Introduction

Bimetallic clusters have attracted enormous attention due to their unusual optical [1, 2], magnetic [3–6], and catalytic properties [7–9], which are significantly different from those of the corresponding bulk materials and monometallic clusters. The unique physical and chemical properties of bimetallic clusters depend on not only the size, geometric structure, but also the composition [10–16]. Among these bimetallic clusters, RhPt is one of the most interesting clusters and has been investigated in experiment [7–14, 17–19] and theory [20–27]. Experimentally, RhPt is regarded as one of the effective three-way catalysts, which is widely used in the converters of the automobiles exhaust gases including carbon monoxide (CO), nitrogen oxides (NO x ), and hydrocarbons (C x H y ) [28]. Moreover, RhPt is the best candidate for the direct methanol and ethanol fuel cells. In the oxidation of ethanol, the oxidative activity has been improved on the RhPt catalysts compared to the Pt/C, since the Rh helps to increase the activity of the C–C bond breaking [9, 17]. On the other hand, some experimental techniques have been performed to characterize the structures of bimetallic RhPt clusters [29–31]. Harada [29] has studied the colloidal dispersions of Pt/Rh bimetallic cluster based on the extended X-ray absorption fine structure spectroscopy (EXAFS). Fang [30] used the X-ray photoelectron spectroscopy (XPS) to characterize bimetallic PtRh cluster and its surface composition in CO electro-oxidation process.

Theoretically, Yuge [21, 22] has studied the segregation behavior of Pt–Rh bimetallic clusters based on the first-principles density functional theory (DFT). Polak and coworkers [23] explored the surface tension effects on segregation of Pt25Rh75 (111) alloy using free energy concentration expansion method (FCEM). Additionally, by combining three different methods including density functional theory, cluster expansion and Monte Carlo, Weller et al. [24] have predicted the segregation profile of Pt25Rh75 (100). Their results have shown that the Pt atoms tend to segregate to the surface of RhPt. Understanding the surface segregation of Pt atom can help us design stable geometric structure. In previous studies, the cuboctahedral and icosahedral structures are regarded as relatively stable structures for bimetallic RhPt cluster. However, there are a few reports about structural characteristics of 55-atom bimetallic RhPt cluster with cuboctahedral structure. On the other hand, from our knowledge, the study on the magnetic properties of bimetallic PtRh cluster is far from satisfactory. The bulk Rh is nonmagnetic which has been confirmed in experiment, while Rh clusters exhibit varying magnetic moments with cluster sizes and structures [32, 33]. However, the studies on magnetic properties are mainly focused on small Rh clusters and a few Rh-based bimetallic clusters by combining with 3d metals, for instance, Co–Rh [3], Ni–Rh [4, 5]. Therefore, a complete exploration to the structural characteristics, magnetic, and electronic properties is very desirable for bimetallic RhPt cluster.

In this work, firstly, a series of geometry optimizations of bimetallic RhnPt55−n (n = 0–55) clusters as varying compositions have been performed and the relatively stable structures are found out. Secondly, structural characteristics including bond length, coordination number and bond numbers of Rh–Rh, Rh–Pt, and Pt–Pt for the stable structures are analyzed in detail. Thirdly, the total and local magnetic properties of RhnPt55−n clusters are systematically discussed. Finally, the electronic properties including d-band center (εd) and partial density of states (PDOS) have been taken into account.

