Theoretical Models of Structures of AuNCs

“Divide and Protect Model” Concept

In 1999, the first theoretical model of the structure of AuNC was suggested [52]. In this model, Au₃₈(SCH₃]₂4 was chosen as an example.

Its metal core was found to be fcc-type through optimization using density functional theory (DFT). The remaining thiol groups were in the -SCH₃ form as monolayer covering the Au_3S core. At that time, this was a rather sound guess of the geometric architecture of AuNCs.

Until 2006, a low-temperature STM experiment performed on Au surface suggested that RSAuSR complex could also be the absorbate, which implied that this complex might also cover the Au core as a shell in AuNC [53]. Independently and almost simultaneously, Hakkinen performed another theoretical study proposing a novel motif covering the Au core [54]. Au₃₈(SCH₃)₂₄ was again used as the sample cluster. (AuSCH₃]₄ was suggested as the shell cap and the rest of the Au₁₄ formed the core of this AuNC. This work represents a significant improvement for understanding AuNCs because it suggests that different bonding statuses of Au atoms may coexist in the outer ligand and the inner core. The "divide and protect" concept just originates from this work. This work also suggests that the overall structure of AuNC may be present in a highly symmetric form, which helps to maximize the number of Au-Au interactions, rather than in a disordered form proposed earlier [71, 72].

Inherent Structure Rule

The proposal of the "divide and protect model” concept was a breakthrough in this field at that time because no experimental evidence of how atoms arrange in the cluster was available then. In 2007, Jadzinsky et al. successfully obtained the crystal structure of Au₁₀₂(p-MBA)₄₄ [55]. This was the first structural determination study that proved the value of the divide and protect model concept. Later, Jin and Murray independently prepared [Au₂s(SR)₁₈]', which was another major advance in understanding the structure of AuNC [18, 73]. These three studies confirmed the structural pattern of core plus protecting-ligand and refreshed the views of what is the shape of Au core and what is the protecting ligand. According to the crystal structures, the motifs may include -RS-Au-RS- or -RS-Au- RS-Au-RS-.

Although major advances have been made, the question of whether there exists a general and underlying formula for AuNCs arouses the interest of researchers. Through analyzing the available structures of AuNCs, Pei et al. proposed an inherent structure rule in 2008 [74]. They claimed that the structures of future possible Au_m(SR)„ AuNCs could be divided into several parts by following a general formula of

where a represents the number of Au atoms in the core, b is the number of Au(SR]₂ motifs, and c and d are the number of Au₂(SR)₃ and Au₃(SR)₄, respectively. This rule is consistent with the previously proposed divide and protect rule. According to this inherent rule, the outer motif can be monomeric, dimeric, trimeric, and so on based on the number of Au atoms in this motif. And m is equal to the sum of a + b + 2*c + 3*d .... This rule fits well with the available Au₁₀₂ and Au₂₅ clusters and has been used in the prediction of new AuNCs [75, 76].

Superatom Complex (SAC) Model

This model adapts from the initial Jellium model. Jellium model was initially used in understanding alkali metal clusters [56] in which the valence electronswere considered to be distribute dunbiasedly within the cluster, instead of over confined to particular atoms. According to Jellium model, the orbitals of these free electrons spread over the whole metal atoms rather than an individual atom [77, 78]. Analogy to the atomic theory, this produces an electron configuration of IS² |1P⁶|1D¹⁰|2S²1F¹⁴|2P⁶1G¹⁸|2D¹⁰3S²¹H²²|..., which means that a stable metal cluster should have the full occupation of the outermost orbital. Orbitals of each shell configuration that distribute over the entire gold core and shape like the atomic orbitals have been found in a lot of AuNCs [79-81]. A notation called the "magic number" is generally used to denote the number of these free electrons. Typical magic numbers include 2,8,18, 20, 40, and 58.

