A grand unified model for liganded gold clusters

A grand unified model (GUM) is developed to achieve fundamental understanding of rich structures of all 71 liganded gold clusters reported to date. Inspired by the quark model by which composite particles (for example, protons and neutrons) are formed by combining three quarks (or flavours), here gold atoms are assigned three ‘flavours' (namely, bottom, middle and top) to represent three possible valence states. The ‘composite particles' in GUM are categorized into two groups: variants of triangular elementary block Au3(2e) and tetrahedral elementary block Au4(2e), all satisfying the duet rule (2e) of the valence shell, akin to the octet rule in general chemistry. The elementary blocks, when packed together, form the cores of liganded gold clusters. With the GUM, structures of 71 liganded gold clusters and their growth mechanism can be deciphered altogether. Although GUM is a predictive heuristic and may not be necessarily reflective of the actual electronic structure, several highly stable liganded gold clusters are predicted, thereby offering GUM-guided synthesis of liganded gold clusters by design.

L iganded gold clusters have attracted intensive interest over the past 10 years owing to their broad and practical applications in catalysis 1 , electrochemistry 2 , quantum electronics 3 and biomedicine 4 . A grand challenge to scientists in this field, however, is the precise determination of atomic structures of liganded gold clusters. To date, atomic structures of tens of liganded gold clusters have been determined via X-ray crystallography  . Nevertheless, these revelations appear serendipitous as the structural determination largely hinges on availability of single crystals for the liganded gold clusters. Although density-functional theory computation has been widely applied to predict the structures of many gold clusters [64][65][66][67][68][69][70][71][72][73][74][75][76][77][78] , ultimate confirmation still requires X-ray crystallography measurements. Theoretical efforts have also been made in the past for more general unification models to comprehend stabilities of liganded gold clusters with apparently very different and seemingly unrelated complex structures. For example, Wade-Mingos counting rules 79,80 can provide a simple rationale of various shapes of 'electron-deficient' polyhedral clusters in terms of the number of skeletal electron pairs, particularly for borane and carborane clusters. However, few gold clusters can be rationalized by the Wade-Mingos counting rules 81 , especially for the gold clusters with a high number of interstitial atoms, whose structures have been precisely resolved 60,61 . The superatom complex (SAC) model proposed by Walter et al. 82 suggests that the high stability of several spherical-like ligand-protected gold clusters 26,55,60 is due largely to the strong electron shell closures, an important concept that stems from the jellium model 83 84 to explain the electronic stability of non-spherical shells of Au 38 (SR) 24 (ref. 53). They suggest that the bi-icosahedral Au 23 ( þ 9) core of Au 38 (SR) 24 can be viewed as a superatomic molecule. Later, the superatom network (SAN) 66 model, coupled with the adaptive natural density partitioning analysis 85 , have been also invoked by Cheng et al. to explain the high stability of certain low-symmetry ligand-protected gold clusters. A key notion in the SAN model is that the core of Au nanoclusters can be viewed as a network of n-centered two-electron (n ¼ 2-6) superatoms. A tetrahedral unit with two valence electrons has also been identified by Jin and co-workers 58 through an account of the number of valence electrons and tetrahedral units in a serial of structurally resolved double-helical gold clusters (Au 28 (SR) 20 , Au 36 (SR) 24 , Au 44 (SR) 28 and Au 52 (SR) 32 ) 44,49,56,58 . Recently, the Borromean-ring diagrams for the Au 25 (SR) 18 (refs 42,43), Au 38 (SR) 24 (ref. 53) and Au 102 (SR) 44 (ref. 60) clusters have been proposed by Pradeep, Whetten and co-workers 86 to explain high stabilities of these clusters. All the theoretical models developed thus far are mainly to address stabilities of a subset of gold nanoclusters, rather than the entire set of 71 reported liganded gold clusters. As such, exceptional cases to these independent models abound. Hence, a grand unified model that can go beyond these previously developed models (SAC, SVB, SAN and so on) for understanding stabilities of all ligand-protected gold clusters is called for. Here we present a grand unified model (GUM) that can offer a universal description of the structures of diverse liganded gold clusters.
