A New Polyhedra Approach to Explain the Structure and Evolution on

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A New Polyhedra Approach to Explain the Structure and Evolution on Size of Thiolated Gold Clusters Alfredo Tlahuice-Flores J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b02265 • Publication Date (Web): 12 Apr 2019 Downloaded from http://pubs.acs.org on April 12, 2019

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The Journal of Physical Chemistry

A New Polyhedra Approach to Explain the Structure and Evolution on Size of Thiolated Gold Clusters A. Tlahuice-Flores * *Universidad Autónoma de Nuevo León, CICFIM-Facultad de Ciencias FísicoMatemáticas, San Nicolás de los Garza, NL 66455, México.

ABSTRACT

Thiolated gold clusters (TGC) have received a lot of attention in the Nanoscience field in the last decade due to their potential applications that include catalysis, image diagnosis, photovoltaics, nanomedicine, etc. Their study has been fueled by the development of new synthesis, separation and analysis methods in the experimental side, and by new algorithms able to propose more reliable initial structures during theoretical predictions. Up to date, there are more than 70 calculated/synthesized structures, displaying a variety in both their shapes and their constituting inner cores. When hundreds of atoms are constituting TGC, a description of their structures is not easy and they request of a conceptual simplification. In this manuscript, is proposed a simplified description of 36 reported structures of TGC in terms of the assembly of

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elementary building blocks (tetrahedral or octahedral units). It is expected that the obtained trends can be exploited during the prediction of new structures.

1. INTRODUCTION Thiolated gold clusters (TGC), such as Au25(SR)18,1-4 Au38(SR)24,5-9 and Au102(SR)4410 clusters (solved by means of X-ray total structure determination), have played an important role in the understanding of the distinct bonding displayed by constituting gold and sulfur atoms. Theoretical calculations based on Density Functional Theory (DFT) reported that certain gold atoms located on the outermost region, can form large Au-Au bonds (Aurophilic bonds of circa 3.5 Å) and they attested the presence of certain S-Au-S units resembling “staples” motifs.11 TGC can be obtained by bottom-up methods where a metal salt is reduced by agents such as sodium borohydride, borane, tert-butylamine complex and citrate.

The

aggregation of the neutral gold atoms might be directed by the experimental ligands, and it depends on factors such as temperature, concentration, solvent, etc. Precisely during the growth of TGC, the presence of atoms or molecular species (building blocks) and their interactions with ligands might produce characteristic type of assemblies.12 Further studies of Face Centered Cubic or FCC-like cores pointed out the importance of the ligands, that preferentially are bulky thiolates (i.e. c-pentanethiolate, chexanethiolate,

adamantine

thiolate,

phenyl-thiolate,

t-butylphenylthiolate,

t-


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butylthiolate, and 3-mercaptobenzoic acid).13-27 Experimentalists have attributed the preference by certain type of structures to kinetic28, steric and thermodynamic aspects of the specific synthesis methods.29-32 The evolution on size of TGC is led by the stable Au15(SR)13 cluster,33 whose inner core represents the smallest building block (Au4 polyhedral block) known so far. In the case of the Au18(SR)14 cluster, its distorted Au9 inner core can be seen as constituted by two distorted octahedral Au6 (O-units) sharing a triangular face.34-36 The distortion produced by ligand effects is also present in the Au25(SR)18 cluster, where its symmetry is reduced (i.e. Ci).37-38 Recently, various FCC-like clusters have emerged, displaying FCC elongated cores along a preferred direction of growth.39-40 There exist attempts devoted to account or rationalize the structure of thiolated gold clusters. A more recent approach is the grand unified model or GUM,41 that describes the anatomy and evolution of thiolated gold clusters in terms of elementary blocks such as charged Au3, Au4 and Au13 units. Therefore, GUM approach is able to explain their inner cores by the aggregation of elementary building blocks, e.g., the electronic levels of Au62+ can be seen as a linear combination of electronic levels of two elementary Au3+1 blocks.41b An experimental study based on the in-source collision-induced dissociation (CID) of the Au144 and Au130 clusters revealed Au4(SR)6, Au3(SR)2, Au6(SR)4, or Au6(SR)6 units as


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components of these sizes,42 and supporting the existence of stable units (intermediate clusters) under experimental conditions. From a structural point of view, the separation of one structure into its polyhedral blocks might imply that bonds in the individual polyhedral blocks are shorter (compact polyhedral blocks) than bonds formed among fused building blocks. However, in a FCC structure, all bonds are equal and there is not a clear distinction among intra-polyhedral or inter-polyhedral bonds. In this manuscript, are considered geometrical factors as nearest-neighbors counting, bond lengths, angles, capping of polyhedral faces, determination of center of polyhedral blocks, etc. to elucidate the structure of thiolated gold clusters. A special attention is pay to the description of the assembly of Au4 and Au6 building blocks. It is remarkable that their assembly might generate more units, for example, five linked Au4 units produces a distorted pentagonal bipyramid (Au7 or decahedron-like), and 20 Au4 units might produce an icosahedron-like structure (Au13). However, the Au12 unit is not obtained by the assembly of Au4 and/or Au6 blocks requesting it as another elementary building block. Obviously, the above-mentioned tetrahedral blocks (T-units) are slightly distorted to assure the filling of the space left in the case of a perfect icosahedra (due to the presence of an angular mismatch). The implementation and application of the polyhedral model in TGC is not only restricted to the description of the inner gold core, because it is shown that even gold adatoms are constituting the distorted Au4, and Au6 blocks. This new approach offers a simplified description of a set of 36 key structures


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based on the arrangement (or assembly) of their polyhedral blocks. However, the analysis can be applied successfully to clusters containing tents or hundreds of gold atoms.

