Reactions of N₂O and CO on neutral Rh₁₀O_n clusters: a density functional study

Abstract

Density functional theory calculations were performed to identify product, reactant and intermediate dissociative/associative structures for the oxygen abstraction and addition reactions: Rh₁₀O_n + CO → Rh₁₀O_n−1 + CO₂, n = 1–5 and Rh₁₀O_n + N₂O → Rh₁₀O_n+1 + N₂, n = 0–4 reactions. In the case of the oxygen abstraction reactions, the energetics of the reaction path were very similar in energy regardless of the number of oxygen atoms on the Rh₁₀O_n cluster, whereas for the addition of oxygen to the Rh₁₀O_n cluster, the reaction was found to become significantly less exothermic with each successive addition of oxygen.

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1 Introduction

Transition metal clusters have been of interest for several decades as both models of surfaces¹ and entities in their own right, where their remarkable size-² and structure-dependent^3–5 properties make them highly attractive as potential heterogeneous catalysts.^6–8 Late transition metal clusters, especially those of rhodium have maintained particular interest over that time with a multitude of studies employing a variety of spectroscopic and computational methods to probe their structure and behaviour.

Many of these studies have been motivated by the use of rhodium in automotive three-way catalytic converters⁹ and the reactions of N₂O,^10–12 NO,^13–22 CO^23,24 and CO₂²⁵ have been extensively investigated on both pure and doped^26–28 rhodium clusters. Given that spectroscopic methods such as infrared multiple photon dissociation spectroscopy (IRMPD)²⁹ use mass spectroscopy to identify the species of interest, they generally provide information only about the products of reactions and reactivity studies therefore often include computational investigation,^30,31 typically using density functional theory (DFT), but sometimes including higher level methods such as CCSD(T).³²

Two particular reactions that are of interest for automotive catalytic converters are the reduction of NO_x to N₂ and the oxidation of CO to CO₂. Both of these reactions have specifically been investigated on rhodium^11,26 and platinum^33,34 clusters. Aviles et al. undertook calculations to elucidate the N₂O reduction mechanism on small Rh_n clusters (n = 2–4), and determined that the N₂–O bond was readily broken by the Rh_n cluster, but the N₂ bond was not.¹² Yamamoto et al. used both IRMPD and DFT methods to determine reaction paths for the reduction of N₂O by Pt₆ and Pt₇ clusters.,³⁵ while Hernández and coworkers used DFT to elucidate the reaction of N₂O on Pt₈.³⁶

Yet despite these successes, computational investigation remains challenging, especially on large and/or late transition metal clusters: Even employing frozen core/effective core potential approaches, clusters still possess a large number of electrons. A wide range of spin multiplicities need to be considered. And many clusters have multiple, very different geometries that all occur within a small energetic range,^5,37,38 that is thermally accessible and/or within the error of computational methods. For some clusters, the manifold of low-energy geometries may be so dense and the ease of interconversion so facile, that the clear identification of a ground state is challenging. Rh₁₀ has been identified as one such fluxional molecule.³⁹ Such a dense manifold of structures, energies and spin multiplicities is additionally challenging when employing reaction path following methods such as Quasi-Synchronous transit and intrinsic reaction coordinate methods, as reaction paths may follow a comparatively broad path, where spin and cluster geometry may differ throughout a reaction.

In 2012, Yamada et al. undertook a mass-spectroscopic study on neutral rhodium clusters, Rh_n (n = 10–28).⁴⁰ By injecting both N₂O and CO into their cluster source, they were able to demonstrate a full catalytic cycle and observed that the distribution of oxide products Rh_nO_m was determined by an equilibrium between the oxygen being transferred to the Rh_nO_mm = 0–4 cluster from N₂O and oxygen being abstracted from the Rh_nO_mm = 1–5, and that the rate constants for the oxygen abstraction reaction were 2–3 orders of magnitude higher on clusters where m ≥ 4 than those clusters with three or less oxygen atoms. They concluded that oxidized rhodium clusters were more effective as catalysts compared to bare clusters. Yamada's study used experimentally derived rate constants and did not include any calculated structures or reaction paths. Indeed the size of the rhodium clusters employed by Yamada et al., Rh_n (n = 10–28) makes DFT challenging.