In this communication, the triangular elementary block Au 3 (2e) and tetrahedral elementary block Au 4 (2e) are identified to describe the stabilities of 71 liganded gold nanoclusters (Supplementary Table 2) reported up to date. On the basis of the GUM, deeper insights into structure evolution of the liganded gold nanoclusters can be obtained, namely, the structure evolution of the gold core cannot be viewed simply as addition of Au atoms, but rather as seamless packing of the elementary blocks. In addition, several stable liganded gold clusters are predicted.

Results
GUM development and quark model analogy. The development of the GUM is based on detailed analysis of the structures of all 71 liganded gold clusters (Supplementary Table 2) either determined from previous experiments (54 crystallized structures) or predicted from density-functional theory computation (17 structures) over the past three decades. The scheme of grand conceptual unification of the diverse structures of these 71 liganded gold clusters is motivated from the quark model in particle physics wherein six types of quarks, known as flavours, are conceptualized as a unification scheme for composite particles, such as protons and neutrons, and exotic hadrons, in terms of their valence quarks. For instance, it is known that protons, neutrons alike are not elementary but are viewed as bound states of the elementary valence quarks and antiquarks. All quarks are characterized by a set of quantum numbers, such as fractional electric charge of ± 2/3 or ± 1/3. In an analogous fashion, here, we assign a gold atom as the 'elementary particle' but with one of three 'flavours' due to its three possible valence states, that is, 1e, 0.5e and 0e. The three flavours are named as bottom, middle and top, respectively. Through close inspection of the 71 known clusters, we identify two 'composite particles', namely, the triangular Au 3 and tetrahedral Au 4 elementary blocks, in analogy to the protons and tetraquarks, respectively. We find that both elementary blocks satisfy the duet rule, that is, the high tendency of having two electrons in the valence shell. As a result, depending on the flavour of each constituent gold atom, the triangular elementary block can exhibit in total 10 variants of valence states (named as D 1 -D 10 ), whereas the tetrahedral elementary block can exhibit in total 15 variants of valence states (named as T 1 -T 15 ) ( Fig. 1). We show that for all 71 liganded gold clusters, once the outer ligands are effectively detached from the inner Au cores (see below), the resulting Au cores are universally packed by the elementary blocks. Hence, the stabilities of the liganded gold clusters are due to the high stability of each individual elementary block.
Duet rule. Note that the duet rule elucidated here is akin to the textbook octet rule, a well-known and the first chemical rule of thumb in general chemistry. The octet rule is a valence-electron counting rule for the explanation or prediction of electronic structure and chemical bonding of molecules made of main-group elements. The other two valence-electron counting rules (that is, the second and third rules of thumb), namely, the 18-electron rule and Wade's rule, are newer chemical rules of thumb for understanding chemical structures of organometallics and polyhedral cluster compounds, respectively. Below, we report that in conjunction with the GUM, the duet rule of the valence shell for the elementary blocks can be treated as the fourth rule of thumb for understanding diverse liganded gold clusters.
Evidences on stabilities of the elementary blocks. In GUM, both elementary blocks triangular Au 3 and tetrahedral Au 4 entail only two valence electrons [Au 3 (2e) and Au 4 (2e)], thereby both having strong electron shell closures. As shown in Fig. 2, the two valence electrons are delocalized in the shell-closing elementary blocks Au 3 (2e) and Au 4 (2e), consistent with the SAC model 82 . Moreover, from an experimental perspective, the Au 3 core of crystallized [Au 3 (IDipp) 3 Table 2) are essentially the same as the elementary blocks Au 3 (2e) and Au 4 (2e), respectively, supporting the high stabilities of the Au 3 (2e) and Au 4 (2e). The high stabilities of Au 3 (2e) and Au 4 (2e) structures can be also shown from ab initio computation. In Supplementary  Table 3, the formation energies of five isoelectronic species, Au 2 (2e), Au 3 (2e), Au 4 (2e), Au 5 (2e) and Au 6 (2e), are listed. Only the triangular Au 3 (2e) and tetrahedral Au 4 (2e) exhibit highly negative formation energies, which provides another piece of strong evidence of their high stabilities. In addition, both elementary blocks exhibit larger highest/lowest occupied/ unoccupied molecular orbital (HOMO/LUMO) gaps than their isoelectronic counterparts (Supplementary Table 3).