2. METHODOLOGY Prior to the study of TGC, is important to stablish why Au4 and Au6 polyhedral blocks are considered as vital to explain their structure. Precious metals, as gold and silver, form a FCC arrangement in the bulk that confers them with compactness and high thermal stability. Interestingly, each conventional FCC unit cell can be seen as constituted by one perfect octahedron and eight T-units capping its faces. In the bulk, the combination of both polyhedral blocks are totally filling the space, for example one FCC Au63 cluster can be described as a cubic arrangement (Figure 1a) of tetrahedra (Figure 1b), which are capping the octahedral blocks (Figure 1c).

Figure 1. A 2x2x2 FCC supercell containing 63 atoms is shown. (a) The centers of 64 Tunits describe a simple cubic arrangement. (b) They are linked by their edges, and (c) 14 complete O-units are sharing edges, but only eight are displayed to facilitate the view. Missing six O-units share one vertex located in the center of the supercell.


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To justify the use of the Au4 and Au6 units as building blocks, they were optimized, and characterized by a vibrational calculation (See supporting information). Au4+2 (2e), and Au6-2 (8e) clusters, have close shell electronic structure and large HOMO-LUMO gaps. A further analysis of one Au4 unit holding a 2+ charge (HL=2.60 eV) reveled it as formed by two cationic Au2 units with a calculated energetic barrier of circa 0.65 eV. Similarly, one Au6-2 unit (HL=1.56 eV) can be formed by two anionic Au2 and one neutral Au2 units, without an energetic barrier (HL=0.06 eV). In TGC, the presence of anionic thiolate ligands affects the charge distribution of gold atoms and one appropriated study requires DFT calculations. The last argument demonstrates that the geometrical structure and the electronic properties of TGC are related in a unique manner. On the other hand, the translational symmetry of the bulk is missed in the metal clusters (0-D). More importantly, the reduction on size must affect the assembly of Au4 and Au6 polyhedral blocks. One question arises: How polyhedral blocks are distorted from their bulk values (bond lengths and angles) when the size is reduced? The methodology to study characteristic TGC is outlined in this section. (1) The study of growth patterns for naked metal clusters, and (2) The consideration of the metal adatoms (outer gold atoms) when the study of TGC is done. To accomplish the first part of this study, it was necessary to implement a computational algorithm able to grow


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FCC-like structures, by capping an inner core constituted with Au4 and Au6 building blocks, preferentially. The second part included the implementation of an algorithm able to calculate a) the number of neighbors and to determine the pairs of linked gold atoms, b) the center position of each polyhedral block, and c) the calculation of the angles sustained by polyhedral faces. In the following are summarized distinct types of growth patterns along this study (Part I). a) An inner gold tetrahedron (Au4), surrounded by four octahedra (Au6 in Figure 2a). The obtained structures are comprised by 4, 16, 28, 44, 68, etc., and they have FCC {111} planes. b) One Au6 core covered with T-units (Figure 2b). It produces clusters constituted by 14, 38, 68, etc, atoms. Their shapes correspond with a truncated octahedron. However partial growth of the structure can produce for example clusters constituted by 7, 10, 16, 19, and 25 atoms. c) One pair of Au4 units sharing an edge produce one Au6 core. Its further covering with octahedra (Figure 2c), might produce clusters comprised by 10, 18, 24 atoms and so on.


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d) The inner cuboctahedron formed by 12 first neighbors in a FCC structure and surrounded by six more atoms, might result in six Au6 sharing edges or vertexes (Figure 2d). At the same time, the 19-atoms structure can be used to build the next clusters (43, 55, 79, etc). e) The combination of an inner icosahedron (Au12 with no center gold atom) and surrounding O-units is found as an important arrangement, because the new generated positions avoid the angular misfit due to separated atomic positions. For example, Au12 surrounded by octahedral produced 60 positions with pairs of atoms separated by circa 0.12 Å (Figure 2e). In this case, each pair of neighboring gold atoms are substituted with one position located in the middle part (obtaining the Au30 cluster), this distortion obtained in the early stage of growth may result in large icosahedral structures with “twinned planes”.43 Worthy of note is that those combinations of seeds and surrounding polyhedral blocks producing overlapping positions (duplicate positions) were discarded, because they will avoid the further growth of regular structures. For example, Au4 covered by Tunits only produces Au8 cluster without overlapping positions, and further growth produces “distorted shapes”. The same trend is obtained in the case of an icosahedron (Au12) covered by gold tetrahedra (larger size is the Au32 cluster); Au12 covered by octahedral blocks (Au30 was an Au60 with very close Au atoms). Another wrong


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combination is an octahedron covered by O-units. In Figure 2 are displayed some of the preferred growth patterns for naked metal clusters.

Figure 2. Growth patterns of naked gold clusters are displayed. (a) One gold tetrahedron covered by four octahedral blocks. (b) One octahedron capped with eight tetrahedral blocks. (c) One Au6 formed by two fused tetrahedra by an edge can be covered by octahedral blocks. (d) One Au12 cuboctahedron can be seen as comprised by six octahedral blocks sharing edges or vertexes. (e) One icosahedron surrounded by 20 octahedral blocks depicts overlapping positions. It is important to mention that Cartesian Coordinates of the X-ray solved structures and those obtained by DFT optimizations are used along this study. For the counting of polyhedral blocks in TGC, all gold atoms were considered.