In this work, we revisit the catalytic cycle presented by Yamada et al. for the smallest rhodium cluster size, Rh₁₀. As reaction path following on such flat potential energy surfaces (PES) is difficult, we employ a fully stochastic search procedure followed by DFT calculations to identify the reactants, products and the bond forming/breaking intermediate for both the oxygen transfer and oxygen abstraction reactions on the Rh₁₀ cluster.

2 Computational method

The geometry of Rh₁₀ was determined using the Kick³ stochastic structure generator supplied with 10 Rh atoms.^41,42 Following optimization, the lowest energy Rh₁₀ geometry was then employed as a fragment in further Kick³ runs with n oxygen atoms in order to determine the lowest energy Rh₁₀O_n structures, for n = 1–5. A key point on each Rh₁₀O_n → Rh₁₀O_n±1 reaction path, is the species where the N₂⋯O bond is being broken, or where the O⋯CO bond is being formed. To identify these intermediate species directly, N₂O and OCO fragments with elongated NO and CO bonds respectively, were used as input to Kick³ to identify the Rh₁₀O_n CO and Rh₁₀O_n N₂ intermediate species. 1000 stochastic geometries were generated for each stoichiometry.

The stochastically generated structures were optimized in a two-stage procedure: Firstly using the semi-empirical GFN1-xTB method,⁴³ maintaining the geometry of all fragments employed in the stochastic structure generation, before a full optimization using BP86/DZP and then re-optimization using the TZ2P basis set. Up to the 4p electrons of rhodium were treated using a frozen core. Structures were initially optimized in the octet multiplicity before re-optimization with each multiplicity from doublet to 20-tet. 18- and 20-tet structures were found to lie very high in energy and were consequently not considered in analysis. All calculations were undertaken using AMS2022.1 using scalar relativivity and applying an electronic temperature of 373.15 K.⁴⁴

To confirm that the BP86^45,46 functional is sufficient, we also tested the TPSS⁴⁷ functional with the D3(BJ)^48,49 dispersion correction. The choice of functional made no significant difference to the relative ordering of different multiplicity Rh₁₀O_n structures, as shown by difference-of-difference values in the ESI.† GGA functionals are expected to only give qualitative information about barrier heights, due to their self-interaction error⁵⁰ and this is evident, with BP86 intermediate relative energies for oxygen abstraction reactions being consistently ≈0.4 eV higher in energy than the TPSS-calculated values and BP86 product energies being consistently ≈0.5 eV lower than the TPSS-calculated values. For the oxygen addition reaction paths, the TPSS relative energies for intermediate and product species are lowered w.r.t. the BP86 paths by ≈0.6 and 0.3 eV respectively.

After optimization of Rh₁₀O_n, Rh₁₀O_n CO and Rh₁₀O_n N₂ structures, Rh₁₀O_n → Rh₁₀O_n±1 reaction paths were determined by geometric RMSD. For each reaction, the lowest energy Rh₁₀O_n reactant was chosen, and the nearest Rh₁₀O_±1 geometric match was determined (Rh₁₀O_n+1 for Rh₁₀O_n + N₂O → Rh₁₀O_n+1+ N₂ and Rh₁₀O_n−1 for Rh₁₀O_n + CO → Rh₁₀O_n−1 + CO₂). The intermediate species was determined in the same manner, by determining the RMSD with the reactant and then confirming RMSD with the product. After the addition of the first two oxygen atoms to Rh₁₀, multiple oxygen abstraction reaction paths are possible, corresponding to abstraction of different oxygen atoms, the path presented in this work is the one containing the most favourable (lowest energy) intermediate geometry.

