Evolution on the Size of Protected Metal Clusters by Assembly of Au₄ and Au₆ Units

Abstract

Perfect tetrahedra and octahedra are the fundamental building blocks of FCC lattices; however, even distorted versions of these polyhedra can assemble into periodic HCP and BCC metal structures. In the case of protected metal clusters (PMCs), protecting ligands such as thiols, alkynyls, carbenes, and phosphines are responsible for the observed distortion of their metal frameworks due to the strong covalency of M–S, M–C, and M–P bonds (M stands for metal atoms). This work examines the assembly of tetrahedral and octahedral units and their relationship to the inner cores of PMCs. For instance, a perfect or distorted tetrahedron can share 1, 2, or 3 vertices with another tetrahedron, forming M₇(D _3d or D _3h ), M₆(D _2h or C _2h ), and M₅(D _3h ) units, respectively. Accordingly, M₇ units containing differently oriented tetrahedra can serve as building blocks for the FCC and HCP structures. Similarly, one octahedron sharing 1, 2, and 3 vertices yields M₁₁(D _4h or D _2h ), M₁₀(D _2h ), and M₉(D _3h ) units, respectively. The assembly of one octahedron and two tetrahedra along one C ₃ axis yields an M₈ unit displaying D _3d , D _2h , or C _2h symmetry. In contrast, the icosahedral M₁₃ cluster is composed of 20 distorted tetrahedra, while an M₁₂ shell can be identified in structures formed by 20 octahedra. Regarding decahedral structures, they contain pentagonal bipyramids (M₇ units with a D _5h symmetry) assembled with five distorted tetrahedra. It is noteworthy that unlike in FCC or HCP atomic arrangements, M₅, M₇, and M₉ units are not present in BCC lattices. This study implements an algorithm that successfully generates atomic positions by capping an inner core with tetrahedra or octahedra, revealing the pivotal role of the type and distribution of polyhedral units in defining the anatomy and bonding of PMCs, and offering new insights into their structural understanding.

1. Introduction

Nowadays, there are around 345 crystallized structures of protected metal clusters (PMCs), and they are studied for potential application in catalysis, biomedicine, and optoelectronic devices. − These novel structures are composed of an inner core of metal atoms surrounded by a protective organic shell. The metal core may consist of Au, Ag, Cu, Pd, Al, Ga, Ge, Ru, Ir, or a mixture of these elements, while the outer shell contains ligands, such as thiols, alkynyls, phosphines, and carbenes. Although periodic face-centered cubic (FCC) metal structures are composed of a mixture of perfect tetrahedra and octahedra, thiolated gold clusters (TGCs) exhibit dispersed metal–metal bonds due to interactions at the metal–ligand interface. The great variety in size, symmetry, inner core shape, and overall structure of these clusters has prompted the development of theoretical methods to explain their stability and bonding. These include the superatomic network (SAN) model, grand unified (GUM) model, polyhedra approach, and tetrahedron-triangle growth (TTG) mechanism. Obviously, their electronic and related properties depend on size, and when PMCs consist of tens of atoms, quantum size effects become more pronounced. The Polyhedra method, introduced in 2019, was developed to identify tetrahedral and octahedral building blocks (units of FCC crystals) within the structure of TGCs. It was stated that distorted tetrahedra and octahedra are components of TGCs, and their bond lengths no longer match those found in the bulk phase. Therefore, the distortion induced by the strong metal–sulfur interaction must be considered, and an algorithm was needed to address this. In 2024, a Fortran version of the Polyhedra approach was released, enabling a more comprehensive study of TGCs. To understand how size influences PMCs properties, it is essential to examine the distribution of polyhedra and their aggregation motifs, including FCC, body-centered cubic (BCC), or hexagonal close-packed (HCP), decahedral, polytetrahedral, and icosahedral arrangements. The code enabled the identification of types and number of polyhedral units (Au₄ tetrahedra and Au₆ octahedra), their correlation with other building blocks in the inner cores of TGCs, and the analysis of the tetrahedra and octahedra distribution across the structure. In this work, a new algorithm is implemented (see Supporting Information) to simulate the growth/assembly of PMCs based on tetrahedra and octahedra, providing insights into their growth pathways. Further analysis of the generated structures enables us to characterize their atomic arrangements. Additionally, it is demonstrated that this analysis can be extended to systems involving silver, copper, aluminum, gallium, palladium, and various other metals.

2. Methodology

In this manuscript, various generated clusters were optimized through DFT-D calculations, as implemented in the ORCA package. All structures were calculated in the gas phase, and a small ligand (–SH) was considered when it was required. The Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional was used. In the case of Au atoms, the def2-TZVPP Ahlrichs basis set and the Coulomb-fitting auxiliary basis def2/J were considered, which is defined due to the usage of the RI approximation. Effective Core Potentials substituted the 19 valence electrons of gold atoms. , The energy and gradient convergence criteria were selected as 1 × 10^–6 Hartree and 3 × 10^–5 Hartree Bohr^-1, respectively. The election of mentioned parameters reduced the computation time without lack of the quality in the obtained results.

3. Results and Discussion

Prior to this study, it is important to note that the polyhedra approach explained the FCC atomic arrangement as a unique mixture of tetrahedra and octahedra. It also revealed various growth patterns through mutual coverage between tetrahedra and octahedra, including configurations such as two octahedra sharing one edge, two octahedra sharing a vertex, and multiple tetrahedra capping an icosahedron. Herein, the implementation of a new algorithm to simulate the growth of an inner core through the sequential addition of tetrahedra and octahedra enables the investigation of various polyhedral assemblies. This study offers a comprehensive/simplified explanation of FCC, BCC, and HCP atomic arrangements as well as polytetrahedral, icosahedral, and decahedral structures.