Computational Methods

The geometry optimizations of RhnPt55−n clusters are carried out using the CP2K program package [34] based on the DFT approaches [35, 36]. The CP2K program adopts a hybrid basis set formalism known as the Gaussian and Plane Wave (GPW) [37]. For Quickstep module, double-ζ valence plus polarization (DZVP) Gaussian basis sets of the Molopt-type [38] are used in combination with the norm-conserving Goedecker–Teter–Hutter (GTH) [39–41] pseudopotentials to describe the interaction of valence electron–ion. The generalized gradient-corrected approximation of Perdew–Burke–Ernzerhof (PBE) [42] is used for the exchange–correlation functional. The SCF convergence criterion is set to be 1.0 × 10−6 Hartree and the energy cutoff for auxiliary plane wave electron density expansion is 300 Ry (1 Ry = 13.606 eV). To accelerate convergence, the Fermi–Dirac smearing function with the electron temperature being 300 K is used. All calculations are implemented in a periodic cubic simulation box of 20 Å. Geometry optimizations have been performed using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) [43] minimization algorithm until the maximum atomic forces lower than 4.5 × 10−4 Ha/Bohr.

The magnetic and electronic properties are calculated in the vienna ab initio simulation package (VASP) [44, 45], with the projector augmented wave (PAW) [46, 47] pseudopotentials. The energy cutoff for plane wave basis set is set to be 300 eV. The energy and force are converged with a tolerance of 1.0 × 10−5 and 10−2 eV/Å, respectively. The Brillouin Zone (BZ) integration is performed using only the Γ-point [48].

The lattice constants (a) and binding energies (E b ) of Rh and Pt bulk solids are obtained through our calculations and shown in Table S1 of Supporting Information. The lattice constants of Pt and Rh agree very well with the experimental results, and the errors of binding energies between DFT calculations and experimental data [49–51] are acceptable, indicating that our computational methods are reliable and accurate.

Results and Discussions

Structural Stability

The cuboctahedral structure has O h symmetry, which is obtained from the face-centered-cubic structure. There are five nonequivalent atoms, i.e., one core atom, twelve atoms in the second shell, and the remaining forty-two atoms in the surface formed by 12 vertexes, 24 edges, and 6 (100) facet sites, which can be seen from Fig. 1. The vertex atoms have the lowest coordination number (five), followed by the edge atoms (seven), and the (100) facet atoms (eight). The stable structures of bimetallic RhnPt55-n clusters including the random alloy structures where n = 9, 11, 13, 19, 27, 36, 42, 44, 46 and ordered structures of three Rh54Pt1 isomers, core–shell Rh13@Pt42 and Pt13@Rh42 are present in Fig. 2.

Fig. 1
figure 1

The cuboctahedral structure of Rh55, and the five nonequivalent sites are core site (1), the second shell site (2); vertex site (3), edge site (4), and (100) facet site (5)

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Fig. 2
figure 2

The stable structures of bimetallic RhnPt55−n (n = 0–55) clusters include the random alloy structures where n = 9, 11, 13, 19, 27, 36, 42, 44, 46 and ordered structures of Rh54Pt1, Rh13@Pt42, and Pt13@Rh42. As for the Rh54Pt1 isomers, i.e., the Rh54Pt1 (1), the Pt located at the vertex site; Rh54Pt1 (2), the Pt located at the edge site; and Rh54Pt1 (3), the Pt located at the (100) facet site

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To make clear which sites for Pt (Rh) prefers to occupy, we calculate the segregation energy. It can be described as the difference between the energy of a cluster with a foreign atom at the surface and at the core site. We consider three types of sites in the surface, i.e., vertex, edge, and (100) facet sites. The E seg equation is described as follows:

$$E_{seg}^{A} = E_{A - surface\,cluster} - E_{A - core\,cluster} ,$$
(1)

where E A-surface cluster is the cluster energy when the foreign atom A is located at the surface, E A-core cluster is the cluster energy when the A atom is located at the core. The negative E seg indicates that the surface segregation of the foreign atom will take place. E seg  < −0.3 eV, suggests strong surface segregation; −0.3 eV < E seg  < −0.05 eV, suggests moderate surface segregation [52]. In contrast, the positive E seg indicates the opposite behavior, i.e., the foreign atom prefers to the located at the core site. The segregation energies of Pt and Rh are listed in Table 1.