Electron counting protocols for effective detachment of ligands. All ligand-protected Au clusters are composed of an inner Au core and a number of outer ligands. The first step toward developing the GUM is to find effective protocols to detach the protection ligands from the Au core so that the vast complex factors of the outer ligands can be removed while all ligand-protected Au clusters can be reduced to bare Au cores for structural analysis. Note that different elements or functional groups that are directly bonded with an Au atom at the core surface are in different valence-electron states. As such, if one is to focus on the valence state of the inner Au core, equivalent electron counting is required to effectively detach all protection ligands from the Au core. The following electron-counting protocols for effective detachment of different types of ligands can be undertaken (see Fig. 3 for graphical illustrations): First, each SR group and Au atom embedded in the gold-thiolate staple motifs possess À 1e and 1e valence electron, respectively. To effectively detach the smallest staple motif from the Au core, the net number of valence electrons of the staple motif should be converted to zero. To this end, each of the two Au atoms (on the Au core) bonded with SR is considered to transfer 0.5e valence electron to the staple motif. As such, equivalent electron counting for effective detachment of the staple motif is achieved (Fig. 3a). Second, for SR group bonded with two Au atoms on the Au core, again, each of the two Au atoms (on the core) is considered to transfer 0.5e valence electron to the SR group so that equivalent electron counting for effective detachment of the SR group is achieved (Fig. 3b). Third, for halogen ligand X (X ¼ F, Cl, Br and I) bonded with a single Au atom on the Au core, each Au atom is considered to transfer 1e valence electron to the X atom so that equivalent electron counting for effective detachment of the X atom is achieved (Fig. 3c). Fourth, the phosphine group is known as a weak ligand, thereby possessing 0e valence electron. As such, each Au atom bonded with the phosphine group still maintain its original 1e valence electron upon detachment with the phosphine group (Fig. 3d).
In summary, depending on the ligands, for example, PR 3 , X, SR or gold-thiolate staple motifs, Au atoms on the Au core can exhibit one of the three flavours: bottom (1e), middle (0.5e) and top (0e) after effective detachment of the protection ligands from the Au core. Note also that when two elementary blocks are fused together via sharing a single Au atom, the shared Au atom contributes 0.5e valence electron to each elementary block. As such, the fused elementary blocks can be effectively separated via the protocol shown in Fig. 3e.  The remaining cases can be analysed in similar fashion.

Prototypical liganded clusters. As shown in Supplementary
The first prototype structure we consider is [Au 6 (dppp) 4 ] 2 þ (dppp ¼ 1,3-Bis(diphenylphosphino)propane) 9 , which is composed of four dppp ligands (each with 0e valence electron) and an Au 6 2 þ core (Fig. 4a). The Au 6 core consists of two triangular Au 3 blocks. According to the electron-counting protocol, the six Au atoms of the Au 6 core are all bonded with phosphine ligands and thus have 1e valence electron or the bottom flavour. Two positive charges are equally distributed in the two triangular Au 3 blocks. Thus, both triangular Au 3 elementary blocks are in the D 1 valence state (Fig. 4b).