The first step in pursuing the polyhedra study of TGCs and PMCs is to realize that in FCC structures, tetrahedral (T-units) and octahedral units (O-units) hold one type of M–M bonds (Figure S1). In such a manner, one tetrahedron holds a T _d symmetry (24 operations of symmetry) and one octahedron displays an O _h symmetry (48 operations of symmetry). However, in both HCP and BCC lattices, there are 2 types of M–M bonds (Figures S2 and S3). Therefore, the tetrahedron found in HCP structures displays C _3v symmetry (6 operations) and the octahedron comprising HCP structures holds a D _3d point group (12 operations). In BCC lattices (see Figure ), the tetrahedron and octahedron reduce their symmetry (8 and 16 operations), respectively. This geometrical fact underpins our rationale for considering “distorted” polyhedra rather than exclusively perfect ones.

1.

Symmetry of tetrahedron and octahedron building blocks of FCC, HCP, and BCC lattices. In bulk FCC, polyhedral units have the same bond length values. In HCP and BCC lattices, the two sets of bonds are displayed with two different colors. In BCC and HCP lattices, polyhedral units hold less symmetric point groups.

The second step is to determine distinct assemblies of tetrahedra and octahedra to generate known and new building blocks. Inspection of the FCC lattice reveals an M₇ unit with a D _3d symmetry, consisting of a pair of tetrahedra sharing a common vertex. The rotation of one tetrahedron changes the M₇ unit toward a D _3h symmetry in the HCP lattice. In Figure , featured FCC and HCP lattices reveal other constituting polyhedral units. Indeed, the distribution of tetrahedra centers forms a cubic arrangement, while the distribution of octahedra centers complies with one FCC structure. Unlike this, the HCP structure shows a hexagonal arrangement of both tetrahedra and octahedra centers (see the tetrahedra center distribution in Figure ). It is important to mention that one M₆ unit (D _2h ), composed of a pair of tetrahedra sharing one edge, is a building block exclusively found in FCC structures. An M₅ unit (D _3h ) consisting of two face-sharing tetrahedra can be found in the HCP lattices. Interestingly, the HCP lattice complies with the stacked octahedra that are sharing one face, and two stacked octahedra produce one M₉ unit (D _3h ). Conversely, M₉ units may also participate in the formation of decahedral structures. Another pair of tetrahedra sharing one vertex (M₁₁ unit with D _4h ) is present only in the FCC lattice, but M₈ and M₁₀ units are featured by both FCC and HCP arrangements.

2.

Building blocks of FCC and HCP lattices. In the FCC arrangement (upper panel), tetrahedra of the M₇ unit are rotated featuring a D _3d symmetry. For HCP arrangement (lower panel), the M₇ unit displays a D _3h symmetry (unrotated tetrahedra). Moreover, the M₆ (D _2h ) unit is exclusively part of FCC, while the M₅ (D _3h ) unit is part of the HCP structure. The M₉ unit produced by the stacking of two octahedra is found in HCP structures, while two octahedra aligned along one axis and sharing one vertex (M₁₁ unit) are found in FCC structures. Red faces are used to define tetrahedra, while green ones correspond to octahedra. A, B, and C stand for atomic planes.

3.

Distribution of tetrahedra and octahedra centers in FCC and HCP lattices. (a) In FCC structures, tetrahedra (centers in gray) are distributed in a cubic arrangement, but octahedra centers (brown color) are distributed following one FCC arrangement. (b) For HCP structures, tetrahedra and octahedra centers form hexagonal lattices.

It can be concluded that FCC features M₄ (T _d -tetrahedron), M₆(O _h -octahedron and D _2h ), M₇(D _3d ), M₈(D _3d ), M₁₀(D _2h ), and M₁₁(D _4h ) units, while HCP structures might be decomposed into M₄ (tetrahedron with a C _3v symmetry), M₅(D _3h ), M₆ (octahedron with a D _3d symmetry), M₇(D _3h ), M₈(C _2h ), M₉(D _3h ), and M₁₀(C _2h ) units.

Regarding BCC lattices, each tetrahedron is distorted, with six bonds arranged as four short and two longer bonds featuring a D _2d symmetry (8 operations of symmetry). The same occurs with the octahedron displaying 5 longer bonds, with reduction of its symmetry toward D _4h (16 operations of symmetry). The distortion is evidenced by triangular faces with angles ranging from 55 to 70°. Another building block in the BCC lattices is M₆(C _2h ) comprised by 2 tetrahedra sharing one edge; M₈(C _2h ) constructed by assembly of 1 octahedron and 2 tetrahedra; M₁₀(D _2h ) corresponding with 2 octahedra sharing one edge; and 2 isomers of M₁₁ (2 octahedra sharing one vertex) with D _4h and D _2h symmetries, respectively. In Figure , building blocks found in BCC lattices are displayed.

4.

Building blocks of one 2 × 2 × 1 supercell of a BCC structure. There are two types of edges that differ by 86.6%. In the visualization, longer bonds are represented in cyan color. In BCC lattices, missing polyhedral units are M₅, M₇, and M₉.

The strong distortion of one octahedral M₆ unit in BCC lattices is attested by its decomposition into 4 distorted tetrahedra. The polyhedra decomposition of one FCC cuboctahedron is of 8 tetrahedra, but it can convert to a BCC-based cuboctahedron by applying one perpendicular force to a pair of its square faces. Therefore, starting from one FCC cuboctahedron (O _h ), it will convert to a BCC structure with a D _4h symmetry. In Figure , a polyhedral decomposition of the FCC/BCC cuboctahedron is shown.

5.

