Nanoclusters usually display exotic physical and chemical properties due to their intriguing geometric structures in contrast to their bulk counterparts. By means of first-principles calculations within density functional theory, we find that heavy noble metal PtN nanoclusters around the size N = 55 begin to prefer an open configuration, rather than previously reported close-packed icosahedron or core-shell structures. Particularly, for PtN, the widely supposed icosahedronal magic cluster is changed to a three-atomic-layered structure with D6h symmetry, which can be well addressed by our recently established generalized Wulff construction principle (GWCP). However, the magic number of PtN clusters around 55 is shifted to a new odd number of 57. The high symmetric three-layered Pt57 motif is mainly stabilized by the enhanced covalent bonding contributed by both spin-orbital coupling effect and the open d orbital (5d96s1) of Pt, which result in a delicate balance between the enhanced Pt–Pt covalent bonding of the interlayers and negligible d dangling bonds on the cluster edges. These findings about PtN clusters are also applicable to IrN clusters, but qualitatively different from their earlier neighboring element Os and their later neighboring element Au. The magic numbers for Os and Au are even, being 56 and 58, respectively. The findings of the new odd magic number 57 are the important supplementary of the recently established GWCP.

Because of their intriguing geometric structures and their atomic scales that promote the quantum size effects, nanoclusters have drastically different physical and chemical properties than their bulk counterparts, and thus leading to many unique applications.1–4 Interestingly, these nanoclusters are energetically more stable at certain sizes and referred as magic clusters.5–8 Clearly, the understanding of this “magic size” phenomenon will help us tune their individual physical/chemical properties and also possibly use them as building blocks to assemble into novel materials that have unprecedented functionalities.2,9–11

Three mechanisms have been successfully identified for the magic-size behavior of a given nanocluster, including the atomic shell closure5 model for inert gas clusters, electronic shell closure6 mechanism for simple metal clusters, and the recently established generalized Wulff construction principle (GWCP)12 for transition metal clusters. The atomic shell closure model is a geometric construction principle that predicts the formation of icosahedral clusters/structures at magic sizes of 13, 55, 147, etc. The new mechanism GWCP12 emphasizes the minimization of the total edge energy of a given transition metal nanocluster due to the excess energy arising from the undesirable d-type dangling bonds located on the cluster edges. Correspondingly, around size 55, the magic numbers of transition metal nanoclusters are predicted to be even, such as 56, rather than the widely accepted odd number 55 as predicted by the atomic shell closure model, which was strongly supported by recent experimental observations.13,14 Since the edge energy is largely governed by the d-type dangling bonds, the GWCP is applicable for almost all nd transition metal elements (n = 3, 4, 5), with the only exceptions being the earliest and latest TM cases.12 This exception can be attributed to the negligible numbers of d-type dangling bonds on the edge atoms for both the earliest and the latest TMs as compared to the central ones.12 

We emphasize that in these aforementioned exceptions, the latest noble elements, such as Au, are particularly novel and intriguing, due to the relativistic effect15–18 which stabilizes the s orbitals and destabilizes the d orbitals, leading to the enhanced s-d hybridizations. Correspondingly, AuN nanoclusters exhibit the enhanced covalent bonding,19,20 the preference of planar open structure in the small size regime (N = 3–13),16 exotic cage structures in a relatively larger regime (N = 16–20),21 symmetry breaking based amorphous structures15 and even magic number of 58 due to both atomic and electronic shell closures around size 55, and size-selective catalysis for an even larger size regime.22 