3. RESULTS AND DISCUSSION The growth patterns depicted in Figure 2 were considered to generate various TGC. In Table 1 are shown clusters built by combination of polyhedral blocks as outlined in the previous section. A more complete table is delivered in the supporting information. Table 1. Polyhedra approach and calculated combinations of building blocks in the naked gold clusters.


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Au13@Au6

2 Au4@Au6

1 6 10 18

12 19

24

43 55

40/ 48 56 68

79 87

76 84 98

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Partial covered Au6@Au4

Au4@Au8

Au6@Au4

nxnxn FCC

Icosahedral shells

3 7 10 16 19

4

6 14

14

13

25 (19 tet) 26 35 47 50 62 74 77 83 98

28

16

38 44 55 68

68

63

80 92 104

114 /118 135

134

116 122 137

116

A further inspection of the provided Table 1 demonstrates that distinct initial seeds can produce same sizes (structural isomers). A key example is the 68 atoms cluster, with three different initial seeds.15 The analysis of polyhedral blocks distribution in a set of key TGC is shown below.

I.

Structure of the Au30(SR)18 cluster described in terms of polyhedral blocks

The Au30(SR)18 cluster, was considered as a hexagonal close packed (HCP).44 It holds one central elongated Au12 core constituted by three octahedral blocks sharing a


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triangular face (Figure 3a), and surrounded by six triangular bipyramids or Au5 units (Figure 3b). The HCP structure (Figure 3c) is easily explained by aligning the octahedra along a C3 axis in such manner that three Au5 units are enclosing to the upper octahedron. The ABA arrangement can be explained by considering one Au5 unit and one octahedron; its upper apex containing one Au atom (forming a triangular face of the octahedron) might locate on the A plane, the middle part (containing three Au atoms) corresponds with a B plane and the lower apex is again located on one A plane.

Figure 3. The anatomy of the Au30(SR)18 cluster described as an assembly of (a) three Au6 units sharing a triangular face forming one column and (b) 12 Au4 units forming six triangular Au5 bipyramids. (c) The Au-S framework holds one Au18 core and six dimer motifs with S- adamantine as the experimental ligand (not included here). Calculated angles forming the middle square-like planes of O-units are included in the range from 40.58 to 54.00 degrees. Similarly, T-units hold triangular faces with angles in the range from 52.35 to 68.54 degrees. It means that the polyhedral blocks are highly distorted with respect to the 45.00 and 60.00 degrees expected for regular faces of octahedral and tetrahedral units, respectively. The same trend is obtained in the bond lengths displayed by the cluster attesting a major dispersion (see SI section).


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2. Structure of the Au36(SR)24 cluster The Au36(SR)24 cluster is one of the first TGC displaying a FCC structure.45 It has one central Au4 unit covered with four Au6 units (same pattern displayed in Figure 2a) and five more external T-units are found. Tetrahedral blocks are sharing their edges or apexes. The distribution of center of Au6 blocks is displayed in Figure 4a. A further analysis shows that the structure contains two incomplete O-units (including gold adatoms). Displayed angles of Au6 units are included in the range from 40.27 to 54.96 degrees. Is important to note that in Figure 4b, the presence of two O-units sharing an edge is consistent with a FCC structure. Moreover, there is a central Au4 unit (Figure 4c) in the structure of the Au36(SR)24 cluster (Figure 4d).

Figure 4. Structure of the Au36(SR)24 cluster. (a) It can be described as an assembly of four O-units surrounding one center Au4 tetrahedron. Their distribution is shown in blue color. (b) The octahedral blocks are displayed in green color and they are sharing an edge. (c) Five more external T-units are sharing a vertex or edge and are surrounding the octahedral block. (d) The full Au-S framework holds seven dimer motifs being S atoms forming bridges among gold atoms. It is evident the presence of two incomplete octahedral blocks in the outermost region.


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3. Structure of the Au38S2(SR)20 cluster The experimental structure of the Au38S2(SR)20 cluster was first described as a Body Centered Cubic (BCC).46 However, this idea needs to be abandoned based on the following considerations. First of all, the Au38S2(SR)20 cluster contains two Au4 blocks forming one Au6 cluster (Figure 2c) and one Au11 cluster (Figure 1c) formed by one pair of O-units sharing a vertex. The presence of those Au6 and Au11 units is consistent with a FCC structure, but in this case, the assembly or the distribution of surrounding octaedral and T-units are not filling the space correctly and the structure is not compact and it should be no longer considered as FCC. Instead, the Au38S2(SR)20 cluster can be considered as a frustrated FCC structure. In the Figure 5, is displayed the tetrahedral blocks arrangement (Figure 5a, Figure 5b), and the octahedral blocks (Figure 5c) in the complete structure of the Au38S2(SR)20 cluster (Figure 5d). The shown arrows indicate one vertex shared among two linked Ounits, and above and under that vertex are located two square planes forming the central part of each octahedron. In FCC structures, the presence of two O-units sharing one corner is evident from figure 1. Au-Au distances along the edges of square planes (2.8 Å) of both octahedra, and the distance among parallel square planes are different (3.3 Å) in figure 5c, in such manner that arrows in that figure indicates a pseudo-BCC structure. In total seven tetrahedral and only two octahedral blocks are forming the


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Au38S2(SR)20 structure. The structure is highly distorted and its square planes have angles included in the range from 40.15 to 54.97 degrees.