One M₁₃ FCC cuboctahedron (shown on the left side) might convert to the BCC M₁₃ structure by compression along a C ₄ axis. Their edges differ by 86.6%. The FCC cuboctahedron can be decomposed into 8 tetrahedra, while the distorted BCC cuboctahedron has 4 octahedra. T and O stand for tetrahedron and octahedron, respectively.

After revealing all polyhedral units of periodic FCC, BCC, and HCP lattices, we can continue toward PMCs study. How can the presence of icosahedral and decahedral structures in the field of TGCs be explained? At first glance, one might assume that they originate from a different “seed” structure. However, they can also be interpreted as distinctive assemblies of polyhedral units. To address this, it is necessary to propose an algorithm capable of generating a tetrahedron or an octahedron from a single triangle, considering that FCC polyhedral units are composed of triangular faces. This algorithm, detailed in the Supporting Information (Figures S4 and S5), enables the growth of a structural “seed” through the successive insertion of tetrahedral and octahedral units.

Although bulk gold exhibits an FCC atomic arrangement, it was not expected to retain this structure upon forming clusters. As a result, alternative arrangements, such as BCC and HCP, were also considered. In this work, it is demonstrated that tetrahedra and octahedra form the inner cores of PMCs, just as they do in FCC lattices. In icosahedral structures, the inner core corresponds with one M₁₃-centered icosahedron built by 20 distorted tetrahedra (see Figure S6). But when the inner core is one icosahedral M₁₂ shell (hollow icosahedron), it can be seen as the inner shell of a hollow M₄₂ cluster (comprised by 20 face-sharing octahedra). The inner M₁₃ icosahedron can be grown through the subsequent assembly of tetrahedra or octahedra (T- or O-units). In the first case, 20 tetrahedra capping the triangular faces of an M₁₃ icosahedron yield an M₃₃ cluster. Over their pentagonal faces, 60 tetrahedra can be added, yielding one M₄₅ cluster that may convert to the M₁₂₉ cluster after octahedra assembly. Its outer M₈₄ shell has 6 rectangular faces. In Figure , the steps to generate icosahedral structures starting from the center icosahedron are shown.

6.

Growth of icosahedral structures by insertion of polyhedral units. The inner core is an M₁₃ unit, and by subsequently capping it with tetrahedra, M₃₃ and M₄₅ clusters are obtained. The octahedra growth of the M₄₅ cluster yields an M₁₂₉ structure containing 6 rectangular faces (defects). Metal atoms are colored to facilitate the visualization of the growth pathway. T and O stand for tetrahedron and octahedron, respectively.

The M₄₂ cluster is obtained by an assembly of 20 face-sharing octahedra. Its outer M₃₀ shell is arranged as an icosidodecahedron, and it can be further grown by the addition of 20 octahedra, yielding an M₁₀₂ cluster. The outer shell of the M₁₀₂ cluster is one rhombicosidodecahedron comprised by 60 atoms, and the subsequent addition of 12 atoms along the C ₅ axis produces the M₁₁₄ cluster. Other related sizes are M₁₂₆, M₁₅₆, M₂₀₆, and M₂₃₆. In Figure , the inner M₁₂ cluster is grown by adding octahedra units.

7.

Growth of the icosahedral M₁₁₄ cluster from an inner M₁₂ shell. It starts with one inner M₁₂ core and by subsequent assembly with octahedra (O-units). The M₁₀₂ cluster is generated, and the addition of 12 Au atoms yields the M₁₁₄ cluster, which is composed of 40 octahedra and 180 tetrahedra.

On the other hand, the inner core of decahedral structures is one decahedral M₁₃ cluster (Inos decahedron holding a C _5v symmetry) that can be obtained from the assembly of 2 pentagonal bipyramids (M₇ units with a D _5h symmetry) sharing one vertex. This decahedral cluster has triangular and square faces, and it represents an important atomic arrangement, because it forms the central part of the Au₂₄₆(SR)₈₀ cluster (stacking of Inos decahedra). , The Inos decahedron can grow through the assembly of tetrahedra or octahedra (Figure ). In the first case, the bi-icosahedral M₂₅ cluster can be generated, and in the second case, one M₃₈ cluster is obtained.

8.

Generation of decahedral structures from the Inos decahedron. The inner M₁₃ core holds a C _5v symmetry, and it is composed of 2 pentagonal bipyramids. Further capping with tetrahedra (T-units) yields the bi-icosahedral M₂₅ cluster. The M₃₈ cluster corresponds with the assembly of octahedral units (O-units) to the Inos decahedron. Polyhedra decomposition is given, and green color stands for octahedral units, while red color corresponds to tetrahedral units.

Another distinctive structure is the decahedral M₄₉ cluster, being the inner core of the Au₁₀₂(SR)₄₄ cluster. It has one Inos decahedron (10 tetrahedra) and one pentagonal bipyramid (5 tetrahedra) along one C ₅ axis. The upper and lower regions of the M₄₉ cluster contain 15 tetrahedra (adding 30 tetrahedra). Between pentagonal bipyramids, there are 2 rings of 10-octahedra (20 extra octahedra). The overall structure of the M₄₉ cluster can be decomposed into 20 octahedra and 45 tetrahedra (Figure ). It can be concluded that stacking pentagonal bipyramids following the geometry of an Inos decahedron results in the formation of decahedral structures.

9.

Polyhedra decomposition of the decahedral M₄₉ structure. (a) Three pentagonal bipyramids are stacked along a C ₅ axis. (a–c) The upper panel displays top views and (d–f) the lower panel displays side views of the structure. (b,e) The middle column displays distribution of 20 octahedra (b) and 45 tetrahedra (e) centers where blue pentagons indicate the presence of pentagonal bipyramids. T and O stand for tetrahedron and octahedron, respectively.