Then, one simple question naturally arises: does the relativistic effect also play the key role in determining the geometric structure, magic sizes, and physical and chemical properties of Pt nanoclusters? First, as the nearest neighbor of Au in the periodic table, noble metal Pt is also a well-known catalyst23–27 and may also possess strong relativistic effects as implied by the comparable relativistic contraction of the 6s shells between Pt and Au.28 Additionally, the d orbital of Pt is of an open-shell configuration, which helps to form stronger d-d directional bonding for Pt nanostructures despite its relatively weaker relativistic effect than Au. However, the synergetic or competitive effects between the relativistic effect and the open-shelled d orbital on the geometric/electronic structures of heavy noble nanoclusters, such as PtN, have not been rigorously established. Importantly, we will show later that Pt actually lies at the boundary of the elements whose atomic stacking rule and magic numbers of nanoclusters may not be guided simply by GWCP or the electronic and geometric shell-closure models.5,6,8,12 Therefore, the atomic stacking rule and magic numbers of PtN are hard to specifically predict from the established principles. Despite recent studies on Pt13 nanoclusters,29,30 Pt55 nanoclusters,12,31–35 and other TM55 nanoclusters,12,13 the above question still remains unclear and awaits for a definite answer in order to determine accurately the specific structures and properties of the nanoclusters consisting of “boundary” element Pt for practical applications such as nanocatalysis.

In this paper, by means of detailed first-principles calculations, we have specifically identified that PtN nanoclusters around size 55 prefer an intriguing three-layered stacking form, mainly due to the synergetic effect from the established GWCP, relativistic effect, and the significant covalent bonding as well. Correspondingly, the magic cluster of PtN around size 55 is identified to be Pt57; meanwhile, the magic cluster now is a highly symmetric (D6h) three-layered-wheel (TLW) configuration, rather than the widely supposed Ih,31,32 Oh,33 or amorphous core-shell motifs.34,35 The differences in atomic stacking and in magic numbers around 55 between PtN and AuN are mainly due to a relatively stronger covalent bonding that results from the open-shell d orbital in the former case, as opposed to the closed-shell of the latter. The present findings are also valid up to iridium. Additionally, when moving towards the central element Os with more unpaired d electrons as compared to Ir and Pt, the present TLW open structures are found to be significantly less stable. Correspondingly, the established GWCP mechanism12 must dominate, due to the critically undesirable d-type dangling bonds12 located on the sharp edges of TLW structures on one hand, and due to the significantly reduced relativistic effect28 on the other hand.

The remainder of the paper is organized as follows. The methodology is described in Sec. II. The results and discussion are presented in Sec. III. The main conclusions are summarized in Sec. IV.

Our calculations adopted the density functional theory (DFT)36 within the spin-polarized generalized gradient approximation (GGA)37 as implemented in the VASP code.38 The interaction of the valence electrons with the ionic core was described by the projector augmented wave (PAW) method39 with PW91 form as the exchange-correlation functional. The wave functions are expanded in a plane wave basis with two sets of energy cutoff (EC), 230.3 and 350.0 eV: the former one (the default value when generating the potential of Pt as adopted by the VASP code) is for the first round of extensive low-energy structural candidates searching, and only those configurations with significantly lower energy structures are re-optimized by using the lager energy cutoff of 350.0 eV. Note that EC of 350.0 eV is confirmed to obtain well converged energetics for different low-lying structures. To identify the accuracy of our calculation method, we have carried out calculations on the properties of both Pt2 molecule and Pt bulk crystal. With the EC of 230.3 (350.0) eV, the calculated vibrational frequency 215.613 (215.408) cm−1 of the Pt2 molecule agrees well with experimental value40 of 222.26 ± 0.31 cm−1, and the calculated bond length 2.379 (2.378) Å of the Pt2 is also very close to the experimental value of 2.333 Å.41 In addition, for the fcc-Pt crystal, the optimized crystal constant 3.991 (3.991) Å agrees well with experimental value 3.92 Å.42 The atomic positions of clusters are optimized in a big simple cubic cell with edge length of 25 Å. The convergence criterion of the electronic loop is adopted up to 10−5 eV and the force convergence less than 0.01 eV/Å per atom. To obtain the ground state configurations of the Pt clusters, we have considered many initial candidate configurations manually constructed or computationally generated via high-temperature first-principles molecular dynamic (MD) simulations, and classical MD simulations performed by LAMMPS43 code as well. We have also carried out structural optimizations by using the particle swarm optimization (CALYPSO) code,44,45 which is very powerful for searching for low energy structures. In addition, the low-energy structural candidates obtained by recent works on TM55 clusters12,34 have also been checked for Pt55 in this work. Finally, the most stable structures are further examined by thermal dynamic simulations in high temperature and vibrational property analysis.