It is not necessarily the case that the lowest energy Rh₁₀O_n will lead to the lowest energy Rh₁₀O_n±1 species, and because of the circular nature of the catalytic reaction, each Rh₁₀O_n species appears as both a product and a reactant. Therefore, a second reaction path, beginning with the lowest energy reactant was also determined following the same geometric RMSD procedure. We denote the two reaction paths as lowest energy reactant (LER) and lowest energy product (LEP).

3 Results and discussion

Rh₁₀

Several computational^38,51 and experimental studies have investigated the structure of the Rh₁₀ cluster in neutral, cationic and anionic charge states, identifying a dense manifold of low-energy structures. Knickelbein et al. measured the static electric dipole polarizabilities of Rh_nn = 5–28 clusters and suggested that Rh₁₀ is a fluxional molecule.³⁹ The lowest energy Rh₁₀ octet species identified in this work is a structure with C_2v symmetry and a binding energy of −1.7686006 a.u. and is shown in Fig. 1. Re-optimization of each identified Rh₁₀ isomer at other multiplicities (doublet to 20-tet) revealed the lowest energy isomer overall, is the same structure, but with 16-tet multiplicity and a binding energy of −1.7909319 a.u. This structure corresponds to the sc3 structure identified for Rh₁₀⁺ by Harding et al.⁵¹

Fig. 1 Lowest energy geometry of Rh₁₀.

Rh₁₀O_n + CO → Rh₁₀O_n−1 + CO₂ reactions

Rh₁₀O + CO → Rh₁₀ + CO₂. Only the predetermined minimum Rh₁₀ structure is considered in this work, therefore there is only one Rh₁₀O + CO → Rh₁₀ + CO₂ reaction path, defined by the lowest energy Rh₁₀O species, which has a μ²-bound oxygen. The reaction path from this species is shown in Fig. 2. The Rh₁₀O⋯C distance in the intermediate state is 2.72 Å. Overall, the reaction path is very flat with the separated products (Rh₁₀ + CO₂) lying only 0.15 eV above the reactants and the intermediate Rh₁₀O⋯CO structure being effectively isoenergetic with the Rh₁₀O + CO reactants. For the reactant and intermediate structures, all multiplicities (doublet to 16-tet) were found within a range of 0.35 eV. Different multiplicities of the Rh₁₀ product span an energetic range of 0.71 eV.

Fig. 2 Rh₁₀O + CO → Rh₁₀ + CO₂ reaction path for the lowest energy Rh₁₀O species. Relative energies are shown for the octet surface in eV. The octet surface is bolded, all other multiplicities are shown with thin lines.

Rh₁₀O₂ + CO → Rh₁₀O + CO₂. Both the LER (Fig. 3) and LEP (Fig. 4) reaction paths arrive at the same μ² Rh₁₀O product – i.e. the lowest energy Rh₁₀O₂ can follow a reaction path that results in the lowest energy Rh₁₀O. The lowest energy Rh₁₀O₂ structure has two μ²-bound oxygen atoms in symmetry-equivalent positions. The intermediate CO-bound species, removing either of these oxygen atoms is −1.93 eV lower in energy than the separated reactants and has a Rh₁₀O⋯C distance of 3.19 Å. Beginning with the lowest energy μ² Rh₁₀O product and searching for a reaction path towards any Rh₁₀O₂ reactant, yields a μ³μ² reactant 0.45 eV higher in energy than the lowest energy Rh₁₀O₂ and a corresponding Rh₁₀OO·CO intermediate. The intermediate structure removes the μ³-bound oxygen atom and is at −1.68 eV, 0.25 eV higher in energy than the intermediate in the LER pathway.

Fig. 3 Rh₁₀O₂ + CO → Rh₁₀O + CO₂ reaction path for the lowest energy Rh₁₀O₂ species. Relative energies are shown for the octet surface in eV. The octet surface is bolded, all other multiplicities are shown with thin lines.

