Structural Predictions of RS-AuNPs

To find the most stable isomer of cluster means to locate the global minimum on the highly sophisticated potential energy surface (PES) using the unbiased or biased method. Considering that the number of local minima increases exponentially with the size of cluster, the unbiased strategy, that is, the heuristic global search method, is believed to be more suitable for small-sized RS-AuNPs. The biased strategy works well for RS-AuNPs with relatively larger size and available structure knowledge from experimental and theoretical research.

Unbiased Prediction Method

The unbiased method refers in particular to the global search from scratch, which can be roughly categorized into two strategies. The first one is random sampling search based on the Monte Carlo move operating on the single individual, for example, basin hopping (BH)

[16] and simulated annealing [17]. Another popular strategy is the algorithm guided under specific evolution rules, typified by genetic algorithm (GA) [18] and particle swarm optimization [19]. We will lay emphasis on two popular global search methods, that is, BH and GA. Considering that unbiased global search is also the initial and important step of the biased prediction method, the principle, advantages, and disadvantages are presented in this section.

BH algorithm: The rather complicated PES with many local minima needs to be explored with the combination of random move and optimization operation on a single individual [16, 20]. In this algorithm, the trial structure is normally generated after a random deformation on the structure of the last step within a set step length, that is, the hopping behavior illustrated in Scheme 4.1. The essence of this algorithm is performing optimization for every

Scheme 4.1 (a) Schematic representation of BH and GA and (b) the flowchart of a hybrid strategy combining BH and GA. The pink dots represent the local minima on PES through the combination of random move and optimization operation in BH. The red lines with double arrowhead represent the evolution paths between local minima through the crossover and mutation operation under the GA.

structure after a random sample, which transforms the original PES into the combination of a series of basins (the square plane in Scheme 4.1} representing local minima. Given that the goal is to find the lowest lying isomer, the Metropolis criterion is utilized to decide whether the current generated geometry is accepted. Specifically, according to the energy difference ДE between the current structure and the structure of the last step, the current structure is accepted as a favored isomer if the Д£ < 0 or the Boltzmann factor ехр[-Д£Дв7] is larger than a generated random number in case that ДE > 0, where kB is the Boltzmann constant and T is the set temperature. Subsequently, a new structure will be generated for either the next step or another trial structure of the current step.

The procedure is repeated until the search process terminates or converges. With the aid of optimization operation, the transition states on the PES are eliminated, which effectively accelerates the search process. On the other hand, due to the fact that the operating object remains to be an individual, the search process will be easily trapped when locating on a deep potential funnel.

GA: In contrast to the BH algorithm, GA represents another typical evolution search where the operating object is a population of individuals instead of one single individual [18, 20]. Being similar to the BH algorithm, a local optimization is performed for each individual within the search cycling. The fitness (/) of those optimized structures is evaluated based on their energies. One popular way of fitness calculation is given in Eq. 4.1:

Here, fj and £,• are the fitness and energy of the /th individual, respectively, and £min and £max represent the lowest and the highest energy of the current population, respectively. In this way, the isomer with lower energy is evaluated to possess higher fitness. Then, on the principle of the survival of the fittest, a subset of individuals with higher stability are chosen and grouped to constitute the parent generation. Through the crossover and mutation operation as in Scheme 4.1a, the structures of the parent individuals are randomly fragmented, packed together, and then deformed to produce the offspring generation. Next, the combinations of local minimization, fitness evaluation, selection, crossover, and mutation are repeatedly operated until the energy/structure converges or the search cycle is completed. With the pivotal selection, crossover, and mutation operation mimicking evolution, the favored structure genes are kept and assembled to generate more fitted offspring structures during the search process. Although the operation on the population is beneficial to globally explore the sophisticated PES, it also gives rise to a problem that the search is dependent on the initial population.

Based on the above description, there is both evident discrepancy and complementarity between two algorithms. Through the Monte Carlo move and geometry optimization on the single individual, the BH algorithm possesses the excellent local search ability. For GA, since the operation object is the isomer population, the global search ability turns to be the superiority. To make full use of both algorithms, the combination is suggested in case that a single algorithm works poorly. As shown in Scheme 4.1, a BH search is first implemented and then it generates diverse local minima that represent different valleys on PES. The local minima are then utilized as the initial population of GA, and the thorough exploration between different local potential energy valleys is completed with the help of evolution operation. This hybridization ensures both local and global search capacities of the search process, enhancing the effect of the search process.

Improvement on the search efficiency: Apart from the choice of the algorithm, one issue of great importance in global search is how to prevent the stagnancy of search process. For BH algorithm, the search process will easily stagnate when a deep potential energy valley is located. As for GA, the evolution operations will keep the favored structure genes and naturally generate population with more and more similar geometry, resulting in invalid search on the restricted potential energy range. The key to prevent stagnancy concerns two parts, that is, an effective geometry similarity characterization and a timely feedback modulation.