The second prototype structure we consider is Au 40 (SR) 24 (ref. 56). The Au core of Au 40 (SR) 24 can be viewed as a combination of an Au 7 and a Kekulé-like Au 18 structure (Fig. 5a). The Au 7 core is composed of two tetrahedral Au 4 blocks, fused together by sharing an Au atom. The latter contributes 0.5e valence electron to each tetrahedral Au 4 block. Other six vertex Au atoms are bonded with the SR group and each contributes 0.5e valence electron to the resident tetrahedral Au 4 block. Thus, each of the four vertex Au atoms in the tetrahedral Au 4 block has the middle flavour so that each tetrahedral Au 4 elementary block is in the T 9 valence state (Fig. 5b). Moreover, the Kekulé-like Au 18 structure can be viewed as six tetrahedral Au 4 blocks fused together in a loop with six sharing Au atoms (Fig. 5c). Again, each of the six tetrahedral Au 4 elementary blocks is in the T 9 valence state.
Finally, eight other representative clusters are considered. The structure decompositions of eight other representative clusters in terms of elementary blocks as well as their corresponding valence states are shown in Supplementary  Figs 1-8. Specifically, the structure decomposition of three largest ligand-protected gold clusters, Au 102 (SR) 44 19 [RS-Au-SR-Au-SR] 2 . Each of the two Au atoms in the Au 102 (SR) 44 shared by two [-RS-Au-SR-] staple motifs can be viewed as the Au atom with 0e valence electron (that is, top flavour). As such, the Au 79 core of Au 102 (SR) 44 is composed of 29 elementary blocks and their corresponding valence states can be described by [5D 1 , D 3 , 16D 5 , 2T 4 , 2T 5 , T 7 and 2T 9 ] (Supplementary Fig. 9), giving rise to 58 valence electrons in total (corresponding to the strong electron shell closure, according to SAC model 82 ) for Au 102 (SR) 44 . The Au 105 core of Au 130 (SR) 50 is composed of 40 elementary blocks and their corresponding valence states include 5D 3 , 11D 5 , 4T 4 , 3T 5 and 17T 9 ( Supplementary Fig. 10), giving rise to 80 valence electrons in total for Au 130 (SR) 50 . The Au 114 core of Au 144 (SR) 60 is composed of 42 elementary blocks and their corresponding valence states include 20D 5 , 13T 4 , 3T 5 and 6T 9 ( Supplementary  Fig. 11), giving rise to 84 valence electrons in total in Au 144 (SR) 60 .

Discussion
In general, the inner Au cores of all the ligand-protected gold clusters are composed of elementary blocks (Supplementary Table 2 and Fig. 6), each having two valence electrons (Fig. 1), whereas the outer ligands with 0e valence electron provide the geometry constraint and arrangement of valence electrons in each elementary block to satisfy the duet rule. In Supplementary  Fig. 12 6 clusters are expected to be less stable due to their violation of the duet rule. Note that a recent experimental investigation of Au 25 (SR) 18 in its three oxidation states, that is, Au 25 (SR) 18 À 1/0/ þ 1 , provides more compiling evidence on the effect of violation of the duet rule (or deviation from the strong electron shell closure) to the symmetry of the Au 13 core and stability of the cluster 89 . The obtained three crystalline structures in the related three oxidation states demonstrate that the structural distortion (in the Au core in particular) increases with the decreased superatomic valence from 1S 2 1P 6 to 1S 2 1P 4 as the Au 25 (SR) 18 À 1 cluster has the eight-electron shell-closing configuration 82 1S 2 1P 6 , and thus the highest thermal stability. The other two oxidation states, Au 25 (SR) 18 0/ þ 1 , are less stable due to the incomplete 1P superatomic orbital. Likewise, the structural distortion observed in the two oxidation states, Au 25 (SR) 18 0/ þ 1 , can be understood based on GUM. The Au 13 core of Au 25 (SR) 18 À 1 can be decomposed into four elementary blocks and their corresponding valence states can be described as [2D 5 and 2T 9 ] (Supplementary Table 2). Clearly, the number of valence electrons in both oxidation states, Au 25 (SR) 18 0/ þ 1 , dissatisfies the duet rule. As a consequence, lower thermal stability and larger structural distortion in the highly symmetric Au 13 core are expected, consistent with the experimental evidence 89 .