We start our systematic optimization of Pt55 structures first without considering the spin-orbital coupling (SOC) effects, i.e., by using the plain GGA calculations. In Figure 1, we summarize six representative low-energy Pt55 candidate structures optimized in the second round of calculations, i.e., with an EC of 350.0 eV, see also the supplementary material46 for their Cartesian coordinates. The most stable Pt55 structure consists of an amorphous core-shell configuration with 9 core atoms and 46 shell atoms, which is denoted as C9-S46 in Fig. 1(a). This structure can be obtained by moving 4 inner Pt atoms from the core of an icosahedral (Ih) Pt55 structure to the surface shell, followed by further optimization. The resulting total energy is lower by 5.238 eV for the structure (a) than that of the perfect Ih structure (f). Note that previously, Baletto et al.31 and Aprà and Fortunelli32 reported that Pt55 prefers Ih structure, essentially, based on classic Wulff construction. However, here, we identified that an Ih-Pt55 is energetically unstable. Actually, the lowest energy structure (a) is also identified as the ground state of Pt55 cluster by Da Silva et al.34 using a different functional. Interestingly, we have also found a relatively ordered three-layered structure in Fig. 1(b), which is almost as degenerate in energy as structure (a). Note that when using an EC of 230.3 eV, the former is only 90 meV less stable than the latter; however, when the EC is gradually increased up to 350.0 eV, the energy difference (ΔE) between these two structures is finally converged to 154 meV, as presented in both Figs. 1 and 2. Thus, in the remainder of this paper, unless otherwise specifically noted, all the data presented are obtained from calculations with an EC of 350.0 eV. Detailed structural analysis reveals that structure (b) is a “double-defected three-layered-wheel” (DDTLW) configuration. By rearranging these two “defect” sites, we obtained another low-lying isomer (structure (c), denoted as DDTLW-2) with these two “defects” now lying diagonally on this cluster. The resulting energy is 0.411 eV higher than that of structure (a).

FIG. 1.

Geometric structures and relative energies of the six representative low energy configurations of Pt55 obtained by GGA calculations. The relative energies in (a)-(f) are measured from that of the C9-S46 structure in (a), given by ΔE = E(Pt55) − E(Pt55(C9 − S46)).

FIG. 1.

Geometric structures and relative energies of the six representative low energy configurations of Pt55 obtained by GGA calculations. The relative energies in (a)-(f) are measured from that of the C9-S46 structure in (a), given by ΔE = E(Pt55) − E(Pt55(C9 − S46)).

Close modal
FIG. 2.

The relative energies of Pt55 with the double-defected three-layered wheel (DDTLW) structure as compared to that of the core-shell configuration of C9-S46, given by E(DDTLW) − E(C9-S46), against different functionals or calculation methods. Significantly, the SOC effect stabilizes the DDTLW structure, by the value of (E(DDTLW) − E(C9-S46))(GGA) − (E(DDTLW) − E(C9-S46)) (GGA + SOC) in energy.

FIG. 2.

The relative energies of Pt55 with the double-defected three-layered wheel (DDTLW) structure as compared to that of the core-shell configuration of C9-S46, given by E(DDTLW) − E(C9-S46), against different functionals or calculation methods. Significantly, the SOC effect stabilizes the DDTLW structure, by the value of (E(DDTLW) − E(C9-S46))(GGA) − (E(DDTLW) − E(C9-S46)) (GGA + SOC) in energy.

Close modal

To compare Pt55 and Au55, we have also optimized Pt55 with the initial coordinates taken from the lowest energy structure of Au55,18 i.e., a configuration with 10 core atoms (denoted as C10-S45). This low-lying isomer of Pt55, as shown in structure (d), is identified to be 0.660 eV higher in energy. We have also used the most stable Os55 configuration12 as the initial structure, which leads to a structure (Fig. 1(e)) with 1.324 eV higher in energy than structure (a). These results indicate that the electronic bonding properties of Pt nanostructures are significantly different from the latest element Au and the central ones such as Os. Finally, we note that the octahedral (Oh) configuration (not shown here) is much less stable, by 6.817 eV, than structure (a), though previously Oh configuration was predicted to be the lowest energy structure for Pt55.33 

