentheses. \(M=2S+1\), q is the electron charge, and \(E_{r}\) the relative energy in eV using the 6-31G* basis set

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Figure 5 shows the geometries for the pure eight-atom Co cluster and interacting with NO and \(N_{2}O\). The geometry of the \(Co_{8}\) system (with a multiplicity of 17) agrees with that reported in the literature [50, 54]. According to Fig. 5a, the results obtained with 6-31G* and 6-311G agree in both M and molecular geometry, suggesting that there is no effect of polarization on this system. It is worth saying that multiplicity increases one unit concerning the system of seven atoms. In addition, the total spin agrees with the result obtained by Hong [52]. The geometry of this system suggests a growth pattern of the cluster from \(Co_{7}\) to \(Co_{8}\) incorporating atoms, one by one, such as that shown by Rodríguez López et al. [53]. \(Co^{+}_{8}\) with a spin multiplicity of 18 keeps the same geometry as the neutral case, which suggests no distortion after removal of one electron of the system. Once again, the results with both basis sets agree. The polarized function finds the geometries in Fig. 5c and d as energy minima for \(Co_{8}(NO)\), both with M=12. The second isomer agrees with the minimum found by 6-311G; the minimum reported has a multiplicity of 12, decreasing considerably regarding the pure system. Also, as shown in Fig. 5, the nitric oxide molecule has no direct bond between N and O, suggesting a fragmentation of the molecule. NO considerably modified the \(Co_{8}\) cluster original arrangement. The calculations made with 6-31G* suggest the geometry shown in Fig. 5e as the most stable for the one shown in Fig. 5f, also differing in the value of the electronic spin multiplicity. However, the 6-311G finds that for the cation, \(Co^{+}_{8}(NO)\) with a multiplicity of 15, slight distortions are noticed relative to the neutral (see Fig. 5). The results obtained for the interaction of \(Co_{8}\) and \(N_{2}O\) show two structures as energy minima with 6-31G* where, according to Fig. 5, there is an energy difference of 0.29 eV. Of these geometries, the one with the highest energy agrees with the structure representing the most stable system for the calculations performed with 6-311G (Fig. 5h). The \(Co_{8}(N_{2}O)\) system with a multiplicity of 11 shows a dissociation of \(N_{2}O\) molecule. A similar structure is \(Co^{+}_{8}(N_{2}O)\), with a spin multiplicity equal to 14, where the dissociative chemisorption of \(N_{2}O\) is clear on different faces of the cobalt cluster; that geometry turns out to be the most stable for both basis sets and even with the same multiplicity.

Fig. 5
figure 5

Geometries obtained for \(Co^{q}_{8}\), \(Co^{q}_{8}(NO)\), and \(Co^{q}_{8}(N_{2}O)\) (q = 0,1). Cobalt atoms are shown in blue green, nitrogen atoms in blue, and oxygen atoms in red. The distances are given in angstroms, BPW91/6-311G without parentheses and BPW91/6-31G* with parentheses. \(M=2S+1\), q is the electron charge, and \(E_{r}\) the relative energy in eV using 6-31G* basis set

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The minimums found on the potential energy surface for \(Co_{9}\), \(Co^{+}_{9}\), \(Co_{9}(NO)\), \(Co^{+}_{9}(NO)\), \(Co_{9}(N_{2}O)\), and \(Co^{+}_{9}(N_{2}O)\) are shown in Fig. 6. The structure obtained for \(Co_{9}\) with a multiplicity of 18 is quite similar to the reported earlier by Zhan, and other works [50, 53]. Anew, regarding neutral systems described before, the growth pattern for this system suggests incorporating atoms one by one [53]. When \(Co^{+}_{9}\) is ionized, results suggest that the structure with more energy stability has a multiplicity of 19 and a similar geometry to the neutral. The results obtained by 6-31G* and 6-311G suggest that both energy minima agree on the value of the electronic multiplicity and the geometry for this cluster size. Anew, the two levels of theory used in this work agree on the energy minimum for the structural arrangement of \(Co_{9}(NO)\) (M = 17). With different structures and multiplicities for \(Co^{+}_{9}(NO)\), the polarized basis set finds the geometries shown in Fig. 6d and e, the latter being the one that agrees with the most stable structure with 6-311G. For both the neutral system and the cation, those systems indicate dissociative chemisorption of NO, with the atoms adsorbed on centers of the faces of the original \(Co_{9}\) geometry. In both cases, O and N atoms are chemisorbed on opposite sides of the cluster, differing considerably in structure (neutral and cation).