Considering the isomers with a minor geometrical difference are normally located on the same potential energy funnel and optimized to the same local minimum, the geometry similarity characterization is needed to prevent repeated local search and enhance the global exploration on PES. The root mean square deviation (RMSD) is frequently adopted as a characterization index. Due to the defect that the RMSD is highly sensitive to the atom order of the structure, the calculation based on the quaternion method is more recommended [21], which minimizes the RMSD deviation originating from the atom sequence with the aid of translation and rotation operation. The quaternion RMSD calculation gives a reasonable evaluation of the radial distance distribution deviation around the mass center, but neglects the topological connectivity between atoms. Instead, a difference quantification between connectivity matrix of varied isomer geometries is more powerful to measure the geometry similarity from the point of the topological connectivity [22]. The incorporation of quaternion RMSD and topological connectivity constitutes an effective geometry similarity characterization.

With the introduction of geometry similarity characterization, the search program can automatically recognize the isomers with similar geometry and, correspondingly, define explored PES zones.

Once the invalid revisits on the marked potential energy funnel occur frequently, the search process should be able to identify the timely feedback and tune the next search orientation. Many strategies have been developed to resolve this issue, in which adopting the adaptive step length or probability is a common way [23, 24]. Exampled as "the BH with occasional jumping" strategy [23], both step length and accept probability of the next trial moves will be tentatively increased until the current energy valley is escaped, illustrated as the "jumping move" in Scheme 4.1. Together, a heuristic global search algorithm embedding geometry similarity characterization and feedback modulation system is capable of self-optimization and guiding the search orientation via adaptive step length and probability adjustment, ensuring both the effect and the reliability of the search process.

Nucleation and growth mechanism of intermediate RS-AuNPs based on global search

The controllable synthesis and characterization of monodisperse RS-AuNPs is a challenging task for researchers, stemming from the lack of size evolution information on intermediates with different sizes. In a synthesis experiment of Ai^sCSR)-^, dozens of intermediates have been identified through tracking the synthesis process with mass spectrum [25]. Based on this research, structures of a series of intermediate clusters Aum(SR)n with m and n ranging from 5 to 12 have been predicted through the hybrid BH/GA search with PBE/DND level in DMol3 package.

The low-lying structures of different sized clusters are shown in Fig. 4.1, and three evident growth patterns can be concluded, that is, the core growth, core dissolution, and staple-motif growth [26]. Taking core-growth pattern for instance, along with the continuously added Au atom, the inner core evolves from the linear Au2 core to triangular or linear Au3, tetrahedral Au4, bipyramid Au5, and vertex-sharing bi-tetrahedral Au7 step by step, while the outer ligands remain unchanged. Notably, this core-growth pattern works for the growth processes starting from diverse precursors, testifying the universality of this nucleation pattern. In case that the number of thiolates is increased but the number of Au atoms remains the same, the added thiolate tends to break a original Au-Au bond in kernel and forms longer ligand, but diminishes the inner kernel. Hence, this pattern is defined as core-dissolution rule. As for the circumstance that the Au atom and thiolate are simultaneously increased, the ligand becomes larger while the inner core remains unchanged, that is, the staple motif-growth rule.

Schematic illustrations of

Figure 4.1 Schematic illustrations of (a) the size evolution of Aum(SR)„ clusters and (b) the typical core structures of intermediate clusters. Yellow and blue balls represent Au and S atoms, respectively, and -R groups bonded to S are eliminated for clarity. Reprinted with permission from Ref. [26], Copyright 2015, American Chemical Society.

The small size of the Au6(SR)6 cluster (0e' system) limits the formation of Au core. Notably, along with a successive increase of two Au atoms from Au8(SR)6 (2e') to Au14(SR)6 (8e~), an evolution of tetrahedron Au4 to vertex-sharing bi-tetrahedron Au7, double vertexsharing tri-tetrahedron Au10, and quadruple vertex-sharing tetra- tetrahedron Au12 is observed in Fig. 4.2 [26]. Every increase oftwo Au atoms is found to be associated with a newly formed Au4 tetrahedron in kernel and 2e" growth, revealing the compact connection between topological and electronic structures. Considering the fact that the Au4 core could be viewed as a superatom with 2e", the kernel of Au25(SR)18 is divided into four Au4 tetrahedrons from the perspective of both topological structure and electric structure.

The Au13 icosahedron core of Au25(SR)18 is presumed to be the sequential assemble of a successive Au4 tetrahedron, which matches the 2e" —> 4e" —> 6e~ —> 8e~ growth rule observed in the experiment [25]. Moreover, based on the calculated average binding energies of intermediate clusters, the intermediates with even valence electrons are found to possess higher stability, which validates the phenomenon that only clusters with even valence electrons are detected in experiment and strengthen the reliability of 2e" growth pattern.

In addition, a further analysis on the average connectivity has been carried out on the low-lying structures of intermediate clusters. From the initial homoleptic clusters, intermediate clusters, to the final product, the average connectivity of three types increases successively, proving that intermediate clusters are highly active and easily participate in the later growth and size-focusing stage. Considering the core similarity between the intermediate cluster and experimentally resolved clusters, it is clear that the intermediate clusters serve as the building blocks and assemble into diverse final kernel structures under diverse experimental conditions. Together, based on global search and DFT, the predicted structures vividly display a preliminary nucleation process and the subsequent size- focusing growth process.

Illustration of the successive 2e~ growth patterns

Figure 4.2 Illustration of the successive 2e~ growth patterns.

 
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