A second way to grow decahedral structures is by starting from a single pentagonal bipyramid (M₇ with a D _5h symmetry) and later assembly of octahedra. The M₇ unit is constructed from five distorted tetrahedra, and further growth through the addition of octahedra yields the M₂₇ cluster. Further octahedra growth yields the M₇₉ cluster, which is the core of the Au₁₀₂(SR)₄₄ cluster. In Figure , the formation of the M₇₉ structure is shown.

10.

Growth of the decahedral M₇₉ structure by insertion of octahedra from one pentagonal bipyramid. Growing an M₇ unit with octahedra (O-units) yields M₂₇ and M₇₉ clusters. Polyhedra decomposition is given. Green represents triangular faces of octahedra, and red faces correspond to tetrahedra.

In terms of constituting building blocks, the FCC atomic arrangement and icosahedral structures are not very different, as can be seen in Figure . Both the icosahedron and cuboctahedron (related FCC structure shown in Figure S7) share a common M₇ unit (D _3d ), and the final anatomy depends on the followed growth pathway (octahedra or tetrahedra growth). The inner M₇ core assembled to octahedra yields one M₂₅ structure that can be trimmed to a cuboctahedron. Conversely, by capping with tetrahedra, the M₇ unit generates the M₁₃ icosahedral structure (after the elimination of 2 atoms located along one C ₃ symmetry axis).

11.

M₇ unit (D _3d ) as a common building block of FCC and icosahedral structures. The assembly of tetrahedra (T-units) to an M₇ unit gives an icosahedral structure, while cuboctahedron implies octahedra (O-units) growth. In both cases, it is necessary to eliminate atoms to obtain 13-atom structures.

It can be concluded that the M₇ building block featuring one D _3d symmetry is the main seed to build FCC and icosahedral structures.

How does the growth pathway influence the structural anatomy of the resulting FCC clusters? FCC structures are grown from an inner tetrahedron or octahedron and subsequent assembly of polyhedral blocks. The order and type of polyhedra determine the final anatomy of generated structures, as will be shown herein. FCC structures can be obtained through the alternating assembly of tetrahedra and octahedra. In Figure , the initial seed is one octahedron that after capping with tetrahedra yields one conventional M₁₄ FCC cell, and later octahedra growth of the M₁₄ cluster generates a truncated octahedral M₃₈ cluster. By following this alternating growth, it is possible to obtain related truncated octahedral structures such as M₆₂ and M₉₂ clusters.

12.

Growth of FCC structures by alternating insertion of polyhedral units. Starting from one octahedron and by capping of its triangular faces, one M₁₄ cluster is obtained (it resembles one conventional FCC cell). Next step is to grow with octahedra to obtain a truncated octahedral M₃₈ cluster. Other related sizes are shown.

However, starting with one tetrahedron and continuing the growth with octahedra yield truncated tetrahedral clusters (FCC-related structure). These structures have 4 truncated triangular corners. The size of the corners depends on the number of divisions of their edges (frequency). A frequency of 1 corresponds to the M₁₆ cluster (see Figure S8 for a polyhedra decomposition of the M₁₆ cluster); a frequency of 2 to the M₄₀ cluster; and a frequency of 3 to the M₈₀ cluster. See Figure for a visualization of truncated tetrahedral clusters obtained from one inner M₄ unit.

13.

Truncated tetrahedral clusters growth by insertion of octahedral units (O-units). They start growing from one tetrahedron and by further assembly of octahedra. Polyhedral decomposition of each generated cluster is shown.

In the case that the inner tetrahedron is grown exclusively by tetrahedra, it yields structures with overlapping atoms in such a manner that elimination of atoms is required. The cause of overlapping atoms is the inability of tetrahedra to fill the space, inducing distortions into the metal framework. In Figure , one tetrahedron is capped with tetrahedra, resulting in an M₈ cluster. The following size (M₁₄ cluster) was obtained from an intermediate M₂₀ cluster holding overlapping atoms (see Figure S9). The same fate befalls the M₂₆ and M₅₄ clusters. Interestingly, this type of growth generates structures known as polytetrahedral clusters. Previously, it was said that due to induced distortion, polyhedral clusters are generally limited to a maximum of about 70 atoms.

14.

Polytetrahedral cluster growth via sequential tetrahedral units (T-units) insertion. Tetrahedra assembly is not capable of completely filling space, and it induces distortion into the atomic arrangement. The lower panel displays the distribution of tetrahedra centers where the presence of dodecahedra reveals that interpenetrated icosahedra can be found in polytetrahedral clusters. T stands for tetrahedron.

One octahedron can grow by later octahedra assembly, but overlapping atoms appear and the M₁₈ cluster is obtained after trimming one M₃₀ cluster (see Figure ). This M₁₈ cluster cannot be grown through an octahedra assembly because bond lengths reduce approximately 55.6%.

15.

Assembly of octahedra to one inner octahedron is not possible because the structure has rapidly overlapping atoms (strong distortions). The trimming of the M₃₀ cluster yields one M₁₆ cluster that can be capped by tetrahedra and produce the M₂₆ cluster (it was found previously in Figure ). Further growth of the displayed M₁₈ cluster through octahedra is not possible because generated atoms are very close.

It is important to mention that once a size is obtained, following growth type (through tetrahedra or octahedra) will help to reduce the distortion of the atomic framework. For example, one truncated tetrahedral M₁₆ cluster can generate either an M₃₂ or an M₄₀ cluster, depending on the growth pathway followed (see Figure ).

16.

Truncated tetrahedral M₁₆ cluster growth by insertion of octahedra/tetrahedra. The polyhedral decomposition of generated M₃₂ and M₄₀ clusters is provided.