Fig. 6
figure 6

Geometries obtained for \(Co^{q}_{9}\), \(Co^{q}_{9}(NO)\), and \(Co^{q}_{9}(N_{2}O)\) (q = 0,1). Cobalt atoms are shown in blue green, nitrogen atoms in blue, and oxygen atoms in red. The distances are given in angstroms, BPW91/6-311G without parentheses and BPW91/6-31G* with parentheses. \(M=2S+1\), q is the electronic charge, and \(E_{r}\) the relative energy in eV using 6-31G* basis set

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Finally, the \(Co_{9}(N_{2}O)\) system has a multiplicity of 14 and is shown in Fig. 6 for the minimum energy with 6-311G. This geometry corresponds to the second minimum energy isomer obtained with 6-31G* since the first, with a multiplicity of 16, shows an entirely different structural arrangement. For both levels of theory, this geometry shows dissociative chemisorption of nitrous oxide, where N and O atoms form bridges on \(Co_{9}\) atoms.

The cation of this composition exhibits a multiplicity of 15, for the results with both basis sets. For this system, Fig. 6 shows a distortion for this cationic state compared to the neutral state, thus indicating the effect of ionizing the system.

Global reactivity indexes

This section summarizes the results obtained for all neutral systems about the global reactivity indexes, such as ionization potential, also called ionization energy (IE), electron affinity (EA), chemical potential (\(\mu\)), donor chemical potential (\(\mu ^{-}\)), acceptor chemical potential (\(\mu ^{+}\)), chemical hardness (\(\eta\)), and electrophilicity index (\(\omega\)), all given in eV (see Fig. 7). The data for this properties can be also found in Table S13. The calculations were performed through the BPW91/6-311G and BPW91/6-31G* levels of theory, which are blue and orange boxes in Fig. 7, respectively.

Fig. 7
figure 7

Ionization energy (IE), electron affinity (EA), chemical potential (\(\mu\)), donor chemical potential (\(\mu ^{-}\)), acceptor chemical potential (\(\mu ^{+}\) ), chemical hardness (\(\eta\)), and electrophilicity index (\(\omega\)) all in eV. Calculations were performed through BPW91/6-311G and BPW91/6-31G* levels of theory, blue, and orange boxes, respectively

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For both basis sets used in this work, all clusters present positive values of ionization potential. Except \(Co_{6}(N_{2}O)\), it is clear that the IE obtained with 6-31G* is more significant than that with the other basis set for each system analyzed. Examining \(Co_{4}\), \(Co_{4}(NO)\), and \(Co_{4}(N_{2}O)\) clusters, there is a diminishing going from the first to the second, but an increase when the cluster is interacting with \(N_{2}O\). In the systems with five cobalt atoms, the lowest ionization potential corresponds to the pure cluster. This property increases when NO and \(N_{2}O\) are present, according to the blue boxes, which indicate the results with 6-311G. In the case of six cobalt atoms, \(Co_{6}(NO)\) has the lowest ionization potential. The \(Co_{6}(N_{2}O)\) presents the most considerable stability since more energy is required to form a cation. Regarding the systems with seven atoms of cobalt, IE increases when \(Co_{7}\) interacts with both NO and \(N_{2}O\). This fact suggests that it is necessary to have more energy to remove an electron of the \(Co_{7}(N_{2}O)\) system, which indicates that the nitrous oxide gives stability to the system concerning this descriptor. According to the results obtained for the clusters of eight atoms of cobalt (pure and mixed with NO and \(N_{2}O\)), it is observed that \(Co_{8}(NO)\) becomes less susceptible to positive ionization. In Fig. 7, the IE of \(Co_{9}(NO)\) is much higher than the other two systems for this size of the cluster, which classifies it as less reactive to losing an electron. In the case of four and six cobalt atoms interacting with \(N_{2}O\), that is, \(Co_{4}(N_{2}O)\) and \(Co_{6}(N_{2}O)\), they have the most considerable ionization potential compared to the other systems studied here, and then, more energy is necessary to remove an electron of these systems.