Table 1 The segregation energies E seg (eV) of Pt and Rh atoms
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Clearly, the E seg of Pt atoms are <~−0.3 eV, indicating that there is a strong tendency for Pt atoms to segregate to the surface, while the positive E seg of Rh atoms mean that the Rh atoms like to occupy the core site. Besides, the lower surface energy (1.34 eV/atom) [25] and larger atomic radius of Pt atom (1.39 Å) in comparison with the Rh atom, of which the surface energy and atomic radius are 1.56 eV/atom [25] and 1.34 Å, respectively, also give rise to the Pt surface segregation. On the other hand, compared the segregation energies of Pt at different surface sites, it is found that, the E seg at the vertex (−0.39 eV) is lower than those at the edge, and the (100) facet sites (−0.37 and −0.29 eV). These results indicated that the most preferential site for Pt to occupy is the vertex site. The site preference of Pt and Rh not only determines structural stability, but also affects catalytic activity. Pt surface segregation of PtRh system makes more (less) Pt (Rh) atoms expose to the surface. It can enhance some reactions that take place on Pt atoms [53, 54].

To explore the structural stabilities of these bimetallic clusters with different compositions, the excess energies (E exc ) of RhnPt55−n (n = 0–55) clusters are calculated and shown in Fig. 3. The excess energy equation can be defined as

Fig. 3
figure 3

The excess energies of RhnPt55−n (n = 0–55) clusters as a function of number of Rh atoms. The red star represents the excess energy of core–shell Rh13@Pt42

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$$E_{exc} = E_{cluster} - \frac{n}{55}E_{{Pt_{55} }} - \frac{55 - n}{55}E_{{Rh_{55} }} ,$$
(2)

where E cluster , E Pt55 , E Rh55 represent the energies of RhnPt55−n, Pt55, Rh55 clusters respectively. The negative excess energy suggests that the mixing of Pt and Rh atoms is thermodynamically favorable.

The binding energy of Pt55 cluster is found to be 5.06 eV/atom. While the binding energy of Rh55 is 4.49 eV/atom, which is in good agreement with the theoretical result of 4.57 eV/atom [27]. Therefore, the introduction of Pt atoms to the Rh55 cluster enhances the interaction of metallic atoms in RhnPt55−n clusters. The excess energies of the Rh54Pt1 isomers are shown in Table S2 of Supporting Information. It is found that the order of excess energies of Rh54Pt1 isomers is as follows: Pt in the vertex site (Rh54Pt1 (1), −0.22 eV) < Pt in the edge site (Rh54Pt1 (2), −0.20 eV) < Pt in the (100) facet site (Rh54Pt1 (3), −0.12 eV). Interestingly, the order is in accord with the results obtained from segregation energy. From Fig. 3, the excess energies present negative values over the entire composition ranges, indicating that bimetallic RhnPt55-n are energetically favorable for alloying. The excess energies gradually decrease at first and then rise with the increase of Rh atoms. The lowest excess energy (−2.49 eV,) is found at approximate Rh:Pt ≈ 1:1 ratio, i.e., Rh27Pt28 cluster, indicating that Rh27Pt28 cluster is the most stable structure in random clusters. Simultaneously, it provides clear evidence that there is a composition-dependent effect on the stability of bimetallic RhnPt55-n clusters. As mentioned above, the Pt atom prefers to occupy the surface. Therefore, the well-known core–shell Rh13@Pt42 structure by putting 42 Pt atoms on the surface, and 13 Rh atoms in the core and second shell is obtained. The core–shell Rh13@Pt42 has quite low excess energy of −5.16 eV compared with the random alloy structure of Rh13Pt42, of which the excess energy is −1.96 eV., The core–shell Rh@Pt clusters are expected to play a crucial role in catalysis from the experimental studies. Rh@Pt bimetallic nanoparticles have been synthesized and present a higher activity in oxidation of CO at 70 °C compared with RhPt (1:1) alloy and pure Pt nanoparticles [14]. However, the core–shell Pt13@Rh42 shows strongly structural instability with a very high positive excess energy of 4.25 eV. As a consequence, we can conclude that structure-dependent effect also plays a significant role in the stability of bimetallic clusters.