We now investigate the energetics of these low energy structures (particularly the most stable ones) when considering relativistic effects, i.e., by using the GGA + SOC calculations, since Pt is also a heavy noble element with open d-orbital. Furthermore, for comparisons, we have also optimized these structures by using the empirical, embedded atom model (EAM) in the LAMMPS code.43 Our central findings are presented in Fig. 2. The relative stabilities of structure (a) and structure (b) in Fig. 1 are significantly modulated in consideration of the relativistic (SOC) effect. First, when using empirical EAM potential, the structure DDTLW is now 0.312 eV higher in energy than structure in Fig. 1(a). The EAM potential is known to be generally unreliable for nanostructure predictions because it neglects the directional nature of d-d interactions and other quantum effects such as spin magnetism, orbital symmetry, and electronic shell closings.6 As already described in Fig. 1, first-principles calculation with GGA (PW91) functional results in a significant reduction in energy from structure Figs. 1(a) and 1(b), i.e., ΔE = 90 meV in the calculation with an EC of 230.3 eV. Amazingly, when spin-orbital coupling is also considered (GGA + SOC), the DDTLW structure now is essentially energetically degenerate with structure (a). Actually, it is also even slightly more stable than structure Fig. 1(a), nevertheless merely by 3 meV. Note that such a small energy difference between these two structures may be already beyond the limit of the accuracy of the present DFT calculations. Therefore, we performed the second round of high accuracy calculations with an EC of 350.0 eV. Now, the DDTLW structure is 50 meV higher in energy than the most stable core-shell structure, C9-S46. First, these interesting results indicate that it is difficult to conclusively identify which structure is more stable due to their small energy difference, particularly in consideration of the entropy effect47 in experiment; however, unambiguously, the role of relativistic effect in modulating the relative stabilities of these two motifs is definitely established. More specifically, in the calculations using ECs of both 230.3 and 350.0 eV, the SOC stabilizes the relative stability of the DDTLW structure, by 93 and 104 meV, respectively, see Fig. 2. Additionally, the SOC effect in stabilizing the DDTLW Pt55 structures is also observed in calculations with PBE functional (not shown here). Furthermore, as introduced before, the relativistic effect, such as SOC, stabilizes the s orbitals and destabilizes the d orbitals, leading to the enhanced s-d hybridizations in the late TMs and reduced electron-electron repulsion in the open layered structures.16 

Motivated by the above findings that the relativistic effects may tune the relative stability of two different Pt55 cluster structures, it is imperative to investigate its role in determining the magic number of Pt nanoclusters around size 55. To do this, we first optimize the most stable structures of the PtN clusters (N = 52–61). Previously, based on the classic Wulff construction and geometric closed-shell model, the number 55 was widely accepted as a magic number for transition metal nanoclusters. Here, by using GGA + SOC calculations, our extensive searches have found the most stable structural candidates for PtN clusters (N = 52–61) as shown in Fig. 3, see also the supplementary material46 for their Cartesian coordinates. The preferred structural growth mode in this size range takes a very interesting route. The most stable Pt52 cluster prefers a low symmetric core-shell structure (C9-S43) over a defected-TLW-like structure, by 0.914 eV. Similarly, Pt53 cluster also favors a core-shell structure, denoted as C9-S44. These two structures were optimized from various arrangements for the numbers of core and shell atoms, but no layered structure with lower energy was found. Interestingly, when the cluster size increases to 54, the TLW-like structure becomes almost degenerate in energy with the most stable core-shell motif. For Pt54 cluster, an elongated-TLW configuration (54-(b) in Fig. 3) with each layer consisting of 18 atoms is found to be merely 5 meV less stable than the most stable one, 54-(a) in Fig. 3. From top view, the inner three-layered core is of A-A-A stacking. The surrounding 36 atoms are arranged in an A-B-A-like stacking and form a closed hexagonal belt without significant atomic defect from the side view. As discussed, for Pt55, the DDTLW structure is 50 meV less stable than the most stable amorphous core-shell forms, probably due to the two defect sites.