Concerning the electron affinity, structures \(Co_{4}\) and \(Co_{4}(N_{2}O)\) show the same value of this descriptor (6-311G basis set). Therefore, it does not affect the inclusion of the oxide in \(Co_{4}\). The calculations made with the basis set without polarization indicate that \(Co_{5}\) has the highest EA compared with \(Co_{5}(NO)\) and \(Co_{5}(N_{2}O)\), indicating less stability for the addition of an electron. However, by using the 6-31G* basis set, an increase is observed for this descriptor when the pure cluster interacts with NO and \(N_{2}O\). Analysis carried out using 6-311G suggests that for six cobalt atoms, \(Co_{6}(N_{2}O)\) has the minor EA suggesting that it is more stable with electron addition in comparison with the naked structure or containing NO. On the other hand, 6-31G* offers an entirely different value for the electronic affinity of \(Co_{6}(N_{2}O)\), suggesting a behavior almost opposite to that described by the polarized basis set. The EA shows no variation between \(Co_{7}\) and \(Co_{7}(NO)\) systems, this indicates a poor effect of inclusion of NO on the surface of \(Co_{7}\). However, this reactivity slightly increases when \(Co_{7}\) interacts with \(N_{2}O\), and then more energy is removed when the system accepts an electron. This behavior is observed for calculations performed with both basis sets. \(Co_{8}(N_{2}O)\) shows a higher value for the EA of the compounds with eight Co atoms, suggesting that it is a system with greater reactivity when gaining an electron (blue boxes); however, it is the system with the lowest value for this reactivity index when calculations are made using 6-31G*, showing a high impact of polarization for this descriptor. Regarding the reactivity to gain an electron (EA), there is a small difference in the values found for both \(Co_{9}\) and \(Co_{9}(NO)\) (see Fig. 7). The EA is relatively higher for \(Co_{9}(N_{2}O)\); consequently, this system shows more tendency to gain an electron. Interestingly, this behavior for these three systems with nine Co atoms is the same for the calculations developed using the two base sets, being even lower for \(Co_{9}(NO)\) with 6-31G*.