In order to understand the structural characteristics of bimetallic RhnPt55−n clusters, the bond length, coordination number, and bond numbers of Rh–Rh, Rh–Pt, and Pt–Pt are thoroughly analyzed. The average bond lengths of Pt–Pt, Rh–Rh, and Rh–Pt of RhnPt55-n in random alloy structures are present in Table 2. It is found that for Pt55 cluster, the average Pt–Pt bond length is 2.72 Å, and for Rh55 cluster, the average Rh–Rh bond length is 2.65 Å. The bond lengths are both contracted than those in their bulks. which are found to be 2.78 and 2.68 Å in Pt and Rh bulks, respectively [31]. The contraction of Pt–Pt and Rh–Rh bond in Pt55 and Rh55 clusters is mainly due to the increase of the surface atoms in cuboctahedral structure compared with bulk structure. From Table 2, the average Pt–Pt bond lengths in bimetallic clusters fluctuate between 2.67 and 2.70 Å with the increase of Rh atoms, slightly shorter than that in Pt55 cluster (2.72 Å). The decrease of Pt–Pt bond length is mainly caused by alloying with a smaller radius atom, i.e., the Rh atom [8]. The average Rh–Rh bond lengths in bimetallic clusters are varied from 2.66 to 2.68 Å with the increase of Rh atoms, slightly elongated with respect to the Rh55 cluster (2.65 Å). In addition, the average Rh-Pt bond lengths are also varied in a small range as a function of the Rh atoms. Our results are in good agreement with the experimental values obtained by the EXAFS technique [29, 31].

Table 2 The average bond lengths (Å) of Pt–Pt, Rh–Rh, and Rh-Pt of RhnPt55-n (n = 0–55) clusters in random alloy structures
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The coordination numbers (CNs) are calculated for the first nearest neighbor shell. The partial CNs (CN Pt–Pt , CN Rh–Rh , CN Pt–Rh , and CN Rh–Pt ) are linear fitted and shown in Fig. 4a, and the total CNs (CN Pt–M , CN Rh–M , and CN M–M ) are shown in Fig. 4b. The relationships between these CNs are as following [55]:

Fig. 4
figure 4

a The partial CNs, b the total CNs for RhnPt55-n (n = 0–55) clusters in random alloy structures as a function of number of Rh atoms

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$$CN_{Pt - M} = CN_{Pt - Pt} + CN_{Pt - Rh} ,$$
(3)
$$CN_{Rh - M} = CN_{Rh - Rh} + CN_{Rh - Pt} ,$$
(4)
$$CN_{M - M} = x_{Pt} CN_{Pt - M} + x_{Rh} CN_{Rh - M} ,$$
(5)

where the CN Pt–Pt , CN Rh–Rh are the CNs of Pt around Pt atom, and Rh around Rh atom, CN Pt–Rh , CN Rh–Pt are the CNs of Rh around Pt atom, and Pt around Rh atom; x Pt, and x Rh are the mole fraction of Pt and Rh atoms to the whole cluster.