Fig. 4 Rh₁₀O₂ + CO → Rh₁₀O + CO₂ reaction path for the lowest energy product (Rh₁₀O) species. Relative energies are shown for the octet surface in eV. Zero energy is the lowest energy Rh₁₀O₂ reactant, as shown in Fig. 3. The octet surface is bolded, all other multiplicities are shown with thin lines.

Rh₁₀O₃ + CO → Rh₁₀O₂ + CO₂. The reaction path from the lowest energy Rh₁₀O₃ structure is shown in Fig. 5. The lowest energy Rh₁₀O₃ structure has the same two μ²-bound oxygen atoms as in Rh₁₀O₂ plus a μ³-bound oxygen atom that shares a Rh atom with one of the μ²-bound oxygen atoms. In the Rh₁₀O₂O·CO intermediate structure, the incoming CO approaches the side of the cluster with both the μ³ and μ² oxygen atoms, but abstracts the μ²-bound oxygen. The intermediate is at −1.67 eV and the Rh₁₀O⋯C distance is 3.39 Å. The Rh₁₀O₂ product is at −0.15 eV. Removal of the μ³-bound oxygen instead of the μ²-bound oxygen yields the lowest energy Rh₁₀O₂ structure at −0.59 eV, and this pathway is shown in Fig. 6. The intermediate in this pathway is at −1.37 eV, 0.30 eV higher in energy than the LER path intermediate and the Rh₁₀O⋯C distance is 2.91 Å.

Fig. 5 Rh₁₀O₃ + CO → Rh₁₀O₂ + CO₂ reaction path for the lowest energy Rh₁₀O₃ species. Relative energies are shown for the octet surface in eV. The octet surface is bolded, all other multiplicities are shown with thin lines.

Fig. 6 Rh₁₀O₃ + CO → Rh₁₀O₂ + CO₂ reaction path for the lowest energy product (Rh₁₀O₂) species. Relative energies are shown for the octet surface in eV. Zero energy is the lowest energy Rh₁₀O₃ reactant, as shown in Fig. 5. The octet surface is bolded, all other multiplicities are shown with thin lines.

Rh₁₀O₄ + CO → Rh₁₀O₃ + CO₂. The lowest energy Rh₁₀O₄ structure has three μ³ -bound oxygen atoms and a μ²-bound oxygen. The reaction path from this species is shown in Fig. 7 and removes one of the μ³ oxygen atoms. The products (Rh₁₀O₃ + CO₂) are nearly isoenergetic with the reactants. As the oxygen atoms in the Rh₁₀O₄ cluster are all bound (in part) to the same four Rh atoms, the approaching CO is likely to interact with more than one oxygen atom. In the lowest energy intermediate structure identified, the CO could possibly abstract one of two a μ³-bound oxygen atoms, with Rh₁₀O⋯C distances of 3.08 and 3.22 Å respectively.

Fig. 7 Rh₁₀O₄ + CO → Rh₁₀O₃ + CO₂ reaction path for the lowest energy Rh₁₀O₄ species. Relative energies are shown for the octet surface in eV. The octet surface is bolded, all other multiplicities are shown with thin lines.

The lowest energy Rh₁₀O₃, as above, possesses two μ²-bound oxygen atoms and one μ³-bound oxygen, so it is not a possible product (without rearrangement) from the lowest energy Rh₁₀O₄ structure. The path ending with the lowest energy Rh₁₀O₃ structure beings with a high energy Rh₁₀O₄ structure (+1.25 eV), which has two μ³-bound oxygen atoms and two μ²-bound oxygen atoms (Fig. 8). The close Rh₁₀O⋯C distances in the intermediate structure, are 3.16 Å for the μ³-bound oxygen that is abstracted by the CO and 3.39 Å for the μ²-bound oxygen that remains.