In the analysis of the results obtained with BPW91/6-311G, it is observed that, about the chemical potential, \(Co_{4}(NO)\) and \(Co_{5}(NO)\) present the highest value suggesting these clusters as the most susceptible to transfer electrons in their respective system sizes. The same applies to \(Co_{6}\), compared with the interaction with nitric and nitrous oxide. For the clusters with four and six cobalt atoms interacting with NO, \(\mu ^{-}\) indicates that these systems can give charge. Conversely, for the cluster with five cobalt atoms, the naked one is more liable to transfer electrons. On the other hand, \(Co_{4}(NO)\) and \(Co_{5}(NO)\) are more liable to accept charge since they have the highest values of \(\mu ^{+}\) compared to naked clusters or when \(N_{2}O\) is added. However, \(Co_{6}(N_{2}O)\) is more prone to accept charge than \(Co_{6}\) and \(Co_{6}(NO)\). The values for the chemical potential of \(Co_{7}\), \(Co_{7}(NO)\), and \(Co_{7}(N_{2}O)\) showed the system becomes less reactive, given the interaction with NO and \(N_{2}O\). Specifically, the results indicate that \(Co_{7}(N_{2}O)\) has less tendency to exchange electrons. This fact agrees with the values for \(\mu ^{-}\) and \(\mu ^{+}\) of \(Co_{7}\), since it is the one that shows the highest values for these descriptors, indicating therefore the most reactive compound. From \(\mu\) values of the clusters with eight Co atoms, it can be concluded that there is a greater tendency of \(Co_{8}\) to react with other substances because it presents the highest value for this reactivity index. The result for eight Co atoms is very similar to \(Co_{7}\), that is, \(Co_{8}\) is the system with the most remarkable tendency both to donate (\(\mu ^{-}\)) and to accept (\(\mu ^{+}\)) electrons. Finally, the cluster of nine cobalt atoms becomes less reactive when interacting with \(N_{2}O\) than when it is pure or interacting with NO according to the energy difference observed in Fig. 7. Newly, \(Co_{9}\) shows the highest values of (\(\mu ^{-}\)) and (\(\mu ^{+}\)), suggesting it as the most reactive for this number of Co atoms. When comparing the results described above with the values obtained when using the polarized functions for \(\mu\), \(\mu ^{-}\), and \(\mu ^{+}\), it is observed in the respective graphs that the effect of the polarization on these reactivity indices shows a trend where the pure state clusters have the highest values, except for \(Co_{9}(NO)\), which offers the highest value for \(\mu ^{+}\).

As the chemical hardness (\(\eta\)) measures the stability of a molecule, that is, the resistance to charge transfer of the system, naked clusters with an even number of cobalt atoms should be the most stable. The results indicate that most of the systems follow the same trend, showing little difference for the calculations made with both basis sets. In the analysis carried out with 6-311G, interestingly, the cluster with five cobalt atoms shows a lower value compared with \(Co_{5}(NO)\) and \(Co_{5}(N_{2}O)\). The latter indicates a particular reactivity when the cobalt number is odd. Generally, according to Fig. 7, \(Co_{9}\) is the system with the lowest \(\eta\), suggesting the lowest stability of all the systems studied in this work. On the contrary, the \(Co_{6}(N_{2}O)\) was the most stable structure, presenting the highest hardness value of the clusters studied. In this way, it is possible to observe that the results obtained with this basis set present a more significant variation between systems of different composition, contrary to the results obtained when performing the calculations using the 6-31G* since it seems to tend to a value.

Finally, Fig. 7 shows the electrophilicity index (\(\omega\)). \(\omega\) is used to measure the energy stabilization when the system acquires additional electron charge coming from the surroundings; that is, it measures the tendency of chemical species to receive electrons. The blue boxes, which indicate the results for calculations with 6-311G, show that electrophilicity index increases for \(Co_{4}\) upon the interaction with \(N_{2}O\) and \(Co_{6}\) when interacting with NO (in comparison with naked clusters, respectively). As this index gives information that can be useful to compare two molecules as an electrophile (which is indicated by a higher \(\omega\)), it is evident that modified clusters are affected by electron-withdrawing groups reflecting the highest values for \(\omega\) such as that which happens for the clusters with 7, 8, and 9 cobalt atoms when interacting with NO and \(N_{2}O\). However, for five cobalt atoms, this index decreases upon the interaction with NO and \(N_{2}O\). When comparing the results of the calculations carried out with both basis sets, it is observed that this system of five atoms of Co together with \(Co_{9}(NO)\) presents lower values for \(\omega\) when using the polarized basis function, with the other systems showing higher values when calculations are performed with the 6-31G* basis.

Local reactivity indexes

Figures 8, 9, 10, 11, 12, and 13 show the Fukui functions of the neutral clusters under the theory level BPW91/6-311G. The purple regions on each atom of the respective cobalt cluster define the regions reactive towards nucleophilic attacks (\(f^{+}\)), electrophilic attacks (\(f^{-}\)), and free radicals (\(f^{0}\)).