From Fig. 4a, with the increase of the Rh atoms, the CN Rh–Rh and CN Pt–Rh increase, whereas the CN Pt–Pt and CN Rh–Pt decrease. This can be understood with the increase of Rh atoms, more Rh atoms distribute around Rh and Pt atoms and less Pt atoms distribute around Rh and Pt atoms. Besides, CN Rh–Rh  > CN Pt–Rh and CN Rh–Pt  > CN Pt–Pt are found under the same Rh composition. In bimetallic clusters, the partial CNs depend on the particle size and compositions, meanwhile, the total CNs can reveal the atomic distribution. The total CNs (CN M–M ) of the whole clusters almost keep the same, 7.85, which is in excellent agreement with the experimental value of 7.86 [56]. The total CNs of Rh-metal atoms (CN Rh–M ) are larger than those of Pt-metal atoms (CN Pt–M ), indicating that the Rh atom has a tendency to occupy the inner shell while the Pt atom segregates to the surface, because the atoms at the surface have lower coordination numbers than those in the inner shell. To make clear the changes of CNs with different structures at the same composition, we take Rh13P42, Rh19Pt36, and Rh27Pt28 with four different isomers respectively for example. The structures as well as excess energies of these three systems are shown in Fig. S1. The total CNs (Pt–M and Rh–M) as a function of excess energy are shown in Fig. S2. It is found that the CNs of Rh-M are increased with the decrease of excess energies, on the contrary, the CNs of Pt–M are gradually decreased. The lower-energy configuration has a larger CNs of Rh–M and a smaller CNs of Pt–M. Therefore, the changes of CNs reveal the atomic distribution of a cluster, and meanwhile, to some degree reflect structural stability.

From Fig. 5, for Pt-rich RhnPt55−n (n < 27) clusters, with the increase of Rh atoms, the number of Pt–Pt bonds is gradually decreased, while the numbers of Rh–Rh and Rh–Pt bonds are mainly increased. It is important to mention that the order of excess energy is also following the same trend, i.e., the negative excess energy is gradually increased as a function of Rh atoms for Pt-rich clusters. However, for Rh-rich RhnPt55−n (n > 27) clusters, although the number of Rh–Rh bond is increased, the number of Rh–Pt bond is sharply decreased. The decrease of Rh–Pt bonds results in the decrease of the stability (the increase of excess energy) which can be seen from Fig. 3. Therefore, we can conclude that there must be some correlations between the structural stability and the numbers of Rh–Rh and Rh–Pt bonds. The strength and number of the nearest inter-atomic bonds are closely associated with the stability of a cluster, which has been reported in many bimetallic clusters [15, 16, 57]. From our calculations, the co-effects of Rh–Rh and Rh–Pt bonds are responsible for the structural stability and Rh–Pt bond makes the main contribution. Previous studies have shown that the strengths of Rh–Rh and Rh–Pt bonds are nearly equal, which are stronger than that of the Pt–Pt bond [58]. Yuge [26] has also pointed out that the Rh–Pt bond is energetically favored and the Pt–Pt bond is energetically unfavorable in RhPt bimetallic nanoparticles. Therefore, with more Rh–Pt bonds, the bimetallic RhnPt55−n clusters will be more stable.

Fig. 5
figure 5

The number of Pt–Pt, Rh–Rh and Rh–Pt bonds of bimetallic RhnPt55−n (n = 0–55) clusters in random alloy structures as a function of number of Rh atoms

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Magnetic Properties

To explore the magnetic properties of bimetallic RhnPt55−n (n = 0–55) clusters, we calculate the total and average local magnetic moments. The magnetic moment of Pt55 cluster is found near to be zero. For Rh55, it has 0.376 μ B /atom magnetic moment, which is in good agreement with theoretical value (0.357 μ B /atom) [4], and slightly larger than that of experimental result (0.2 ± 0.1 μ B /atom) [33]. The magnetic moments at nonequivalent sites show a great difference. The magnetic moment of core Rh atom is very small, only −0.070 μ B . While the magnetic moments of surface Rh atoms located at the vertex, edge and (100) facet sites are found to be 0.156, 0.434, and 0.658 μ B /atom, respectively. It should be mentioned that the nonequivalent atoms have different coordination numbers. The (100) facet atom, holding the largest coordination number (seven) among the three surface sites, has the largest magnetic moment (0.658 μ B /atom). On the other hand, although the core atom is 12-coordinated, it has a small magnetic moment (−0.070 μ B ). Therefore, the coordination number together with atomic distribution is responsible for the varying magnetic moments.