Figure 5. Structure of the Au38S2(SR)20 cluster. (a) The distribution of the center of tetrahedral blocks are shown, and (b) they are linked by an edge (forming an inner Au6) or by a vertex. (c) There are two complete octahedra sharing a vertex as in the case of a FCC arrangement, and two incomplete octahedra are located at the outer part. (d) The Au-S framework displays S atoms forming bridges in its middle region (Indicated with arrows).

4. Structure of the Au68(SR)32 cluster The Au68(SR)32 cluster contains a 68 gold atoms core.15 A further inspection of Table 1, suggests three types of growth for this specific size (Figure 6). Experimentally, the isomer with a major number of {111} planes was found as part of the Au68(SR)32 cluster and their presence is sustained by their lowest surface energy. In figure 7, the Au68 cluster evolved from one Au4 seed (Figure 2a), and it can be seen as constituted by 69 tetrahedral and 29 surrounding octahedral units.


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Figure 6. Three calculated isomers of the Au68 cluster seen as constituted by tetrahedral units. (b) The middle one contains a major number of {111} planes, and it was determined as constituting the structure of Au68(SR)32 cluster.

Figure 7. Naked Au68 cluster constituted by a) triangles ({111} planes), b) tetrahedra and c) gold octahedral units. Missing atoms in the octahedra in c) correspond with partial polyhedral blocks. Clearly, the middle image shows tetrahedral building blocks and the octahedral holes.

5. Structure of the Au102(SR)44 cluster In the study of TGC constituted by more than hundreds of gold atoms, the Au102(SR)44 cluster is mandatory (Figure 8).10 This structure has five tetrahedral blocks forming one decahedron unit (Au7), and its central part contains a column built by three decahedra orientated along a C5 axis (Figure 8a and Figure 8b). The space left among decahedra is filled up with octahedral blocks (Figure 8c and Figure 8d). On other words, the structure has three linked decahedra that are encircled by five columns of O-units (adding up to 20). They are sharing edges or faces among them. Ten more octahedral


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units are distributed among the upper and lower part of those five columns. Finally, in the middle part of the Au102 structure are located ten octahedral blocks. Tetrahedral units are located at those spaces left by the stacked O-units. The Au102 cluster contains 40 octahedral, and 70 tetrahedral blocks (their triangular faces have angles in the range from 57.80 to 65.14 degrees). Distortions in the Au-Au bonds of octahedral blocks produces polyhedral square planes contain angles in the range from 40.14 to 54.94 degrees.

Figure 8. The Au102(SR)44 cluster and their Au4, and Au6 polyhedral blocks. (a) The distribution of centers of tetrahedral units is shown in blue color, and (b) its representation with red triangular planes. In a similar manner, (c) the distribution of the octahedral centers has a C5 axis, (d) and O-units are displayed in green color. (e) The central part has three decahedra (Au7) forming a column, and five columns of octahedral units are surrounding it. (f) The perpendicular view along a C5 axis of the full structure shows the staple motifs.


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6. Structure of the Au130(SR)50 cluster The structure of the Au130(SR)50 cluster resembles the Au102(SR)44 cluster,47 it can be explained in a simplified manner by considering the distribution of its octahedral blocks (Figure 9a and 10a). It holds a column of four linked decahedra orientated along one C5 axis (Figure 9b). Among each pair of decahedra are located three octahedral blocks (adding up to 15), but among each decahedron-decahedron union (Figure 9c) are located ten more encircling octahedral (adding up 20). It has ten octahedral units distributed in both endings and five more are located in the equatorial region of the structure. The centers of O-units are shown as blue spheres in the figure 10a and their assembly in Figure 10b. There are 58 octahedral and 98 tetrahedral blocks (Figure 10c) forming the structure of the Au130(SR)50 cluster (Figure 10d), and the counting has been done by considering angles of square-like faces of octahedral in the range from 40.27 to 54.87 degrees and in the case of tetrahedral units, angles on triangular faces are in the range from 57.05 to 69.11 degrees.


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Figure 9. Distribution of both decahedral and octahedral blocks constituting the Au130(SR)50 cluster. (a) There are four decahedra forming a column and (b) it is encircled by five columns of three octahedral blocks linked each other by their edges. (c) This type of arrangement is found in structures displaying a C5 axis. (d) A full view of the assembled building blocks is shown.

Figure 10. Distribution of polyhedral blocks constituting the Au130(SR)50 cluster. (a) The centers of octahedral blocks are displayed as blue spheres. (b) Octahedral blocks are displayed in green color and clearly the front square-like face is an example of an incomplete octahedron (five neighboring gold atoms instead of six). (c) Tetrahedral faces are shown in red color. (d) The whole structure is displayed as a combination of polyhedral blocks, and it contains the staple motifs bridging them.


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7. Structure of the Au133(SR)52 cluster It represents a distinct arrangement of Au6 blocks. The inner core (Au13) is built by 20 slightly distorted T-units (Figure 11a) forming an icosahedron-like unit (Figure 11b), and 20 stacked octahedra are located over each of its triangular faces (Figures 11c, 11d and growth pattern 2e).48 Another manner to visualize the full structure (Figure 11e) is by one inner icosahedron sharing each vertex with 12 gold decahedra orientated along C5 axes (Figure 11b), and the space left among them being filled out with twenty octahedral blocks. By symmetry, a total of 60 T-units are forming those 12 decahedra. The Au133(SR)52 cluster is comprised by 42 O-units and 103 T-units. Further addition of extra gold atoms, might occur along six C5 axes, increasing the size of the cluster (Au145 cluster). In figure 11e, all displayed polyhedral in the Au133(SR)52 cluster have angles of squarelike faces in the range from 40.04 to 54.75 degrees (O-units) and angles in triangular faces (T-units) in the range from 57.05 to 69.96 degrees.