Fig. 8
figure 8

Fukui functions obtained for \(Co_{4}\), \(Co_{4}(NO)\), and \(Co_{4}(N_{2}O)\). Cobalt atoms are shown in blue green, nitrogen atoms in blue, and oxygen atoms in red. The purple regions define the reactivity towards nucleophilic attacks (\(f^{+}\)), electrophilic attacks (\(f^{-}\)), and radical attacks (\(f^{0}\)). Calculations were performed through BPW91/6-311G level of theory

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In \(Co_{4}\) cluster, it is evident that the four atoms are equally able to receive a nucleophilic attack, but not for an electrophilic attack, where atoms 1 and 3 are the most susceptible to be electrophilically attacked. Because of the asymmetry between \(f^{+}\) and \(f^{-}\), \(f^{0}\) shows atoms 2 and 4 as the most probable sites to receive radical attacks as is expected since \(f^{0}\) is an average of \(f^{+}\) and \(f^{-}\) indexes. For \(Co_{4}(NO)\) cluster, cobalt atom labeled as 1, which is between nitrogen and oxygen atoms, is the ablest to have a nucleophilic attack. Cobalt 1 and 3 are susceptible to electrophilic and radical attacks. Regarding the \(Co_{4}(N_{2}O)\) cluster, the cobalt atoms present the same probability of receiving a nucleophilic reaction. Moreover, nitrogen atoms could also be susceptible to presenting an electrophilic attack, in virtue of the \(f^{-}\) Fukui function. Finally, reaction to a radical could be performed mainly in cobalt 2 and 4.

About systems with five cobalt atoms, the geometry of \(Co_{5}\) seems essential in its reactivity (see Fig. 9). The distances from cobalt atoms 1 and 2 to number 5 are more prominent than the distance from cobalt atoms 3 and 4 to cobalt 5. This apparent reduction of symmetry is likely the reason for the fact that \(f^{+}\) Fukui function is the same in atoms 1 and 2 or atoms 3 and 4. However, when the NO molecule interacts with the cluster, it breaks the symmetry resulting in cobalt 2 as the most reactive to this attack. Additionally, \(N_{2}O\) induces a strong reactivity in cobalt atoms 1 and 4. Concerning the electrophilic attack, Fig. 9 shows a similar reactivity in all cobalt atoms of \(Co_{5}\) cluster, atom 5 being the one with slight differences with the other atoms. The electrophilic attack to \(Co_{5}(NO)\) occurs most likely in the proximity to O and N; that is, atom 3 is the most susceptible to give electrons. It is also evident that the most significant electrophilic reactivity for the cluster \(Co_{5}(N_{2}O)\) is located in the cobalt atom 1 and 2, far away from the oxygen atom. On the other hand, atoms 1 and 2 have a considerable \(f^{0}\) Fukui function in the pure cluster. In the case of the system \(Co_{5}(NO)\), it is evident that cobalt 3 is the most probable atom for receiving a radical attack. Besides, it is worth to note the most proximate atoms to oxygen are the most reactive to radicals. Thus, atoms 1 and 2 are also the most reactive sites to radicals in \(Co_{5}(N_{2}O)\) system.

Fig. 9
figure 9

Fukui functions obtained for \(Co_{5}\), \(Co_{5}(NO)\), and \(Co_{5}(N_{2}O)\). Cobalt atoms are shown in blue green,  nitrogen atoms in blue, and oxygen atoms in red. The purple regions define the reactivity towards nucleophilic (\(f^{+}\)), electrophilic (\(f^{-}\)), and radical attacks (\(f^{0}\)). Calculations were performed through BPW91/6-311G level of theory