The total magnetic moments of RhnPt55−n clusters and average local magnetic moments of Rh, Pt and core atoms (mainly Rh atom) of RhnPt55−n clusters in random alloy structures as a function of Rh atoms are present in Fig. 6. From Fig. 6, it is clear that the total magnetic moments of bimetallic RhnPt55−n clusters are larger than that of Pt55 but smaller than that of Rh55 cluster, except the Rh27Pt28. Of which the total magnetic moment is larger than that of Rh55 by 0.15 μ B . Therefore, the introduction of Pt atoms to Rh55 cluster weakens the total magnetic moments of the cluster. However, the average local magnetic moments of Pt atoms in bimetallic clusters are enhanced with respect to the Pt55. The Pt atoms are induced small magnetic moments ranging from 0.08 to 0.16 μ B /atom. Similarly, the average local magnetic moments of Rh atoms are also mainly enhanced except Rh44Pt11 and Rh46Pt9. The main reason is that there are different atomic distributions in Rh44Pt11 and Rh46Pt9 clusters, because the position of Rh atom and the relative position of Rh and Pt atoms can strongly affect the magnetic moment. The similar phenomenon can also be seen in Ni–Rh bimetallic cluster [5]. We also compared the magnetic moments of two core–shell structures, namely, Rh13@Pt42 and Pt13@Rh42, and found that the Pt13@Rh42 has a very large total magnetic moment of 32.859 μ B . However, the total magnetic moment of Rh13@Pt42 is very small (5.453 μ B ). The reasonable explanation is that in Rh13@Pt42, the Rh atoms aggregate to the inner shell, which is similar to the bulk structure. As mentioned above, the bulk Rh is nonmagnetic and Pt55 cluster is also nonmagnetic. Therefore, the small magnetic moment of Rh13@Pt42 is understandable. From all these results, we can conclude that the local atomic environment including coordination number and atomic distribution, structure, and composition are closely associated with the magnetism of bimetallic RhnPt55-n clusters.

Fig. 6
figure 6

The total magnetic moments of bimetallic RhnPt55−n and average local magnetic moments of Rh, Pt and core atoms of RhnPt55−n (n = 0–55) clusters in random alloy structures as a function of number of Rh atoms

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Electronic Properties

To gain more insight into the electronic properties of bimetallic RhnPt55-n (n = 0–55) clusters, the d -band centers of the whole clusters (εd), Pt and Rh atoms (ε *d ) are calculated and shown in Table S3. Results have shown that the d-band centers of the whole clusters are shifted gradually closer to the Fermi level with the increase of Rh atoms. However, the d -band centers of Pt atoms in bimetallic clusters are almost shifted away from the Fermi level with respect to Pt55 cluster. The downshift of Pt d -band center alloying with Rh atoms has been reported in previous studies [59, 60]. On the other hand, the difference of the d-band center (|ε *d |) of Rh and Pt atoms of RhnPt55-n (n = 0–55) are also calculated. The order of |ε *d | of Rh54Pt1 isomers is as follows: Pt in the vertex site (Rh54Pt1 (1), 0.021 eV) < Pt in the edge site (Rh54Pt1 (2), 0.195 eV) < Pt in the (100) facet site (Rh54Pt1 (3), 0.366 eV). Interestingly, this order is consistent with the order of excess energy of Rh54Pt1 (see Table S2). The smaller |ε *d | helps to form more stable structures. Moreover, the |ε *d | of bimetallic RhnPt55−n (n = 0–55) in random alloy structures also keeps the same trend as the excess energy from Fig. 7. To explain this phenomenon, we take the orbital interaction into account. The average d -band centers of the whole cluster, Rh and Pt atoms represent the average energies of their d orbital. In bimetallic RhnPt55−n, the d -band centers of Rh and Pt have taken great changes compared with their pure clusters. These changes are mainly due to redistribution of electrons and orbital overlapping between Rh and Pt atoms. In transition metals, the d orbital interaction plays a dominate role in bonding process. The strong bonding process between metallic atoms can help to structural stability. When the |ε *d | is smaller, more Rh 4d and Pt 5d orbitals overlapping will occur, and consequently, the structure is more stable.