Fig. 8 Rh₁₀O₄ + CO → Rh₁₀O₃ + CO₂ reaction path for the lowest energy product (Rh₁₀O₃) species. Relative energies are shown for the octet surface in eV. Zero energy is the lowest energy Rh₁₀O₄ reactant, as shown in Fig. 7. The octet surface is bolded, all other multiplicities are shown with thin lines.

Rh₁₀O₅ + CO → Rh₁₀O₄ + CO₂. The lowest energy Rh₁₀O₅ structure has four μ³-bound oxygen atoms and a μ²-bound oxygen. The reaction path from this species is shown in Fig. 9 and removes one of the μ³ oxygen atoms. Similarly to the Rh₁₀O₄ to Rh₁₀O₃ LER reaction path, the products (Rh₁₀O₄ + CO₂) are nearly isoenergetic with the reactants, the relative energy of the Rh₁₀O₅·CO intermediate species is also similar to that of the Rh₁₀O₄·CO intermediate (Fig. 7), at −1.61 eV. The approaching CO abstracts a μ³ -bound oxygen atom from 3.04 Å.

Fig. 9 Rh₁₀O₅ + CO → Rh₁₀O₄ + CO₂ reaction path for the lowest energy Rh₁₀O₅ species. Relative energies are shown for the octet surface in eV. The octet surface is bolded, all other multiplicities are shown with thin lines.

The LEP reaction path is shown in Fig. 10. The Rh₁₀O₄ product has a relative energy of −0.96 eV and the lowest energy reaction path removes a μ¹-bound oxygen atom that is bound to a previously bare Rh atom.

Fig. 10 Rh₁₀O₅ + CO → Rh₁₀O₄ + CO₂ reaction path for the lowest energy product (Rh₁₀O₄) species. Relative energies are shown for the octet surface in eV. Zero energy is the lowest energy Rh₁₀O₅ reactant, as shown in Fig. 9. The octet surface is bolded, all other multiplicities are shown with thin lines.

Rh₁₀O_n + N₂O → Rh₁₀O_n+1 + N₂ reactions

Rh₁₀ + N₂O → Rh₁₀O + N₂. Only the predetermined minimum Rh₁₀ structure is considered in this work, therefore there is only one Rh₁₀ + N₂ → Rh₁₀O + N₂ reaction path, defined by the lowest energy Rh₁₀O species, which has a μ²-bound oxygen. The reaction path from this species is shown in Fig. 11. The energetic profile of the reaction path is quite different from the reduction reactions, with the Rh₁₀O.N₂ intermediate structure being −4.54 eV below zero energy and the Rh₁₀O + N₂ products being found at −3.57 eV. The Rh₁₀O·N₂ separation is 3.07 Å.

Fig. 11 Rh₁₀ + N₂O → Rh₁₀O + N₂ reaction path for the lowest energy product (Rh₁₀O) species. Relative energies are shown for the octet surface in eV. Zero energy is the defined Rh₁₀ reactant, as shown in Fig. 1. The octet surface is bolded, all other multiplicities are shown with thin lines.

Rh₁₀O + N₂O → Rh₁₀O₂ + N₂. The lowest energy Rh₁₀O structure, as previously seen, has a μ²-bound oxygen. The reaction path from this species is shown in Fig. 12 and deposits a μ³ oxygen atom such that no Rh atoms bind both oxygen atoms. The intermediate structure has a relative energy of −3.83 eV and the Rh₁₀O·N₂ distance is 3.96 Å. The products are found at −2.62 eV.

Fig. 12 Rh₁₀O + N₂O → Rh₁₀O₂ + N₂ reaction path for the lowest energy Rh₁₀O species. Relative energies are shown for the octet surface in eV. The octet surface is bolded, all other multiplicities are shown with thin lines.

The LEP reaction path is shown in Fig. 13. The Rh₁₀O₂ product has a relative energy of −3.07 eV and deposits the second of the two symmetry-equivalent μ²-bound oxygen atoms.