FIG. 3.

The optimized minimum energy structures of PtN cluster around size 55 (N = 52–61), by calculations considering spin-orbital coupling with PW91 functional. For both Pt54 and Pt55 clusters, the first low-lying isomers noted by (b) are also presented respectively.

FIG. 3.

The optimized minimum energy structures of PtN cluster around size 55 (N = 52–61), by calculations considering spin-orbital coupling with PW91 functional. For both Pt54 and Pt55 clusters, the first low-lying isomers noted by (b) are also presented respectively.

Close modal

From size 56, larger clusters favor stable structures that can be constructed by growing additional atoms onto the DDTLW structure. For example, the lowest energy structure of Pt56 is a single-defected TLW configuration. As expected, Pt57 prefers a perfect TLW with high symmetry of D6h, which is now 1.800 eV more stable than the most stable core-shell structures obtained. Even larger clusters investigated here can be optimized by growing additional atoms on the perfect TLW configuration to form another larger outer belt. Note that the large smooth facets and the sharp edges of these PtN nanostructures may serve as effective catalytic sites, such as for O2 activation and H2O splitting.48 Here, we emphasize that the transition of structural growth mode from the amorphous core-shell to the ordered layered one occurs in the vicinity of 54 and 55, at which size these two structural motifs become essentially degenerate in energy. Specifically, this correlates with the observation that the SOC effect plays a crucial role in reversing the relative stability of these two different symmetric structures. Note also that such an energetic transition between these two structures driven by SOC effect cannot be observed for both early and central elements, as also reported in previous work.12 Additionally, the DDTLW structure is not preferred by Pd55, confirming again the importance of the relativistic effect in tuning the growth modes (from core-shell to TLW motifs) of the PtN clusters in the vicinity of the critical size, around 55.

To identify the magic numbers of PtN nanocluster around size 55, we further calculated the average binding energy per atom, Eb(N) = − [E(PtN) − N × E(Ptatom)]/N, and its second-order difference, Δ2Eb(N) = Eb(N + 1) + Eb(N − 1) − 2Eb(N) in Fig. 4. Intriguingly, the odd number of 57 is established as the magic number, rather than the widely accepted number 55 from geometric construction. This is consistent to the perfect TLW structure in Fig. 3. Previously, we reported that for the TMn clusters of central elements, around size 55, the magic numbers are even (such as 56) which is guided by the generalized Wulff construction principle. This principle considers the significant contribution from edge energy minimization, so the magic size of 56 is a natural outcome of the symmetry restrictions by the corresponding even-layered fcc- or hcp-like crystal fragment configurations.12 Here, we stress that the new odd magic number of 57, rather than 55, is also a natural outcome of an odd (rather than even) layered highly symmetric hexagonal atomic arrangement. To access the contribution of the relativistic effect in stabilizing this high symmetric magic cluster (Pt57), we calculated the energy difference (ΔE) between the most stable core-shell amorphous configuration and the D6h magic motif without considering the spin-orbital coupling. The calculated ΔE is only slightly reduced to 1.629 eV, from 1.800 eV obtained in the relativistic calculation, namely, the relativistic effect may merely stabilize the layered D6h structure by 171 meV, which is close to the value of ∼104 meV in the case of Pt55. Note that for the TLW-Pt57, no significant gap between the highest occupied and the lowest unoccupied molecular orbitals is observed, i.e., electronic shell closure6 is excluded.

FIG. 4.

Average binding energy per atom, Eb(N) = − [E(PtN) − N ×  E(Ptatom)]/N, and its second-order finite difference, Δ2Eb(N) = Eb(N + 1) + Eb(N − 1) − 2Eb(N), for different PtN clusters with the lowest energies presented in Fig. 3. The data points marked by circles are for the binding energies, Eb(N), and stars represent the second-order derivatives, Δ2Eb(N).

FIG. 4.

Average binding energy per atom, Eb(N) = − [E(PtN) − N ×  E(Ptatom)]/N, and its second-order finite difference, Δ2Eb(N) = Eb(N + 1) + Eb(N − 1) − 2Eb(N), for different PtN clusters with the lowest energies presented in Fig. 3. The data points marked by circles are for the binding energies, Eb(N), and stars represent the second-order derivatives, Δ2Eb(N).