Fig. 7
figure 7

The excess energies (black) and the difference of d -band center (|ε *d |) (blue) of Rh and Pt for RhnPt55−n (n = 0–55) clusters in random alloy structures as a function of number of Rh atoms

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To make clear the relationship between the orbital interaction and structural stability, the s-, p-, d-projected partial density of states (s-, p-, d-PDOS) of Rh and Pt atoms for Pt55, Rh55, Rh13Pt42, Rh13@Pt42, Rh27Pt28, Rh36Pt19, and Rh44Pt11 clusters are shown in Fig. 8. From the PDOS, relatively strong d–d orbital and slight s–s orbital interactions are observed, but not any overlapped peaks are formed by p–p orbital. Compared the random alloy structure of Rh13Pt42 with core–shell structure of Rh13@Pt42, it is found that Pt 5d and Rh 4d orbitals become well-resolved in Rh13@Pt42, while the peaks in Rh13Pt42 are low and multiple. Additionally, in Rh13@Pt42, relatively strong s–s and s–p orbital interactions occur at about −7.0 and −5.0 eV, respectively. Simultaneously, more Rh 5p orbital is overlapped with Pt 6s and 6p orbitals in the energy ranging from −4.0 to −2.0 eV. Such kind of strong interaction is not found in Rh13Pt42 structure. In the case of Rh27Pt28, Pt 5d and Rh 4d orbitals are overlapped from −6.0 eV to the Fermi level, showing the strongest d–d orbital hybridization. Meanwhile, Rh 5s and Pt 6s orbitals are overlapped at about −5.0 eV. Additionally, Pt 6s and Rh 5p orbitals developed a very small overlapped peak at −2.5 eV mainly due to slight charge transformation from Pt 6s orbital to Rh empty 5p orbital. Consequently, these orbital interactions make the Rh27Pt28 to be the most stable cluster in random alloy structures. As for Rh36Pt19 and Rh44Pt11 clusters, gradually reduced Pt 5d orbital directly weakens the d–d orbital interaction. The weak interaction results in the increase of excess energy and the decrease of stability. Additionally, the s-, p-, d-PDOS of Rh and Pt atoms for RhnPt55−n (n = 0–55) over the entire composition ranges are present in Fig. S3.

Fig. 8
figure 8

The s-, p-, d-projected partial density of states (s-, p-, d-PDOS) of Rh and Pt atoms for Pt55, Rh55, Rh13Pt42, Rh13@Pt42, Rh27Pt28, Rh36Pt19, and Rh44Pt11 clusters (the Fermi level is presented by dashed vertical line)

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To understand the relationship between the magnetic and the electronic properties, we thoroughly analyze the d-projected partial density of states (d-PDOS). The interaction of d electrons for both majority spin and minority spin states makes the main contribution to magnetic properties [61, 62]. The d-PDOS are shown in Fig. 9. In Fig. 9a, for Pt55, the majority spin states and minority spin states are completely symmetric, suggesting its magnetic moment to be zero. As for Rh55, the minority spin states pass through the Fermi level and the majority spin states move down the Fermi level, which indicates a very large total magnetic moment (20.654 μ B ). In Fig. 9b, clearly, the d-PDOS for Rh atoms at the vertex, edge, and (100) facet sites are mainly localized at the Fermi level, which plays a significant role on the magnetic moment. The d-PDOS of the vertex atom has a slight exchange splitting between the −1.5 and −1.0 eV indicating a relatively small magnetic moment (0.156 μ B /atom). While for the edge atom, the majority spin peak becomes very sharp and move down the Fermi level (about 0.2 eV below the Fermi level) and the minority spin states pass through the Fermi level, which suggests a very large magnetic moment (0.434 μ B /atom). In addition, the d-PDOS of the (100) facet atom becomes broader than those of vertex and edge atoms. Also its minority spin states completely pass through the Fermi level and become stronger above the Fermi level than that below the Fermi level. This enhanced exchange splitting leads to a much larger magnetic moment (0.658 μ B /atom).