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Figure 11. (a) Distribution of tetrahedral blocks centers (shown as blue spheres) in the Au133(SR)52 cluster. (b) One dodecahedron is built by 12 T-units and the outer shell is comprised by tetrahedral blocks located along six C5 axes. (c) 20 O-units are forming a dodecahedron like arrangement around the inner icosahedron, and they are located in the space left by tetrahedra. (d) Octahedral blocks are shown with green faces. (e) The full structure displays Au and S atoms forming the staple motifs bridging among polyhedral blocks.

8. Structure of the Au144(SR)60 cluster The structure of the Au144(SR)60 cluster49 does not have a central Au atom as in the case of the Au133(SR)52 cluster. It is built by O-units distributed on vertexes of one dodecahedron like arrangement (Figure 12a and Figure 12b) and having three of their six atoms located on triangular faces formed by the 12 tetrahedra unions. Moreover, 60 stellated tetrahedra are forming an icosahedral arrangement (Figure 12c). It means that


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12 gold decahedra hold five T-units on their triangular faces. Another 30 O-units are located on the “edges” of the formed dodecahedron, and these are the more distorted because they include gold adatoms. In total there are 120 tetrahedral and 50 octahedral blocks constituting the structure of the Au144(SR)60 cluster (Figure 12d). Calculated angles of octahedral blocks are included in the range from 41.90 to 54.99 degrees and angles of tetrahedral blocks are in the range from 57.05 to 70.00 degrees.

Figure 12. A simplified view of the Au144(SR)60 cluster in terms of Au4 and Au6 units. (a) The distribution of octahedral blocks and their centers (in blue) are describing a dodecahedron. (b) They are sharing an edge and displayed in green color. (c) Twelve decahedra units (Au7), with apexes located on the inner Au12 shell are forming an icosahedral like arrangement. On top are located 20 gold octahedra forming a dodecahedron like arrangement. (d) The full structure includes the monomeric staples bridging among O-units. The image does include 60 tetrahedra located on the top of triangular faces of the decahedra (stellated triangular faces). 9. Structure of Au146(SR)57 cluster The Au146(SR)57 cluster is not properly an FCC like structure,50 and one more appropriated explanation of its structure is provided. To describe the polyhedral blocks


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arrangement and their center distribution are provided some figures (Figures 13a-13d). It is notable that in the Au146(SR)57 cluster (Figure 13e), tetrahedral blocks are not forming a regular FCC like structure, and it is evident the presence of one stacking fault (displayed along the horizontal direction in Figures 13a, 13b). ¿How is the polyhedral blocks arrangement in the planes forming the stacking fault? A deep structural analysis reveals that three planes of atoms are describing an ABA stacking or HCP structure contain the stacking fault (Figure 14a). Moreover, in this region are found two columns of gold decahedra (each column has three Au7 units), forming an angle of approximately 60.8 degrees. The union among columns has a pair of incomplete octahedral blocks that are sharing a triangular-like face. Planes over the stacking fault form an angle of circa 45.2 degrees with respect to it. The presence of three decahedra forming a column is also found in the case of the Au102(SR)44 cluster, and it is interesting that the Au146(SR)57 cluster holds a pair of them. Figures 14a-14d, indicates that only upper and lower part of the Au146(SR)57 cluster complies with a FCC structure. The structure is constituted by 119 tetrahedral and 75 octahedral blocks, and it holds 14 incomplete gold octahedra (They only include five neighbors) located in the outer part of the cluster.


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Figure 13. (a) Distribution of Au4 blocks through the structure of the Au146(SR)57 cluster. (b) The center of assembled tetrahedral forming Au7 units, and the presence of a stacking fault (indicated by arrows) is evident. Three participating planes in the stacking fault contains two columns of decahedra forming an angle of circa 60 degrees. (c) Octahedral blocks are forming a HCP arrangement in the planes containing the stacking fault. (d) They are in green color and they have angles of square-like faces in the range from 40.07 to 54.95 degrees and angles in triangular faces of tetrahedral are in the range from 57.03 to 69.98 degrees. (e) A full view of the structure including staple motifs is also shown.

Figure 14. Three central planes, containing the stacking fault, in the structure of the Au146(SR)57 cluster. (a) There are two columns and each column contains three linked


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decahedra (along the dotted lines), they are forming an angle 𝜃 of 60.8 degrees approximately. Perpendicular to the stacking fault planes, tetrahedral blocks are linked by one vertex forming an ABA or HCP arrangement. (b) The distribution of octahedral blocks is also shown. (c) Those planes located at the upper part of the cluster are shown as constituted by tetrahedral blocks (in red color). The presence of T-units sharing an edges as indicated by the asterisk (figure 2c). (d) The presence of one pair of octahedral blocks (as in FCC structures) linked by an edge is shown by a rectangle. 10. Structure of the Au187(SR)68 cluster The structure of the Au187(SR)68 cluster51 contains seven linked Au7 units and encircled alternately by columns of tetrahedral and octahedral blocks (Figure 15a). There are five columns built by six Au6 units linked by their vertexes (adding up 30 in total) enclosing the central column (central part of the figure 15 b). Fifteen columns of Au7 units (75 Tunits) are orientated along a C5 axis. The next surrounding columns are radially distributed as: 10 columns of five O-units (50 octahedra in Figure 15c), and five columns of five T-units (25 tetrahedra). The full Au187(SR)68 cluster (Figure 15d) is comprised by 135 tetrahedral and 80 octahedral blocks. Polyhedral blocks found in the Au133(SR)52 cluster have angles of square-like faces in the range from 40.21 to 54.98 degrees (octahedra) and angles in triangular faces are in the range from 57.82 to 65.09 degrees.