Close modal

Therefore, there must be some other mechanism taking over the relativistic effect in the structure and magicity of these layered structures. We now elucidate in more detail the underlying mechanism of the magic cluster Pt57 and the TLW-PtN motifs. Note that magic cluster Pt57 is of highly symmetric layered planar configuration, whereas Au57 and Au58 prefer low symmetric amorphous form that is grown on distorted Ih-Au55 structure.15 To explain this contrast, we performed a calculation to relax Au57 cluster with the optimized D6h-Pt57 structure as the initial configuration. Unexpectedly, we observed an amazing structural transition from the planar TLW to an interesting core-shell structure which can be viewed as a planer Au7 core encapsulated in an elliptic D6h-Au50 shell whose configuration is similar to the Au50 cage as previously reported.49 In this process, the two large planar surfaces of D6h-Au57 structure significantly and smoothly arched, leading to large Au–Au bond distances along the high symmetric axis. This phenomenon indicates that the three-layered D6h-Pt57 magic cluster may be stabilized by much stronger interlayer binding in contrast to that of Au clusters.

The above deduction has been further validated by the electronic charge-difference (Δρ) analysis defined by Δρ = ρ(SC) − ρ(SP). Here, ρ(SC) is obtained by a self-consistent calculation method and ρ(SP) by the superposition of the atomic charge for the same structure. First, taking the optimized D6h-Pt57 as an example, we have presented the two-dimensional Δρ charge contour projected onto the high symmetry plane bisecting the D6h-Pt57 cluster as shown by the top panel in Fig. 5. Considerable charge density accumulations have been identified in the bond centers, revealing a significant level of d-type covalent bonding19,20,48 of the interlayers. This argument is also further supported by the electronic density of state (DOS) analysis. In Fig. 6, we compared the DOS of the structures D6h-Pt57 and the most stable core-shell motif C9-S48 presented in Fig. 3 and Fig. S146 of the supplementary material (wherein the optimized Cartesian coordinates of the C9-S48 isomer is also listed), respectively. Clearly, the strong covalent interlayer bonding in the D6h-Pt57 results in a significant pseudo-gaps around both 0.5 and 0.1 eV below the Fermi level, simultaneously reducing the DOS as compared to the amorphous configuration. On the other hand, the sharp edge atoms of the TLW structure dominate the peaks around −0.2 eV and the Fermi level in the DOS (see Fig. 6), which is also comparable with that of the amorphous motif in the vicinity of the Fermi level, indicating no excess undesirable d-type dangling bonds exist on the sharp edges of the TLW as compared to C9-S48.

FIG. 5.

The two-dimensional (top panel) and one-dimensional (bottom panel) electronic charge-difference (Δρ) plots for Ir57, Pt57, and Au57, with Δρ = ρ(SC) − ρ(SP). Here, ρ(SC) is obtained by a self-consistent GGA + SOC calculation method and ρ(SP) by the superposition of the atomic charge for the same structure. The one-dimensional Δρ is obtained along the axis of the bonds of the interlayers for different systems, and the bond lengths are shown in normalized scale. See the corresponding labels of “A, B, C” in these two panels.

FIG. 5.

The two-dimensional (top panel) and one-dimensional (bottom panel) electronic charge-difference (Δρ) plots for Ir57, Pt57, and Au57, with Δρ = ρ(SC) − ρ(SP). Here, ρ(SC) is obtained by a self-consistent GGA + SOC calculation method and ρ(SP) by the superposition of the atomic charge for the same structure. The one-dimensional Δρ is obtained along the axis of the bonds of the interlayers for different systems, and the bond lengths are shown in normalized scale. See the corresponding labels of “A, B, C” in these two panels.

Close modal
FIG. 6.

Electronic density of states (DOSs) of Pt57 clusters. Filled data represent the DOS of the high symmetric TLW-Pt57 while the red lines represent the DOS of the amorphous core shell C9-S48 structures, respectively.