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Fukui functions in Fig. 10 for \(Co_{6}\) show reactivity towards nucleophilic attacks centered mainly on the atoms that are the base of the square bipyramid, that is, numbers 1 to 4. On the other hand, atoms 5 and 6 also show reactivity but at a lower level. In the case of the \(Co_{6}(NO)\) system, it is clear that nucleophilic reactivity is more significant in atoms 3 and 4, indicating that the oxygen atom plays an essential role in redistributing densities in the cluster with six cobalt atoms. Concerning the nucleophilic attack at \(Co_{6}(N_{2}O)\) system, atoms 1, 3, and 4 show the most reactive sites. Regarding the electrophilic attack, \(Co_{6}\) cluster exhibits a negligible reactivity in the sites located at the base of the bipyramid, suggesting that the electron density available to give electrons is sparse. In the \(Co_{6}(NO)\) system, atoms forming the base of the bipyramid present the most considerable \(f^{-}\); however, nitrogen and oxygen atoms also have this Fukui function. In the case of \(Co_{6}(N_{2}O)\), all cobalt atoms and both nitrogen atoms are susceptible to give electrons. For radical attacks, the naked cluster has almost no \(f^{0}\) Fukui function. For \(Co_{6}(NO)\), atoms 1, 2, and 6 present a small contribution of the Fukui function; moreover, cobalt atoms closer to oxygen have the lowest reactivity toward radical attacks. Finally, atoms 1, 3, and 4 of cobalt in \(Co_{6}(N_{2}O)\) have the most significant amount of \(f^{0}\), indicating these sites as the most probable to receive a radical attack.

Fig. 10
figure 10

Fukui functions obtained for \(Co_{6}\), \(Co_{6}(NO)\), and \(Co_{6}(N_{2}O)\). Cobalt atoms are shown in blue green, nitrogen atoms in blue, and oxygen atoms in red. The purple regions define the reactivity towards nucleophilic (\(f^{+}\)), electrophilic (\(f^{-}\)), and radical attacks (\(f^{0}\)). Calculations were performed through BPW91/6-311G level of theory

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The local reactivity calculated by Fukui functions for \(Co_{7}\) shows areas with higher nucleophilic probability; this reactivity is on the atoms 6 and 7 corresponding to the atoms that form the axis of the pentagonal bipyramid. Furthermore and according to Fig. 11, all atoms of the system are susceptible to gaining charge density (\(f^{+}\)). On the other side, atoms 1 and 4 show a more considerable affinity to electrophilic attacks. In addition, these atoms concentrate a higher reactive proportion toward a radical attack. The \(Co_{7}(NO)\) composition shows that the regions with nucleophilic susceptibility are in cobalt atoms 1, 2, and 6. These atoms are further away from the oxygen atom (see Fig. 11). In the same table, cobalt atom 3 has the most electrophilic area in this cluster, thus indicating the region with the highest charge concentration. Reactivity toward radical attacks also focuses on the cobalt atom 3, where the largest reactivity area is exhibited. For \(Co_{7}(N_{2}O)\) composition, the Co atom 2 has the area most likely to gain electrons. This atom is between the two nitrogen atoms; and therefore, this reactivity could be dictated by them. The electrophilic reactivity on \(Co_{7}(N_{2}O)\) is on the cobalt atoms 4 and 6, which are far away from the oxygen atom. Figure 11 shows that the reactivity toward radicals is most likely in atoms 2 and 4 of cobalt and lesser amounts for cobalt atom 3.