Fig. 13 Rh₁₀O + N₂O → Rh₁₀O₂ + N₂ reaction path for the lowest energy product (Rh₁₀O₂) species. Relative energies are shown for the octet surface in eV. Zero energy is the lowest energy Rh₁₀O reactant, as shown in Fig. 12. The octet surface is bolded, all other multiplicities are shown with thin lines.

Rh₁₀O₂ + N₂O → Rh₁₀O₃ + N₂. Both the LER (14) and LEP (15) reaction paths form the lowest energy Rh₁₀O₃ product, which has two μ² and one μ³-bound oxygen atom. This product (Rh₁₀O₂ + N₂) has an energy of −2.83 eV relative to the lowest energy Rh₁₀O₂ + N₂O. The intermediate structure connecting these two lowest energy Rh₁₀O₂ and Rh₁₀O₃ structures, must therefore deposit the μ³ -bound oxygen atom. This intermediate is found at −3.83 eV and the Rh₁₀O·N₂ distance is 3.96 Å. If instead, of depositing the μ³ -bound oxygen atom, the intermediate is chosen to deposit one of the two μ²-oxygen atoms, the reaction path in Fig. 15 is observed. The Rh₁₀O₂ reactant and Rh₁₀O₃·N₂ intermediate structures are found at +0.44 and −3.83 eV respectively - i.e. the intermediate structure is 0.3 eV higher in energy than the LER intermediate and has a Rh₁₀O·N₂ distance of 2.99 Å.

Fig. 14 Rh₁₀O₂ + N₂O → Rh₁₀O₃ + N₂ reaction path for the lowest energy Rh₁₀O₂ species. Relative energies are shown for the octet surface in eV. The octet surface is bolded, all other multiplicities are shown with thin lines.

Fig. 15 Rh₁₀O₂ + N₂O → Rh₁₀O₃ + N₂ reaction path for the lowest energy product (Rh₁₀O₃) species. Relative energies are shown for the octet surface in eV. Zero energy is the lowest energy Rh₁₀O₂ reactant, as shown in Fig. 14. The octet surface is bolded, all other multiplicities are shown with thin lines.

Rh₁₀O₃ + N₂O → Rh₁₀O₄ + N₂. The LER reaction path for the Rh₁₀O₃ + N₂O → Rh₁₀O₄ + N₂ reaction is shown in Fig. 16. The Rh₁₀O₄ product at −2.16 eV has two μ² and two μ³-bound oxygen atoms and the corresponding intermediate structure at −3.23 eV has a Rh₁₀O⋅N₂ distance of 3.20 Å, depositing the second μ³ oxygen atom. The path leading to the lowest energy Rh₁₀O₄ product is shown in Fig. 17. The Rh₁₀O₄ product has three μ³ and one μ²-bound oxygen atom, and the lowest energy intermediate structure deposits the third μ³-bound oxygen atom. The geometrically matching Rh₁₀O₃ reactant with two μ³ and one μ²-bound oxygen, is very close in energy to the lowest energy Rh₁₀O₃ structure (two × μ³ and 1 × μ²), with a relative energy of only 0.07 eV. The Rh₁₀O·N₂ distance in the intermediate structure is 3.56 Å.

Fig. 16 Rh₁₀O₃ + N₂O → Rh₁₀O₄ + N₂ reaction path for the lowest energy Rh₁₀O₃ species. Relative energies are shown for the octet surface in eV. The octet surface is bolded, all other multiplicities are shown with thin lines.

Fig. 17 Rh₁₀O₃ + N₂O → Rh₁₀O₄ + N₂ reaction path for the lowest energy product (Rh₁₀O₄) species. Relative energies are shown for the octet surface in eV. Zero energy is the lowest energy Rh₁₀O₃ reactant, as shown in Fig. 16. The octet surface is bolded, all other multiplicities are shown with thin lines.