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Figure 15. Distribution of the polyhedral blocks through the structure of the Au187(SR)68 cluster. (a) The distribution of tetrahedral blocks displayed in red (b) and the distribution of octahedral blocks (c) attests the stacking of distinct polyhedral. (d) The total structure displays staple motifs. Some of the adatoms are part of polyhedral blocks.

11. Structure of the Au246(SR)80 cluster The Au246(SR)80 cluster holds a central column comprised by five decahedra (adding up to 25 tetrahedra) linked in a similar manner to the inner core of the Au102(SR)44 cluster (containing three decahedra).52 Ten Au4 blocks are distributed along a C5 axis (Figures 16a and 16b), and located at both endings of the central column (adding up to ten). Surrounding the central column are five columns of Au6 units (20 octahedra, Figures 16c and 16d). The following circle is comprised by ten columns of four and 15 tetrahedral units. The next three radial shells are comprised by columns containing one octahedral linked to eigh tetrahedra (in total are 30 octahedra and 75 tetrahedra), and


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this distribution corresponds with a FCC arrangement on each side of the pentagonal faces of one decahedra. There are another external polyhedral blocks but their arrangement is not following an ordered pattern. The whole structure of the Au246(SR)80 cluster (Figure 16e) is built by 221 tetrahedral and 110 complete octahedral blocks.

Figure 16. Distribution of the (a) and (c) tetrahedral blocks and (c) and (d) octahedral blocks through the structure of the Au246(SR)80 cluster. (e) The structure containing the staple motifs is displayed. Displayed polyhedral blocks have angles of square-like faces in the range from 40.84 to 54.81 degrees and angles in triangular faces (tetrahedra) in the range from 57.83 to 65.06 degrees. 12. Structure of the Au279(SR)84 cluster It has been determined as the first thiolated gold cluster displaying a plasmon resonance. The structure of the Au279(SR)84 cluster obeys clearly to a FCC structure (Figure 17a).53 In the case of tetrahedral, the calculated angles of triangular faces are in


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the range from 75.82 to 65.17 degrees. Octahedral units (Figure 17b) are defined by square like faces with angles in the range from 40.00 to 54.75 degrees.

Figure 17. (a) Cubic distribution of the centers of tetrahedral blocks and b) octahedral blocks through the structure of the Au279(SR)84 cluster. The total Au-S framework is displayed and the staple motifs are included in the ball and sticks representation. Green blocks are octahedral and red ones corresponds with tetrahedral blocks. It is evident that more external gold atoms are following a random distribution attributed to the presence of staple motifs (gold adatoms).

The structure of the Au279(SR)84 cluster (Figure 17c) is comprised by 288 tetrahedra and 126 octahedral units, with the presence of 28 no complete octahedral units located at the outermost part of the cluster. It is interesting to see that some outermost O-units are not following the FCC pattern and they are forming a different arrangement on the cluster. This must be due to the induced distortion by the protecting experimental ligands. The new polyhedral approach is able to quantify the distortion grade of the building blocks constituting thiolated gold clusters, and it can be extended to the study of clusters comprised by tents or hundreds of gold atoms. In Table 2 are provided 24 additional structures in order to attest its range of application.


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Table 2. Number of Polyhedral building blocks of 24 Ligand Protected Gold Clusters.a The calculated angles of triangular faces and square faces is provided. Cluster

Octahedra/ Octahedra Angles Tetrahedra Angles Tetrahedra (Degrees) (Degrees) -1 [Au12(SR)9] 1/6 42.51-46.70 50.26-71.28 Au15(SR)13 0/1 58.76-61.81 Au18(SR)14 2/1 42.07-46.74 50.54-71.77 Au20(SCH2CH2Ph)16 0/2 58.02-63.48 Au20(TBBT)16 0/2 58.32-61.09 Au22(SR)18 0/2 58.80-62.25 -1 [Au23(c-C6)16] 0/2 59.12-60.58 Au24(SAdm)16 0/2 57.61-63.17 [Au24(SCH2Ph-tBu)20]0 0/4 58.46-62.07 Au24(SR)20 0/4 51.43-69.86 Au25(SEt)18 0/23 52.98-67.41 +2 [Au25(PPh3)10(SC2H4Ph)5Cl2] 0/40 56.02-67.22 Au28(TBBT)20 2/4 41.43-49.47 58.26-62.06 Au28(SR)20 0/6 52.39-66.51 Au36(SC5H9)24 4/11 40.29-49.18 53.30-67.33 Au38(SR)24 0/35 52.88-68.60 Au38-T 2/31 43.61-47.12 52.56-72.64 Au40(SR)24 0/41 55.00-67.63 Au40(TBBT)24 3/14 40.68-49.48 54.48-67.65 Au44(SR)28 3/10 41.48-54.94 53.07-50.95 Au44(TBBT)28 5/16 41.42-48.07 53.69-64.91 Au52(TBBT)32 8/24 40.70-49.02 53.26-68.10 Au68(SR)34 25/36 39.30-49.84 50.62-75.73 Au76(SR)44 8/36 41.05-49-87 54.74-64.65 a The list includes calculated (-SR ligand) and experimental ligands.