Tetrahedra centers of generated clusters can describe rings (Figure ). Starting from one octahedron and capping with tetrahedra yield one M₁₄ (FCC conventional cell) cluster. Further capping with tetrahedra results in an M₂₆ cluster. This cluster holds its tetrahedra centers forming rings. The M₂₆ cluster can increase its size to the M₅₆ cluster based on octahedra growth. This type of growth explains the reported structure for the [Au₂₄L₆Cl₄]²⁺ cluster (with L denoting 2,3-bis­(diphenylphosphino)­butane).

17.

FCC cell (inner octahedron) growth by insertion of tetrahedra yields tetrahedra centers describing rings. The lower panel provides the distribution of tetrahedra centers. Number of polyhedral units is provided.

Once the growth types were discussed, we continued with the polyhedral decomposition of distinct clusters. The existence of clusters with FCC, BCC, and HCP atomic arrangements underscores the critical influence of protecting ligands on their structural formation. Our polyhedra analysis reveals FCC clusters (located in the 1–3 nm range): Au₂₇₉(SPh-tBu)₈₄. , Au₁₈₈(SCH₃)₆₀, [Net₃H]₂[Au₁₁₀S₂(p-CF₃C₆H₄C₂)₄₈], Au₁₀₈S₂₄(PPh₃)₁₆, Au₉₂(SR)₄₄, Au₇₀S₂₀(PPh₃)₁₂, Au₄₃(CHT)₂₅, Au₄₂(CHT)₂₆, Au₄₀(o-MBT)₂₄, Au₃₈S₂(C₁₀H₁₅)₂₀, Au₃₆(4-EBT)₂₄, Au₃₆(TBBT)₂₄, Au₃₆(SC₅H₉)₂₄, Au₃₆(SPh)₂₄, Au₂₉(SAdm)₁₉, Au₂₈(TBBT)₂₀, Au₂₄(SAdm)₁₆, Au₂₄(SCH₂Ph-tBu)₂₀, [Au₂₄(C₂Ph)₁₄(PPh₃)₄]­(SbF₆)₂, Au₂₄(SeC₆H₅)₂₀, Au₂₃(SC₆H₁₁)₁₆ ^–, Au₂₃(CCBu ^t )₁₅-iso1, Au₂₀(TBBT)₁₆, [Au₁₈(dppm)₆ Cl₄]⁴⁺.

Moreover, a set of FCC-based structures (see Figure ) protected by 4-tert-butylbenzenethiolate (TBBT ligand) were reported previously, and herein, the polyhedral analysis supports this classification. The set includes Au₂₈(TBBT)₂₀, Au₃₆(TBBT)₂₄, Au₅₂(TBBT)₃₂, and Au₇₆(TBBT)₄₄ clusters. ,

18.

FCC clusters are grown along a preferential direction through the addition of octahedra. It is evident that the increasing number of octahedra elongates FCC clusters. R stands for TBBT or 4-tert-butylbenzenethiolate. Green planes correspond to triangular faces of octahedra. The octahedra centers are displayed in brown color. Number of polyhedral units is provided (O and T letters).

One building block of HCP structures is the M₅ unit with a D _3h symmetry, and it can be grown through assembly of tetrahedra. Figure shows the addition of tetrahedra to one trigonal bipyramid, yielding M₁₁ and M₂₃ clusters. The capping with tetrahedra generates overlapped icosahedra, as attested by the distribution of tetrahedra centers (gray framework in Figure ). This can be considered as a similar growth when a tetrahedron is used as an initial seed. Worthy of mentioning is that alternating growth of the M₅ (D _3h ) unit (through octahedra and tetrahedra) yields HCP atomic arrangement.

19.

Addition of tetrahedra (T-units) enables the growth of the M₅ unit. The capping of its triangular faces yields M₁₁ and M₂₃ clusters. Red planes correspond to triangular faces of tetrahedra, while their centers are provided in gray color. T stands for tetrahedron.

When the polyhedra approach was applied to certain TGCs clusters, they were revealed as HCP: for example, [Au₁₀Cl₃(PCy₂Ph)₆]⁺ having Au₇ and Au₅ units with a D _3h symmetry, Au₂₃(CCBu^t)₁₅-iso2, Au₁₈(SC₆H₁₁)₁₄, , Au₃₀(s-Adm)₁₈. Despite the report of BCC structures in TGCs, the polyhedra analysis does not support this type of arrangement for the Au₃₈S₂(C₁₀H₁₅)₂₀ cluster.

There are TGCs structures displaying both regions with FCC and decahedral atomic arrangements. For example, Au₁₄₆(p-MBA)₅₇ and Au₁₉₁(SPh-tBu)₆₆ clusters feature a mixture of pentagonal bipyramids and FCC structures. In Figure , inner cores of Au₁₀₂ and Au₂₄₆ clusters are given. The distribution of tetrahedral centers enables a clear distinction between FCC and decahedral structures. In such a manner, FCC arrangements depict tetrahedra centers forming cubes, while the presence of pentagonal rings (shown in blue color) indicates that pentagonal bipyramids form the structure.

20.

Large sizes of TGCs. The upper panel shows inner cores and their decomposition into tetrahedra (red faces) and octahedra (green faces). The lower panel shows center distribution of tetrahedra. The FCC arrangement is attested by the presence of cubes, while decahedral structures have pentagonal bipyramids (blue color). It is evident that Au₁₉₁ and Au₁₄₆ clusters have a mixture of decahedral and FCC atomic arrangements. Number of polyhedral units is provided (O and T letters).