The GUM does not only offer a universal structural characterization of all 71 liganded gold clusters, but it also provides deeper insights into structure evolution of the Au clusters. In Fig. 6, a structure evolution map for the Au cores with increasing number of elementary blocks is presented. Indeed, the structure evolution of the Au cores can be understood through various routes of packing the elementary blocks. For example, two elementary blocks with T 9 valence state can yield either the Au 8 core of Au 24 (SR) 20   In summary, the structure evolution of the Au core cannot be viewed simply as addition of Au atoms ( Supplementary Fig. 13), but rather as seamless packing of the elementary blocks while obeying the duet rule. In addition to structure unification, the GUM can be utilized for predicting new structures of liganded gold clusters. From the structure evolution map shown in Fig. 6, one can see that there are many vacant spaces, suggesting many missing liganded gold clusters yet to be synthesized. We construct a series of ligand-protected gold clusters (shown in dotted squares in Fig. 6) on the basis of GUM to fill some vacant spaces in Fig. 6. The elementary blocks and their valence states for the Au cores of all constructed clusters are depicted in Supplementary Fig. 14. These clusters exhibit large computed HOMO-LUMO gaps (Supplementary Table 4), suggesting potentially high chemical stabilities.
In particular, a class of hitherto unreported ligand-protected hollow Au clusters are presented here. For example, Au 36 (SR) 12 can be constructed by using the C 12 fullerene as a template. C 12 fullerene exhibits eight polygons: four quadrilaterals and four pentagons (Fig. 7a). Replacing all C atoms of the C 12 fullerene with 12 fused tetrahedral Au 4 gives rise to the Au 30 core (Fig. 7b) of Au 36 (SR) 12 , followed by adding [-RS-Au-SR-] staple motifs on the unfused Au atoms to build the complete Au 36 (SR) 12 cluster (Fig. 7c). The Au 30 hollow cage in Au 36 (SR) 12 is composed of 12 fused elementary blocks all at the T 9 valence state. The Au 36 (SR) 12 cluster exhibits a large computed HOMO-LUMO gap of 2.20 eV and has no imaginary vibrational frequencies. An ab initio molecular dynamics simulation of the Au 36 (SR) 12 cluster at 355 K for 10 ps suggests high thermal stability of the Au 36 (SR) 12 cluster (Supplementary Fig. 15 and Supplementary Methods). Figure 7f shows another example of ligand-protected hollow Au cluster, namely, Au 42 (SR) 14 , constructed by using the C 14 fullerene (Fig. 7d) as a template. The Au 35 hollow cage (Fig. 7e) in Au 42 (SR) 14 is composed of 14 fused elementary blocks all at the T 9 valence state. The computed HOMO-LUMO gap of Au 42 (SR) 14 is 2.00 eV, suggesting high chemical stability. Interestingly, the Au 36 (SR) 12 and Au 42 (SR) 14 can be rewritten as Au 30 [Au(SR) 2 ] 6 and Au 35 [Au(SR) 2 ] 7 , respectively, consistent with the 'divide-and-protect' formulation 87,88 .
In conclusion, a grand unified model that can incorporate previously developed independent models (SAC, SVB, SAN and so on.) is developed to address stabilities of all ligand-protected gold clusters. On the basis of the GUM, all 71 liganded gold nanoclusters can be decomposed into several elementary blocks of triangular Au 3 (2e) and tetrahedral Au 4 (2e). Although GUM is a predictive heuristic and may not be necessarily reflective of the actual electronic structure, a series of highly stable liganded gold clusters are predicted, which provides a guide to synthesizing new ligand-protected gold clusters. Hence, the GUM can offer not only new insights into the packing and structure evolution of the 71 liganded gold clusters known as of today, but also a systematic route toward rational design and characterization of liganded metal clusters to inspire future experimental synthesis.
Data availability. The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information files, and all relevant data are available from the authors.