Ref. 56 33 36 40 57 58 22 59 6 60 61 62 13 63 27 8 64 65 66 67 39 66 68 69

In this manuscript it has been proposed a polyhedral approach that is based on geometrical aspects of the considered clusters. However, it is important to take into account that the relation among geometrical structure and electronic structure is not simple and it requires of more elaborated methodologies. For example, the Au20 has been longer considered as an important cluster given its large HOMO-LUMO gap value that results in a chemical inert system. It can be explained as a structure of concentric


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layers and its stability needs to be rationalized in terms of the interaction among those layers.70 For a review on superatomic cluster readers are encouraged to revise a recent manuscript.71 To summarize the obtained results. a) Tetrahedral and octahedral building blocks and their assembly are able to describe/explain the structure of 36 considered key examples of thiolated gold clusters. Their versatility to built another amply known polyhedral blocks as Au5, Au6, Au7, and Au13 is remarkable. b) In general, the calculated angles of square faces are comprised in the range from 40 to 55 degrees for octahedral blocks and tetrahedral blocks have angles included from 57 to 66 degrees for most of the studied clusters. Only the small Au12(SR)9 contains angles comprised in the range of 50-72 degrees. c) Proposed growth patterns of FCC structures were found in the framework of various TGC. In the case of the Au38S2(SR)20 cluster the polyhedral approach explain it as a frustrated FCC structure. d) The assembly of decahedra forming columns are very common in TGC comprised by hundreds of gold atoms and holding a C5 axis.


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e) A simplified view of the TGC structures is obtained by displaying the distribution of the center of each polyhedral block. For example, it is helpful to determinate the presence of a stacking fault in the structure of the Au146(SR)57 cluster. f) TGC with up to 25 atoms have only one type of the polyhedral building blocks, preferentially. When their size increases then a mixture of both polyhedral blocks are found. 4. CONCLUSIONS The amply structural information available in circa 70 structures needs to be understood in order to guide future experiments. It means that the known sizes hold geometrical features that are linked to experimental conditions as temperature, concentration, solvent, etc. The interactions among intermediate atomic or molecular species (building blocks) formed during experiments might result in characteristic type of assemblies of elementary building blocks. Interestingly the algorithm used to account for neighboring atoms and bond lengths, can be adapted to the bimetallic Ag-Au systems, given the similarity of both atomic radius. The author is currently working on this improvement.


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In this work, the distinct assembly of elementary building blocks in 36 key TGC structures was revealed attesting the power of the polyhedral approach. It is expected that the obtained trends can be used to guide future experimental efforts and to propose more reliable models.


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FIGURES

Figure 1. A 2x2x2 FCC supercell containing 63 atoms is shown. (a) The centers of 64 T-units describe a simple cubic arrangement. (b) They are linked by their edges, and (c) 14 complete O-units are sharing edges, but only eight are displayed to facilitate the view. Missing six O-units share one vertex located in the center of the supercell.

Figure 2. Growth patterns of naked gold clusters are displayed. (a) One gold tetrahedron covered by four octahedral blocks. (b) One octahedron capped with eight tetrahedral blocks. (c) One Au6 formed by two fused tetrahedra by an edge can be covered by octahedral blocks. (d) One Au12 cuboctahedron can be seen as comprised by six octahedral blocks sharing edges or vertexes. (e) One icosahedron surrounded by 20 octahedral blocks depicts overlapping positions.

Figure 3. The anatomy of the Au30(SR)18 cluster described as an assembly of (a) three Au6 units sharing a triangular face forming one column and (b) 12 Au4 units forming six triangular Au5 bipyramids. (c) The Au-S framework holds one Au18 core and six dimer motifs with S- adamantine as the experimental ligand (not included here).


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Figure 4. Structure of the Au36(SR)24 cluster. (a) It can be described as an assembly of four O-units surrounding one center Au4 tetrahedron. Their distribution is shown in blue color. (b) The octahedral blocks are displayed in green color and they are sharing an edge. (c) Five more external T-units are sharing a vertex or edge and are surrounding the octahedral block. (d) The full Au-S framework holds seven dimer motifs being S atoms forming bridges among gold atoms. It is evident the presence of two incomplete octahedral blocks in the outermost region.

Figure 5. Structure of the Au38S2(SR)20 cluster. (a) The distribution of the center of tetrahedral blocks are shown, and (b) they are linked by an edge (forming an inner Au6) or by a vertex. (c) There are two complete octahedra sharing a vertex as in the case of a FCC arrangement, and two incomplete octahedra are located at the outer part. (d) The Au-S framework displays S atoms forming bridges in its middle region (Indicated with arrows).

Figure 6. Three calculated isomers of the Au68 cluster seen as constituted by tetrahedral units. (b) The middle one contains a major number of {111} planes, and it was determined as constituting the structure of Au68(SR)32 cluster.


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Figure 7. Naked Au68 cluster constituted by a) triangles ({111} planes), b) tetrahedra and c) gold octahedral units. Missing atoms in the octahedra in c) correspond with partial polyhedral blocks. Clearly, the middle image shows tetrahedral building blocks and the octahedral holes.

Figure 8. The Au102(SR)44 cluster and their Au4, and Au6 polyhedral blocks. (a) The distribution of centers of tetrahedral units is shown in blue color, and (b) its representation with red triangular planes. In a similar manner, (c) the distribution of the octahedral centers has a C5 axis, (d) and O-units are displayed in green color. (e) The central part has three decahedra (Au7) forming a column, and five columns of octahedral units are surrounding it. (f) The perpendicular view along a C5 axis of the full structure shows the staple motifs


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Figure 9. Distribution of both decahedral and octahedral blocks constituting the Au130(SR)50 cluster. (a) There are four decahedra forming a column and (b) it is encircled by five columns of three octahedral blocks linked each other by their edges. (c) This type of arrangement is found in structures displaying a C5 axis.