The following section explains the anatomy of synthesized TGCs in terms of polyhedral assembly. Cluster growth can occur by capping triangular faces with either one atom, completing a tetrahedron, or three atoms, forming an octahedron. For example, the early detected size with an icosahedral inner core was the Au₂₅(SR)₁₈ cluster. − It can be seen as comprised of one Au₁₃ inner core and protected by 6 dimer motifs (12 gold adatoms). The addition of a single metal atom to each of the 20 triangular faces of the Au₁₃ structure yields an Au₃₃ cluster made up of 40 tetrahedra. However, to accommodate sulfur atoms, it may be necessary to remove 8 metal atoms arranged in a cubic pattern, leaving only 25 metal atoms. The distribution and spacing between metal atoms may allow the attachment of 18 sulfur atoms, which form bridges between pairs of Au atoms, resulting in a Au₂₅(SR)₁₈ cluster. It means that ligands (size, bulkiness, chemical composition, etc.) are the driving force to yield this stoichiometry. Figure shows a schematic representation of the described capping and elimination stages.

21.

Explanation of the growth of icosahedral structures by means of tetrahedra assembly. Starting from one icosahedral Au₁₃ cluster and further capping of its 20 triangular faces, it is possible to obtain the Au₃₃ cluster. Subsequent elimination of 8 atoms located in the vertices of one cube yields the Au₂₅ cluster. The incorporation of 3 S atoms over each of 6 staple motifs oriented on a cubic symmetry generates the Au₂₅(SR)₁₈ cluster. The number of tetrahedra is indicated with a number followed by the letter T. Red planes indicate tetrahedra faces.

The explanation of the structure of the Ag₄₄(SR)₃₀ cluster , can be elaborated from the structure of the Au₂₅(SR)₁₈ cluster (Figure ). The inner shell of Ag₄₄(SR)₃₀ corresponds to an Ag₃₂ shell with the central Ag atom missing. The capping of its 20 triangular faces yields the Ag₄₄ cluster, and a pair of Ag atoms are located over each face of one Ag₈ cube (shown in yellow color). By pulling out 12 Ag atoms, the distinctive bonding of Ag₄₄ is obtained. Five sulfur atoms can bridge each of the six prominent Ag–Ag pairs, resulting in a total of 30 sulfur atoms.

22.

Explanation of the Ag₄₄(SR)₃₀ cluster. (a) Starting from one icosahedral Ag₃₂ cluster and further capping of its 12 pentagonal faces, it is possible to obtain the Ag₄₄ cluster. (b) Pulling out 12 atoms following a cubic arrangement in the Ag₃₂ shell (c) gives the anatomy of the Ag₄₄ cluster. (d) The addition of 5 S atoms over each of 6 staple motifs yields the Ag₄₄(SR)₃₀ cluster. T stands for tetrahedron. Red planes indicate tetrahedra faces.

What is the process to generate two icosahedra sharing one, two, or three atoms? By capping an Au₁₃ Inos decahedron with tetrahedra, it is possible to obtain one bi-icosahedral Au₂₅ cluster. See Figure for a schematic representation of the growth pathway.

23.

Generation of one bi-icosahedral Au₂₅ cluster. The first step is the capping of the triangular faces located at the endings of the Au₁₃ cluster (T-units addition), and the second step is the capping of the pentagonal faces of the Au₂₃ cluster.

An Au₂₄ cluster, consisting of two icosahedra sharing an edge, can be obtained by capping the triangular faces of an Au₆ unit (D _2h symmetry) formed by a pair of tetrahedra sharing an edge. The radial growth of the Au₆ unit will produce one Au₄₀ cluster, but after elimination of certain atoms, the anatomy of the Au₂₄ cluster is obtained. In Figure , the growth of a bi-icosahedral Au₂₄ cluster is depicted.

24.

Generation of a pair of icosahedral Au₁₃ clusters sharing an edge. (a) Radial growth by capping one inner Au₆ cluster yields the Au₄₀ cluster, but further trimming or a preferential growth along one direction results in the Au₂₄ cluster (b,c). In red are displayed a pair of tetrahedra sharing one edge (Au₆ unit holding a D _2h symmetry). (d) Centers of tetrahedra, comprising each icosahedral units, are forming dodecahedral units.

An Au₂₃ cluster (Figure ), formed by two icosahedra sharing a triangular face, can be generated by using an Au₁₁ cluster (D _3h symmetry) assembled from four tetrahedra as the seed. This radial growth produces an Au₂₅ cluster that after elimination of 2 atoms yields the expected one.

25.

Generation of a pair of icosahedral Au₁₃ clusters sharing a triangular face. (a) One Au₁₁ cluster suffers radial growth by adding tetrahedra and yielding the Au₂₅ cluster. (b) Elimination of 2 Au atoms results in the Au₂₃ cluster. (c) Two views of the Au₂₃ cluster. (d) Centers of tetrahedra are forming dodecahedral units.

Mentioned Au₂₅, Au₂₄ and Au₂₃ clusters are of important sizes because they are part of other structures, for example, bi-icosahedral Au₃₈(SR)₂₄ cluster having an Au₂₃ inner core protected by 6 dimer motifs and 3 monomer motifs. − Other crystallized structures correspond to bi-icosahedral units, where a pair of icosahedra share one vertex ([Au₂₅(PPh₃)₁₀(SR)₅Cl₂]²⁺) or even form tri-icosahedral structures ([Au₃₇(PPh₃)₁₀(SR)₁₀Cl₂]⁺).

The icosahedral Au₁₄₄(SR)₆₀ cluster can be explained in terms of a combination of octahedral and tetrahedral units. It contains an inner Au₁₂ shell that after octahedra growth results in the Au₅₄ cluster. The Au₁₁₄ cluster is built by assembling octahedra over triangular faces of the outer shell of the Au₅₄ cluster (see Figure ). This growth adds 20 octahedra and 120 tetrahedra (located on 12 pentagonal faces). To complete the Au₁₄₄ cluster, it is necessary to add 30 metal adatoms over the square faces of the outer Au₆₀ shell (rhombicosidodecahedron), which implies the addition of 30 extra octahedra to the structure and of 30 monomer (−S–Au–S-) motifs. We can conclude that the Au₁₄₄(SR)₆₀ structure is composed of 70 octahedra and 180 tetrahedra. See Figure that displays the described radial growth.