Fig. 11
figure 11

Fukui functions obtained for \(Co_{7}\), \(Co_{7}(NO)\), and \(Co_{7}(N_{2}O)\). Cobalt atoms are shown in blue green, nitrogen atoms in blue, and oxygen atoms in red. The purple regions define the reactivity towards nucleophilic (\(f^{+}\)), electrophilic (\(f^{-}\)), and radical attacks (\(f^{0}\)). Calculations were performed through BPW91/6-311G level of theory

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The \(Co_{8}\) system shows no significant reactivity toward nucleophilic attacks, according to Fig. 12. However, for the electrophilic attack, this system has susceptible areas in all cobalt atoms. The regions that most likely react toward attacks with radical species are on atoms 5 and 8. The \(Co_{8}(NO)\) composition shows that the nucleophilic attack could be more likely in atoms 1, 3, and 8. Electrophilic zones are present on atoms 1, 5, and 6, further away from the atoms of O. The highest reactivity toward radicals on \(Co_{8}(NO)\) system is on cobalt atoms 1, 3, and 5. The \(Co_{8}(N_{2}O)\) composition shown in Fig. 12 indicates that the nucleophilic activity is on the cobalt atoms 3 and 7; interestingly, these atoms have closer proximity to the pair of N atoms. The electrophilic reactivity for the same system is on distant regions to the zone where the nitrogen atoms have been chemisorbed, that is, on cobalt atoms 1 and 6. The average of the two reactivity indexes discussed above has higher intensity on atoms 1, 3, and 6.

Fig. 12
figure 12

Fukui functions obtained for \(Co_{8}\), \(Co_{8}(NO)\), and \(Co_{8}(N_{2}O)\). Cobalt atoms are shown in blue green, nitrogen atoms in blue, and oxygen atoms in red. The purple regions define the reactivity towards nucleophilic (\(f^{+}\)), electrophilic (\(f^{-}\)), and radical attacks (\(f^{0}\)). Calculations were performed through BPW91/6-311G level of theory

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Figure 13 shows for \(Co_{9}\) that the area with the most possibility to receive nucleophilic and electrophilic attacks is on atoms 1, 3, 4, and 6, which are at the vertices of this hexagonal structure. Nonetheless, atoms 2 and 5 have equivalent regions that are susceptible to these reactivities. Additionally, the nine atoms of the Co cluster should not be reactive toward radicals since the surface does not show susceptible areas. Fukui functions for the \(Co_{9}(NO)\) system can also be observed in Fig. 13; it is clear that the susceptible regions toward nucleophilic attack are present in all cobalt atoms, being in the smaller amount on atoms that are close to the oxygen atom. For \(f^{-}\), it is shown that the atoms 2, 4, 8, and 9 are more likely to lose charge density. Finally, for reactivity towards radicals, all cobalt atoms of \(Co_{9}(NO)\) system show susceptible areas, except atom 3, which is close to the oxygen atom. Figure 13 further shows the Fukui functions for \(Co_{9}(N_{2}O)\). Atoms 1, 7, and 9 show the largest areas susceptible to receive both nucleophilic and electrophilic attacks, being also the most favored for a radical attack.

Fig. 13
figure 13

Fukui functions obtained for \(Co_{9}\), \(Co_{9}(NO)\), and \(Co_{9}(N_{2}O)\). Cobalt atoms are shown in blue green, nitrogen atoms in blue, and oxygen atoms in red. The purple regions define the reactivity towards nucleophilic (\(f^{+}\)), electrophilic (\(f^{-}\)), and radical attacks (\(f^{0}\)). Calculations were performed through BPW91/6-311G level of theory

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Binding energies for \(Co^{q}_{n}(NO)\) and \(Co^{q}_{n}(N_{2}O)\) where n=4–9 and q=0–1

For a system, the most negative binding energy is the most energy required to separate the system; and therefore, the system has high stability. In Table 3, it is shown the binding energies (in eV) for the different cluster sizes interacting with NO and \(N_{2}O\). The calculations were developed under the theory levels BPW91/6-311G and BPW91/6-31G*. For all cluster sizes interacting with both nitrous oxides, the binding energies are negatives, suggesting that clusters with nitrous oxides are more stable than Co\(^q\), N, and O separated atoms. In general, neutral systems showed larger \(E_{b}\)’s than cation systems, which imply less stability. In addition, for these neutral systems, an even-odd behavior is observed, where systems with an odd number of Co atoms exhibit more negative \(E_{b}\)’s, indicating more stability compared to those with an even number of Co atoms. For cation systems, the \(E_{b}\)’s are lower with the increasing number of Co atoms, suggesting the dependence of stability on the size of the cluster, and hence the high probability of formation of the systems studied here. It is worth mentioning that this is observed in the experimental literature [15, 18].