In 2008, Hakkinen further extended the Jellium model to explain the high stability of spherical gold clusters, including Au₁₀2(SR)₄4, Au₃9CPR3)₁₄Cl6, Au₁₁(PR₃)₇C1₃, and Au₁₃(PR₃)₁₀Clf⁺ [82]. In these ligand-protected AuNCs, the number of free electrons n* correlates to the number of Au atoms or the number of AufOs¹) electrons N, the number of ligands M, and the charge of the AuNC q. This can be described using a simple formula

Note that the protecting ligands can be either withdrawing (e.g., thiol groups or halogen) or coordination groups like PH₃. For the latter, the binding of PH₃ to the Au core is via weak coordination and it is not included in the calculation of the number of free electrons.

Superatom Network (SAN) Model

Although the SAC model holds for most of AuNCs, some of the AuNCs with four free electrons, such as [Au₁₈(SR)₁₄] (4e), Au₂₀(SR)i6 (4e), and Au₂₄(SR)₂o (4e), do not seem to follow this model [30,83,84]. To understand the stability of these AuNCs, the SAN model was proposed by Yang and Cheng et al [85]. SAN can handle only 4e clusters if they are shell-closed electronic structures. Based on the SAN model, the 4e clusters can be depicted as a network of two-electron superatom gold cores. The authors analyzed the Au-Au distance within and between the two-electron superatom gold cores, the delocalized multicentered bonding, and the nucleus-independent chemical shift to support this model.

Grand Unified Model (GUM)

Understanding the stability of AuNCs has always been an interesting topic since it is one of the most fundamental problems in the research.

Each of the models for interpreting AuNCs' stabilities proposed so far is applicable only to a certain subset of AuNCs. Thus, it is demanding to understand their stabilities through a unified model.

Through analyzing crystal structures of the available 71 AuNCs, Gao et al. proposed grand unified model [86]. In this model, inspired by the quark model, gold atoms can be assigned to be three flavors to represent the possible valence states l.Oe, 0.5e, and Oe. Then, these three flavors constitute two composite particles of this model. These two composite particles include triangular elementary block Au₃(2e) and tetrahedral elementary block Au₄(2e). Both of them satisfy the duet rule, which is similar to the full occupation of the valence shell. In conceiving this model, the authors also tried different possible valence states and different potential isoelectronic species to make the fundamental concept of this model solid. The results of formation energies show that Au₃(2e) and Au₄(2e) seem to be the only two possible elementary blocks because they have highly negative formation energies. To satisfy the duet rule and the confined number of gold atoms of these two elementary blocks, different numbers of flavors could be combined. This produces 10 variants for triangular elementary blocks and 15 variants for tetrahedral elementary blocks. Later, to conveniently explain the stability of those AuNCs with the icosahedral Au₁₃ motif, a secondary block Au₁₃(8e) was introduced into this model [87]. This elementary block also owns highly negative formation energy. Considering this block, another eight variants should be included.

GUM aims at understanding the stability of AuNCs from the perspective of Au core. Having built the underlying architecture of GUM, the next step is to apply it in understanding a specific AuNC. To achieve this, the protecting ligands should be detached first. It is clear that the bonding between the ligands and the Au core can be either chemical bond or coordination based on the chemical nature of the bonding atom. Here, thiolate ligands, phosphine ligands, and halogen ligands were considered as examples. For these three ligands, the Au core would transfer 0.5e valence electron, le valence electron, and Oe valence electron when detaching them. Xu et al. provided several prototypical liganded clusters for interpreting how to apply GUM stepwise. It will also be introduced briefly in the next section. This is rather helpful to gain the knowledge of the beautiful structural pattern of AuNCs. Through GUM, it would be also much easier to understand the stability of AuNCs. In addition, GUM can also be used for other purposes. Xu et al. used GUM in understanding the structural evolution of AuNCs and in predicting new AuNCs as well [88]. Au₆₀(SR]₃₆, Au₆₈(SR)₄₀, and Au₇₆(SR)₄₄ have been successfully predicted based on GUM [88]. GUM can also be used in understanding the structural isomerism of AuNCs [89]. Besides,

GUM can also rationally design new ligand-protected gold clusters [90,91].