26.

Radial growth of the icosahedral Au₁₄₄(SR)₆₀ cluster by assembling tetrahedral and octahedral units. (a) The initial Au₁₁₄ cluster (shown in Figure ) has one outer Au₆₀ shell featuring 30 square faces. (b) The addition of the external Au₃₀ shell (30 gold adatoms) completes 30 octahedral units when linked to these square faces. (c) Each gold adatom is linked to a pair of S atoms yielding 30 monomer motifs.

The growth of decahedral structures can be understood as the assembly of aligned pentagonal bipyramids (Au₇ units with a D _5h symmetry). The Au₁₀₂(SR)₄₄ cluster contains one Au₄₉ Marks decahedron (Figure ) composed of 20 octahedra and 45 tetrahedra. In the Au₇₉ cluster, 30 additional atoms are distributed as a pair of 5 and 10 Au rings over the Au₄₉ cluster to complete a total of 10 octahedra. The presence of 23 staple motifs protecting the Au₇₉ core completes the Au₁₀₂(SR)₄₄ cluster. Its equatorial section (Au₄₄ cluster) displays a circle of 10 octahedra distributed in 5 groups of Au₉ units (2 face-sharing octahedra) linked through their edges. The total number of constituting polyhedra is 20 octahedra and 95 tetrahedra. See Figures and for a detailed description of the structure.

27.

Au₇₉ inner core of the decahedral Au₁₀₂(SR)₄₄ cluster. (a) The left column features 2 views of the inner Au₇₉ core and its polyhedral decomposition (T and O letters). (b) Central column shows tetrahedra centers distribution where one blue pentagon indicates the presence of one pentagonal bipyramid. (c) The right column displays octahedra centers distribution. The Au₇₉ cluster contains 3 pentagonal bipyramids (blue color) along one C ₅ axis, which are surrounded by 5 octahedra located at both endings and 10 octahedra located in its middle region. Moreover, a pair of 5 bipyramids are located among octahedra circles. Red planes represent tetrahedra faces, while green ones represent octahedra faces.

28.

Central region of the decahedral Au₁₀₂(SR)₄₄ cluster with one pentagonal bipyramid surrounded by 5 tetrahedra and encircled by 5 Au₉ units linked through their edges. (b) Side views of the Au₄₄ cluster. The red arrow indicates the shared triangular face, while the blue arrow indicates one shared edge. Red planes represent tetrahedra faces, while green ones represent octahedra faces.

The decahedral Au₁₃₀(SR)₅₀ cluster , is grown by four Au₇ units located along a C ₅ axis. It contains, in the middle region, a Marks decahedron whose central part (Figure a) is composed of 5 octahedra sharing their triangular faces (Figure b). In addition, it contains 10 pentagonal bipyramids located at its endings. Moreover, 5 cuboctahedra encircle the column of Au₇ units (Figure e). In total, it holds 55 octahedra and 155 tetrahedra. It is easy to visualize the Au₁₃₀(SR)₅₀ cluster from the distribution of its centers of octahedra and tetrahedra (Figure b,e). In Figure , the distribution of polyhedra centers of the whole structure is provided.

29.

Central part of the decahedral Au₁₃₀(SR)₅₀ cluster. (a,b) It has 5 octahedra sharing triangular faces and encircled by 5 cuboctahedra (center shown in (d) and (e)). (c,f) Side views evidencing the distorted square faces of cuboctahedra (red faces). (b,e) Octahedra and cuboctahedra centers. (d) Polyhedra decomposition (T and O letters).

30.

Distribution of polyhedra constituting the decahedral structure of the Au₁₃₀(SR)₅₀ cluster. (a,b) Centers of octahedra (brown color) show that they are encircling a column of pentagonal bipyramids. (c) Triangular faces of pentagonal bipyramids are numerated in black color (black arrow), while cuboctahedra are numerated in light blue color (blue arrow). (d,e) Side views of centers of pentagonal bipyramids (blue and green), cuboctahedra (red), and tetrahedra (gray).

Interestingly, 20 gold adatoms of 25 monomer motifs protecting the Au₁₀₅ core complete all distorted octahedra.

The decahedral Au₁₈₇(SR)₆₈ cluster has 7 pentagonal bipyramids aligned along one C ₅ axis, and this column is encircled by octahedra. It comprises 6 circles of 5 octahedra and 5 circles of 10 octahedra. Moreover, they hold interpenetrated cuboctahedra. In total, there are 20 cuboctahedra around the central column of Au₇ units. The structure, in general, contains 80 octahedra and 185 tetrahedra. In Figure , the distribution of centers of tetrahedra and octahedra displayed by the Au₁₈₇(SR)₆₈ cluster is shown.

31.

Distribution of polyhedra constituting the decahedral structure of the Au₁₈₇(SR)₆₈ cluster. (a,b,d) The distribution of tetrahedra centers (gray color) indicates the presence of a column of 7 pentagonal bipyramids (blue pentagonal rings). (b) Cubes formed by tetrahedra centers correspond with cuboctahedra, and the structure displays rows of these. (e) The distribution of octahedra centers reveals octahedra rings surrounding the central column of pentagonal bipyramids. (c,f) Two views of the Au₁₈₇(SR)₆₈ cluster are shown with its polyhedral decomposition. Pentagonal bipyramids are featured in blue color, while centers of octahedra are displayed in brown color.

The polyhedra analysis carried out herein attests other icosahedral structures such as Au₄₀(SR)₂₄ and Au₁₃₃(SPh-tBu)₅₂.