Table 3 Binding energy (\(E_{b}\)) for \(Co^{q}_{n}(NO)\) and \(Co^{q}_{n}(N_{2}O)\) (n = 4–9 and q = 0,1)
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Summary and conclusions

Molecular dissociation of nitric oxide and nitrous oxide is one of the crucial tasks that would help reduce and prevent environmental problems, and cobalt clusters show an essential contribution due to their catalytic activity. From these results, the dissociative chemisorption of NO and \(N_{2}O\) on small cobalt clusters is evident, and no cluster fragmentation occurs, which is in line with reported data. The present study shows an alternative for obtaining information on these clusters and, in this way, makes possible the design of catalysts that involve the segmentation of these molecules. Also, according to the results obtained in this work, it is suggested that the methodology used is adequate for searching minimums on the potential energy surface. Due to the remarkable similarity between the pure systems and the reported geometries in the literature, that is, since the geometries found for the clusters with 4, 5, 6, 7, 8, and 9 Co atoms are in good agreement with other works, it inspires confidence that the methodology followed here is adequate and affirms that the rest of the structures correspond to a minimum. In this way, one can suggest that cobalt clusters act dissociating NO and \(N_{2}O\) molecules.

Interestingly, global reactivity indexes show an even-odd effect. IE is one of the physical properties that are currently amenable to experimental measurement. Also, their dependence on cluster size provides information about the electronic behavior of these species. According to the results with BPW91/6-311G, it is observed that cobalt clusters with an even number of atoms show an increase in IE descriptor as the cluster interacts with \(N_{2}O\) and decreases when it interacts with NO. For 7, 8, and 9 cobalt atoms, the clusters showed the lowest IE compared with the species containing NO and \(N_{2}O\). Concerning the IE results with BPW91/6-31G*, except for the clusters with five Co atoms, in all other systems, ionization potential increases when comparing the cluster pure with the interaction with NO and \(N_{2}O\). EA also suggested an even-odd behavior for the naked structures, where clusters with an odd number of cobalt atoms have higher electron affinity values. About the chemical potential, for all system sizes, except \(Co_{8}\), this descriptor suggests less reactivity when the clusters are interacting with \(N_{2}O\). In this sense, these systems present more negative values of \(\mu\). This fact agrees with the \(\mu ^{-}\) values except for the \(Co_{8}\) and \(Co_{9}\) sizes, where the clusters with the slightest tendency to transfer electrons are when the \(Co_{8}\) and \(Co_{9}\) systems interact with NO. \(Co_{7}\), \(Co_{8}\), and \(Co_{9}\), shows the highest values for \(\mu ^{-}\) and \(\mu ^{+}\), indicating, therefore, that they are more reactive than when they interact with NO or \(N_{2}O\). The chemical hardness values also show an even-odd behavior for the naked structures. \(Co_{9}\) is the system with the lowest resistance to charge transfer, suggesting the lowest stability of all the systems studied here. Finally, under the level of theory BPW91/6-311G, the clusters with 7, 8, and 9 cobalt atoms have a better tendency to stabilize an electron acquired from their environment when interacting with NO and \(N_{2}O\), due to the highest values of \(\omega\).

Through the Fukui function, it is possible to predict the local reactivity of the systems. For all the studied systems in this work, nucleophilic, electrophilic, and radical sites were investigated to predict the next possible site to attack on each system.

We believe that the results presented here may be valuable for a better understanding of Co clusters, their reactivity, and their use in dissociating toxic environmental molecules such as NO and \(N_{2}O\).