FIG. 6.

Electronic density of states (DOSs) of Pt57 clusters. Filled data represent the DOS of the high symmetric TLW-Pt57 while the red lines represent the DOS of the amorphous core shell C9-S48 structures, respectively.

Close modal

However, for the optimized core-shell D6h-Au57 structure, no significant charge accumulations are observed at the interlayers, displaying a weak covalent bonding of the inner core and the surface shell. Note that although D6h-Au57 possess weak covalent bonding between the core and shell atoms, significant charge accumulations are still observed at the Au–Au bond centers of the surface shell, which exhibits strong covalent bonding characteristic.19,20 We have also plotted the one-dimensional Δρ along the six-fold axis of these structures, as presented in the lower panel of Fig. 5. A significantly stronger covalent bonding feature in Pt57 is further identified than that in Au57, so that, correspondingly, open and planar TLW structure is preferred by the former. Our calculations show that in a four-layered Pt57 structure, the covalent inter-layer binding is significantly reduced as compared with TLW-Pt57, due to the limited number of unpaired d electrons in Pt. This is another origin of the resulted odd magic number of PtN nanocluster around size 55, which prefers odd-layered (three-layered) high symmetric configurations.

Note also that amorphous configurations35 for Au and Pt clusters were previously interpreted in terms of a rosette-like reconstruction of Ih structures, which essentially reduces the number of high energetic edge atoms and gives rise to local fcc (111) facet, and thus can be explained well by the recently established generalized Wulff construction principle.12 However, here, we reveal that PtN nanoclusters around size 55 prefer highly ordered planar configuration due to SOC, particularly the intrinsic open d-orbital resulted covalent bonding, as supported by the electronic structure comparison between Pt and Au performed above.

Here, we emphasize that such an intriguing TLW structure and resulted magic number, as dominated by the collective effect of a relativistic effect and the open d-orbital enhanced covalent bonding, are also valid to the nearest neighbor element Ir. Further extensive calculations have confirmed that IrN cluster around size 55 also prefers the new TLW structural forms over the previously reported fcc-like crystal fragment form, by 1.967 eV, for Ir55 (for more details, see also the supplementary material of Ref. 12), and Ir57 also exhibits magic cluster properties. The electronic charge contour presented in Fig. 5 verifies that Ir57 possesses even stronger covalent bonding characteristic as compared to its later neighbors, Pt and Au. Thus, we can conclude that, to stabilize these TLW structures, the contribution from the covalent bonding due to the intrinsic open d-orbital in Ir, Pt, and Au is qualitatively of the following sequence: Ir>Pt>Au, though the enhanced covalent bonding due to a pure relativistic effect is completely reversed: Ir<Pt<Au.28 In conclusion, the magicity of highly symmetric D6h-Pt57 nanocluster is facilitated not only by the SOC effects which effectively enhance the s-d hybridization by the Fermi level but also by the intrinsic open d-orbital which significantly enhance the covalent bonding of the inter-layers.

To the end, we emphasize that for the cases of PtN and IrN around size 55, the GWCP still plays an important role in determining their preferred structures, as manifested by the enlarged surface areas and reduced edge length.12 Note that for Pt57, four-layered planar configurations are found to be dramatically unstable, due to significantly enlarged areas of high energy facets and particularly the increased number of low-coordinated edge atoms. Importantly, moving towards the central element Os with more unpaired d electrons as compared to Ir and Pt, the GWCP mechanism dominates and the present TLW forms are significantly less stable than those structures predicted by GWCP,12 due to the critically undesirable d-type dangling bonds12 located on the sharp edges of TLW structure, as manifested by the relatively enhanced DOS peaks on the Fermi level, see also Fig. 6. Additionally, we have summarized the four established mechanisms in determining the atomic stacking rule and magic numbers of a given elemental nanocluster around the size 55 in a “magicity table,” to schematically highlight the position- or electronic configuration-dependent mechanisms. As have been discussed, these four representative mechanisms are atomic shell closure for inert gas, electronic shell closure for simple metal (both atomic and electronic shell closures for fullerene), generalized Wulff construction principle for transition metal, and synergetic effects (including SOC) for heavy noble elements, such as Au, respectively, Fig. 7. To do this, we have comparatively presented the geometric structures of the representative magic nanoclusters around size 55, such as Na58, Y55,12 Cu55,12 Ag55,12,18 Ar55,5 Ru56,12 Ag58,12,18 Pt57, Au58,15 and C60, of which the magic mechanisms are indicated by different models. For example, atomic close-shelled magic cluster Ag55 is presented by a space-filling model, and the magic clusters of Ag58, Au58, and C60 of both atomic and electronic shell closures are shown in ball-and-stick structure embedded in electronic charge, respectively. We wish this magicity table is highly instructive for readers to readily catch the characteristic mechanism for a given elemental nanocluster.