Rh₁₀O₄ + N₂O → Rh₁₀O₅ + N₂. The reaction path from the lowest energy Rh₁₀O₄ structure is shown in Fig. 18. The Rh₁₀O₅ product lies at −2.41 eV has two μ² and three μ³-bound oxygen atoms and the corresponding intermediate structure, depositing the second μ² oxygen atom, lies very close in energy at −2.48 eV and has a Rh₁₀O·N₂ distance of 3.38 Å. The lowest energy Rh₁₀O₅ structure has one μ² and four μ³-bound oxygen atoms and the path leading to this product is shown in Fig. 19. The lowest energy matching intermediate structure at −3.75 eV corresponds to depositing the fourth μ³-bound oxygen atom, but the corresponding Rh₁₀O₄ reactant is high in energy, at +1.00 eV. The Rh₁₀O·N₂ distance in the intermediate structure is 3.94 Å.

Fig. 18 Rh₁₀O₄ + N₂O → Rh₁₀O₅ + N₂ reaction path for the lowest energy Rh₁₀O₄ species. Relative energies are shown for the octet surface in eV. The octet surface is bolded, all other multiplicities are shown with thin lines.

Fig. 19 Rh₁₀O₄ + N₂O → Rh₁₀O₅ + N₂ reaction path for the lowest energy product (Rh₁₀O₅) species. Relative energies are shown for the octet surface in eV. Zero energy is the lowest energy Rh₁₀O₄ reactant, as shown in Fig. 18. The octet surface is bolded, all other multiplicities are shown with thin lines.

Comparison of the Rh₁₀O_n + CO → Rh₁₀O_n−1 + CO₂ reaction paths for n = 1–5 reveals a largely similar energetic landscape regardless of n, as shown in Fig. 20 and 21 for the LER and LEP reaction paths respectively. For the pathways beginning with the lowest energy reactant, with the exception of the last abstraction of oxygen, resulting in a bare rhodium cluster (i.e. Rh₁₀O + CO → Rh₁₀O + CO₂, Fig. 2), where the intermediate structure lies very close to the energy of both the products and reactants, the four reaction paths have almost the same energetics, with the Rh₁₀O_n·CO intermediate structure being found in a narrow range between −1.67 and −1.93 eV relative to the reactants. The Rh₁₀O_n−1 products are found in a similarly narrow energy range, from −0.35 eV for the Rh₁₀O₂ reaction to +0.06 eV for the Rh₁₀O₄ reaction. Considering the reaction paths arriving at the lowest energy products, shows similar behaviour, with intermediate structures consistently 2 eV lower in energy than the corresponding reactants. The ΔE (products-reactants) shows some deviation with values of 0.80, 0.59, 1.26 and 1.17 eV for n = 2–5 respectively, indicating that the Rh₁₀O_n−1 products are more stable for the Rh₁₀O₄ and Rh₁₀O₅ reactants. This is consistent with the higher rates observed for the Rh_mO_n + CO → Rh_mO_{n− 1} + CO₂ reactions for n = 4,5 by Yamada et al.⁴⁰

Fig. 20 Rh₁₀O_n + CO → Rh₁₀O_n−1 + CO₂, n = 1–5 LER reaction paths overlaid. The Rh₁₀O +CO₂ path is black, the Rh₁₀O_n paths are coloured blue, green, yellow and red, for n = 2–5 respectively.

Fig. 21 Rh₁₀O_n + CO → Rh₁₀O_n−1 + CO₂, n = 1–5 LEP reaction paths overlaid. Zero energy is the lowest energy Rh₁₀O reactant, as shown in Fig. 2. The Rh₁₀O + CO₂ path is black, the Rh₁₀O_n paths are coloured blue, green, yellow and red, for n = 2–5 respectively.