Fig. 9
figure 9

The d -projected partial density of states (d -PDOS) a Pt55, Rh55 clusters; b Rh atoms at vertex, edge, and (100) facet sites in Rh55 cluster; c Rh and Pt atoms in Rh9Pt46, Rh46Pt9, Rh27Pt28, and Pt13@Rh42 clusters (the Fermi level is presented by dashed vertical line)

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On the other hand, the total and average local magnetic moments of bimetallic RhnPt55−n clusters are induced by the hybridization of Rh 4d and Pt 5d orbitals. For Rh9Pt46, the d -PDOS is mainly contributed by Pt 5d orbital. Because of nearly symmetric Pt 5d orbital and weak Rh 4d orbital, a slight exchange splitting is found at the Fermi level, indicating that the Rh9Pt46 cluster has a very small total magnetic moment (8.513 μ B ). While for Rh46Pt9, Rh 4d orbital becomes stronger, and its minority spin states pass through the Fermi level and majority spin states move down the Fermi level. This characteristic suggests a slightly enhanced total magnetic moment of Rh46Pt9 (11.750 μ B ). When the compositions of Rh and Pt atoms are equal, namely, Rh27Pt28 cluster, the Pt 5d and Rh 4d orbitals are overlapped from about −0.5 eV to the Fermi level. In addition, the minority spin states of Rh 4d orbital completely pass through the Fermi level and become stronger above the Fermi level. Moreover, a small exchange splitting of Pt 5d orbital is found at the energy of −1.5 eV. All these results contribute to a much enhanced total magnetic moment (21.495 μ B ). In the case of Pt13@Rh42, the d -PDOS for Rh 4d majority spin states move down the Fermi level (about 0.5 eV below the Fermi level), and become very sharp below the Fermi level, meanwhile the minority spin states almost completely pass through the Fermi level. Moreover, Pt 5d minority spin states move up through the Fermi level, and its majority spin states are below the Fermi level. The co-effects of Rh and Pt atoms help to generate a much larger magnetic moment (32.859 μ B ).

Conclusions

In this study, DFT calculations have been carried out to investigate the structural, magnetic, and electronic properties of bimetallic RhnPt55−n (n = 0–55) clusters. The segregation energies of Pt and Rh atoms indicate that Pt atom prefers to segregate to the surface while Rh atom tends to occupy the core site. Furthermore, the stability order of Rh54Pt1 isomers is as follows: Pt at the vertex site >Pt at the edge site >Pt at the (100) facet site. As for the random alloy structures, the excess energies are decreased at first and then increased. The lowest excess energy of −2.49 eV is found at Rh27Pt28 cluster. However, the ordered core–shell Rh13@Pt42 has much lower excess energy (−5.16 eV). The average bond lengths of Pt–Pt, Rh–Rh, Rh–Pt of RhnPt55-n clusters all fluctuate in a small range and our results are in good agreement with the available experimental values. The coordination numbers and bond numbers in the first nearest neighbors reveal the distribution of Pt and Rh atoms in random alloy structures and to some degree reflect structural stability.

The magnetic moments of RhnPt55−n are varying. The total magnetic moments are mainly weakened except Rh27Pt28 with respect to Rh55 cluster, while the average local magnetic moments of Pt and Rh atoms are mainly enhanced with respect to their pure phases. The d-PDOS explains the varying magnetic moments. In addition, the d-band centers and the s-, p-, d-PDOS of Rh and Pt atoms of RhnPt55−n clusters reveal the relationship between electronic properties and structural stabilities. It is found that the order of |ε *d | keeps the same tendency with that of the excess energy. From s-, p-, d-PDOS, the more orbital interactions there are, the more stable the structure is.