FIG. 7.

Schematic “magicity table” of the four mechanisms established in determining the atomic stacking rule and magic numbers of nanoclusters, including atomic shell closure for inert gas, electronic shell closure for simple metal, generalized Wulff construction principle for transition metal, and relativistic effects for heavy noble elements, respectively. The relative importance of a given mechanism is marked by the color shade. Correspondingly, we have also schematically presented the geometric structures of the representative elemental magic-sized nanoclusters around size 55, such as Na58, Y55, Cu55, Ag55, Ar55, Ru56, Ag58, Pt57, Au58, and C60, respectively. Refer to text for detailed interpretations of these structures.

FIG. 7.

Schematic “magicity table” of the four mechanisms established in determining the atomic stacking rule and magic numbers of nanoclusters, including atomic shell closure for inert gas, electronic shell closure for simple metal, generalized Wulff construction principle for transition metal, and relativistic effects for heavy noble elements, respectively. The relative importance of a given mechanism is marked by the color shade. Correspondingly, we have also schematically presented the geometric structures of the representative elemental magic-sized nanoclusters around size 55, such as Na58, Y55, Cu55, Ag55, Ar55, Ru56, Ag58, Pt57, Au58, and C60, respectively. Refer to text for detailed interpretations of these structures.

Close modal

Before closing, we emphasize that the present findings obtained by theoretical calculations are based on the gas phase nanoclusters, and wish these interesting results may motivate future experimental efforts. Additionally, clusters deposited on surfaces naturally lead to variations of the cluster-surface contact and bindings, charge transfer may also occur between the clusters and substrate, thereby leading their structures, magicity,50 and catalysis to change, which are of our great interest in future investigations.

Previously, electronic/geometric closed-shell models and generalized Wulff construction principle have been established to predict the atomic stacking rule and magic numbers of early/late and central elemental TM nanoclusters, respectively. However, for heavy noble elemental nanoclusters, such as PtN and IrN, it is difficult to predict their atomic stacking rule and magic numbers using these established principles, because these elements lie on the boundaries of different domains of the periodic table that are governed by different principles. In this paper, by means of first-principles calculations, we have specifically identified that PtN nanoclusters exhibit significant geometric transition from core-shell amorphous structure to an intriguing three-layered structural growth mode around size 55, a result of the synergetic effect from the generalized Wulff construction principle, relativistic effect, and particularly the open-d-orbital enhanced covalent bonding. Correspondingly, the magic cluster of PtN around size 55 is unexpectedly shifted from the widely accepted number 55 to 57, leading to a TLW structure with D6h symmetry. The present findings are found to be valid up to the case of Ir (note: these interesting results deserve of further experimental examinations, such as photoelectron spectra analysis). The contrast atomic stacking form and magic numbers around 55 between PtN (IrN) and AuN clusters are mainly due to the relatively stronger covalent bonding stemmed from the open-shelled d orbital in the former case, as opposed to the closed-shell of the latter, although the relativistic effect is weaker for the former. The present findings are important supplementary of the recently established GWCP which is expected to play an instrumental role in future design of novel metal based nanostructures with desirable functionalities for potential applications such as in nanocatalysis.

This work was supported by the Natural Science Foundation of China (Nos. 11074223 and 11034006) and the US National Science Foundation (Nos. CMMI 0900027, CMMI 1300223, and DMR 0906025).

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Supplementary Material