Figure 10. Distribution of polyhedral blocks constituting the Au130(SR)50 cluster. (a) The centers of octahedral blocks are displayed as blue spheres. (b) Octahedral blocks are displayed in green color and clearly the front square-like face is an example of an incomplete octahedron (five neighboring gold atoms instead of six). (c) Tetrahedral faces are shown in red color. (d) The whole structure is displayed as a combination of polyhedral blocks, and it contains the staple motifs bridging them.


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Figure 11. (a) Distribution of tetrahedral blocks centers (shown as blue spheres) in the Au133(SR)52 cluster. (b) One dodecahedron is built by 12 T-units and the outer shell is comprised by tetrahedral blocks located along six C5 axes. (c) 20 O-units are forming a dodecahedron like arrangement around the inner icosahedron, and they are located in the space left by tetrahedra. (d) Octahedral blocks are shown with green faces. (e) The full structure displays Au and S atoms forming the staple motifs bridging among polyhedral blocks.

Figure 12. A simplified view of the Au144(SR)60 cluster in terms of Au4 and Au6 units. (a) The distribution of octahedral blocks and their centers (in blue) are describing a dodecahedron. (b) They are sharing an edge and displayed in green color. (c) Twelve decahedra units (Au7), with apexes located on the inner Au12 shell are forming an icosahedral like arrangement. On top are located 20 gold octahedra forming a dodecahedron like arrangement. (d) The full structure includes the monomeric staples bridging among O-units. The image does include 60 tetrahedra located on the top of triangular faces of the decahedra (stellated triangular faces).


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Figure 13. (a) Distribution of Au4 blocks through the structure of the Au146(SR)57 cluster. (b) The center of assembled tetrahedral forming Au7 units, and the presence of a stacking fault (indicated by arrows) is evident. Three participating planes in the stacking fault contains two columns of decahedra forming an angle of circa 60 degrees. (c) Octahedral blocks are forming a HCP arrangement in the planes containing the stacking fault. (d) They are in green color and they have angles of square-like faces in the range from 40.07 to 54.95 degrees and angles in triangular faces of tetrahedral are in the range from 57.03 to 69.98 degrees. (e) A full view of the structure including staple motifs is also shown.

Figure 14. Three central planes, containing the stacking fault, in the structure of the Au146(SR)57 cluster. (a) There are two columns and each column contains three linked decahedra (along the dotted lines), they are forming an angle 𝜃 of 60.8 degrees approximately. Perpendicular to the stacking fault planes, tetrahedral blocks are linked by one vertex forming an ABA or HCP arrangement. (b) The distribution of octahedral blocks is also shown. (c) Those planes located at the upper part of the cluster are shown as constituted by tetrahedral blocks (in red color). The presence of T-units sharing an edges as indicated by the asterisk (figure 2c). (d) The presence of one pair of octahedral blocks (as in FCC structures) linked by an edge is shown by a rectangle.


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Figure 15. Distribution of the polyhedral blocks through the structure of the Au187(SR)68 cluster. (a) The distribution of tetrahedral blocks displayed in red (b) and the distribution of octahedral blocks (c) attests the stacking of distinct polyhedral. (d) The total structure displays staple motifs. Some of the adatoms are part of polyhedral blocks.

Figure 16. Distribution of the (a) and (c) tetrahedral blocks and (c) and (d) octahedral blocks through the structure of the Au246(SR)80 cluster. (e) The structure containing the staple motifs is displayed. Displayed polyhedral blocks have angles of square-like faces in the range from 40.84 to 54.81 degrees and angles in triangular faces (tetrahedra) in the range from 57.83 to 65.06 degrees.


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Figure 17. (a) Cubic distribution of the centers of tetrahedral blocks and b) octahedral blocks through the structure of the Au279(SR)84 cluster. The total Au-S framework is displayed and the staple motifs are included in the ball and sticks representation. Green blocks are octahedral and red ones corresponds with tetrahedral blocks. It is evident that more external gold atoms are following a random distribution attributed to the presence of staple motifs (gold adatoms)

ASSOCIATED CONTENT Supporting Information. Details of calculations on Au4 (2e) and Au6 (8e) units. Extended Table of growth patterns of TGC. AUTHOR INFORMATION Corresponding Author Alfredo Tlahuice-Flores. e-mail: [email protected] Universidad Autónoma de Nuevo León, CICFIM-Facultad de Ciencias FísicoMatemáticas, San Nicolás de los Garza, NL 66455, México ACKNOWLEDGMENT The author thankfully acknowledges the computer resources, technical advice and support provided by the Laboratorio Nacional de Supercómputo del Sureste de México (LNS), a member of the CONACyT network of national laboratories. Special thanks to Prof. Rongchao Jin, Prof. Amala Dass and Prof. Robert L. Whetten for providing with .cif files of various structures.


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ABBREVIATIONS Thiolated Gold clusters (TGC); Octahedral blocks (O-units); Tetrahedral blocks (T-units). REFERENCES (1) (2) (3) (4) (5) (6)

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Figure 1 84x29mm (300 x 300 DPI)


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Figure 15 64x64mm (300 x 300 DPI)


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Figure 17 84x29mm (300 x 300 DPI)


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