Comparison of the Rh₁₀O_n + N₂O → Rh₁₀O_n+1 + N₂ reaction paths for n = 0–4, shows a very different pattern to the abstraction of oxygen reactions. Considering first the LER paths, shown overlaid in Fig. 22. As the Rh₁₀O_n reactant is increasingly oxidized (i.e. as n increases), the relative energy of both the intermediate and product increases. With increasing oxygen content, the energy of the intermediate species rises by approximately 0.7 eV. The relative energy of the Rh₁₀O_n+1 product also rises, but less linearly, with the Rh₁₀O₄ and Rh₁₀O₅ products switching order. A change in behaviour is clearly seen for the Rh₁₀O₄ + N₂O → Rh₁₀O₅ + N₂ reaction, where the intermediate and product species are almost isoenergetic, compared to the n = 0–3 reactions, where the products are less stable than the intermediate by approximately 1 eV. The LEP reaction paths, shown overlaid in Fig. 23 show broadly similar energy profiles. The most notable feature is the difference in the relative energies of the intermediate and product species, which increases from 0.195 eV for the Rh₁₀O + N₂ reaction, to 0.71, 0.93 and 1.29 eV for the n = 2–4 paths respectively.

Fig. 22 Rh₁₀O_n + N₂O → Rh₁₀O_n+1 + N₂, n = 0–4 LER reaction paths overlaid. The Rh₁₀ + N₂O path is black, the Rh₁₀O_n paths are coloured blue, green, yellow and red, for n = 1–4 respectively.

Fig. 23 Rh₁₀O_n + N₂O → Rh₁₀O_n+1 + N₂, n = 0–4 LEP reaction paths overlaid. Zero energy is the lowest energy Rh₁₀ reactant, as shown in Fig. 11. The Rh₁₀ + N₂O path is black, the Rh₁₀O_n paths are coloured blue, green, yellow and red, for n = 1–4 respectively.

4 Conclusion

We have determined the key points on the reaction paths of Rh₁₀O_n + CO → Rh₁₀O_n−1 + CO₂ and Rh₁₀O_n + N₂O → Rh₁₀O_n+1 + N₂ reactions at multiplicities ranging from doublet up to 16-tet. The vast majority of structures were found to lie within a relatively small energetic range, indicating that most multiplicities would be accessible in reaction conditions.

For the Rh₁₀O_n + CO → Rh₁₀O_n−1 + CO₂ reactions, the relative energy of the O⋯CO intermediate. Structure was found to be remarkably consistent as the oxidation of the Rh₁₀ cluster increased, lying between −1.93 and −1.67 eV below the energy of the reactants. Similarly, the relative energy of the Rh₁₀O_n−1 product was found to lie in a narrow range of only 0.35 eV.

The oxygen addition reactions, Rh₁₀O_n + N₂O → Rh₁₀O_n+1 + N₂, showed significantly different behaviour, with the energy of the O⋯N₂ intermediate rising by approximately 0.7 eV with each addition oxygen on the Rh₁₀O_n cluster. The relative energy of the Rh₁₀O_n+1 product also rose significantly, by ≈0.3 eV, though a linear relationship was not seen for these species.

These reaction paths combined suggest that the rate of oxygen addition to the cluster surface is governed by both thermodynamic and kinetic effects, whereas, the rate of oxygen abstraction reaction is largely governed by access to oxygen on the cluster surface. Paths defined by the lowest energy reactant and lowest energy product show similar energetic profiles, suggesting that the experimentally observed reaction rates are unlikely to be significantly affected by the presence of a mixture of Rh₁₀O_n isomers.

Author contributions

ATD, VM and OA: investigation MAA: conceptualization of this study, methodology, writing – review and editing.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

MAA is grateful to the Engineering and Physical Sciences Research Council for funding through grant no. EP/S015868/1 and HPC time via the UK Materials and Molecular Modelling Hub on Young, via grant no. (EP/T022213).

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Footnote

† Electronic supplementary information (ESI) available: Spreadsheet of reaction path energies, zipfile of reaction path structures. Structures are additionally available at https://zenodo.org/record/8428605. See DOI: https://doi.org/10.1039/d3cp04929a

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