Among decahedral clusters that can be characterized by polyhedral analysis, the following are found: Au₁₀₃S₂(S-Nap)₄₁, Au₁₀₂(p-MBA)₄₄, Au₁₀₈S₂₄(PPh₃)₁₆, Au₁₃₀(p-MBT)₅₀, , Au₁₃₈(2,4-DMBT)₄₈, Au₁₄₄(SCH₂Ph)₆₀, Au₁₈₇(SR)₆₈, and Au₂₄₆(p-MBT)_80. ,

Finally, it is important to explain the growth of one structure described as non-FCC in 2019: the Au₄₂(TBBT)₂₆ cluster, which has one Au₂₆ core composed of 20 tetrahedra and 4 octahedra. Certainly, the polyhedral arrangement does not correspond with the FCC structure. Instead, it can be seen as an octahedron surrounded by octahedra, and accordingly, it requires elimination of atoms due to overlapping atoms. Therefore, instead of 6 surrounding octahedra, the structure assembles 4 octahedra producing one Au₁₈ cluster (as shown in Figure ). The size is increased by tetrahedra growth adding 8 more atoms. The protecting ligands are distributed as 4 tetramer motifs, 4 monomer motifs, and 2 S atoms in bridge positions. In Figure , the growth of the Au₄₂(TBBT)₂₆ cluster is provided.

32.

Growth of the Au₄₂(SR)₂₆ cluster by means of polyhedral assembly. The inner octahedron is surrounded by 4 octahedra producing the Au₁₈ cluster. To increase the size of the Au₁₈ cluster, 8 atoms are added by insertion of tetrahedra. The addition of protecting ligands yields the total structure. Colors are used to facilitate visualization. Yellow atoms in the structure of the Au₁₈ cluster correspond with one Au₈ unit located in the middle part of the cluster. Atoms in cyan color correspond with a pair of caps located at the endings of the Au₂₆ cluster. Red atoms stand for S atoms located in bridge positions. Yellow and blue atoms are used to distinguish S atoms forming part of tetramer motifs. Octahedra faces are shown in green color.

4. Conclusions

In this work, the building blocks of periodic FCC, HCP, and BCC lattices were identified within the structures of the TGCs and MPC clusters. Various assemblies of tetrahedron Au₄ and octahedron Au₆ capable of forming FCC, HCP, BCC, polytetrahedral, decahedral, non-FCC, and icosahedral structures were discussed.

To summarize all the results obtained:

• aThe M₁₃ cuboctahedron can be decomposed into 8 tetrahedra in FCC structures. Conversely, an M₁₃ cuboctahedron in BCC arrangements is composed of 4 distorted octahedra. This change in the type of composing polyhedral units is attributed to the “distortion” of the BCC lattice.
• bCuboctahedra can be found encircling pentagonal bipyramids as observed in decahedral structures.
• cThe alternating growth of an inner tetrahedron or octahedron by tetrahedra and octahedra leads to the formation of FCC clusters.
• dPolytetrahedral clusters consist of a central tetrahedron surrounded by additional tetrahedra. Their growth leads to structures containing interpenetrating icosahedra. However, polytetrahedral clusters typically do not grow to large sizes due to overlapping atoms causing significant distortion of the metal framework.
• eNon-FCC clusters can be formed by octahedral growth from an inner octahedron. This growth mode has been described and linked to the structure of the Au₄₂(SR)₂₆ cluster.
• fThe Au₇ unit, possessing a D _3d symmetry, serves as a common seed for the growth of icosahedral and FCC structures, with the resulting anatomy dependent on the specific polyhedral assembly pathway.

It is important to mention that small units have been found in experimental structures in various MPCs. Therefore, an inner triangle has been reported in the carbene-protected Au₃ ¹⁺ cluster, forming the inner core of the Au₁₃(SCH₃)₁₁ cluster, and inside one alkynyl-protected Cu₅₃ cluster. FCC building blocks were also identified, and tetrahedron was contained in a phosphine-protected Au₄ ²⁺ cluster, silyl-protected Al₄, and a silyl-protected Ga₄ cluster. In a similar way, octahedron was reported in a phosphine-protected Au₆ cluster. Pentagonal bipyramid was reported in a phosphine-protected Au₇ ¹⁺ cluster. Interestingly, one M₇ unit with a D _3d symmetry has been reported in a protected Au₇ ^1– cluster.

On the other hand, one M₈ cluster, as shown in Figure , was reported in a monocationic hydride-centered Cu₈ cluster. Worthy of mention is the carbonyl-protected Pd₁₄₅ cluster, whose structure served as a model for the structure of the Au₁₄₄(SR)₆₀ cluster.

In this work, a new computational algorithm is implemented to successfully generate atomic positions by capping an inner core with tetrahedra or octahedra, enabling the simulation of different growth pathways that effectively explain the diverse structural anatomies observed in various experimentally crystallized thiolated gold clusters (TGCs) and protected metal clusters (PMCs). This study demonstrates that tetrahedra and octahedra are not only fundamental components of FCC lattices but also serve as building blocks in cluster structures. The key distinction lies in the specific way these units are assembled, and discussed growth pathways can be extended to the study of PMCs. Moreover, understanding their growth mechanisms is essential for optimizing synthesis conditions, as previously reported for FCC-type clusters. ,

Finally, it is mandatory to know and generate the atomic positions of TGCs/PMCs (molecular models) before their electronic properties can be calculated to gain deeper insight into their reactivity and potential chemical applications.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.5c05447.

• The algorithm to grow inner cores of MPCs by means of generation of tetrahedra and octahedra and polyhedra analysis of various MPCs (PDF)

A. Tlahuice-Flores was primarily responsible for computational data analysis and manuscript preparation.

The author declares no competing financial interest.