First-Principles Calculations of Stability, Electronic Structure, and Sorption Properties of Nanoparticle Systems

Abstract

Nanoparticles have a wide range of applications as catalysts. Their catalytic and electronic properties differ from those of materials with flat surfaces and bulk materials. First-principles calculations of real system nanoparticles, which use nanoparticle models based on real shapes extracted from experimental observations, are essential for studying these properties to facilitate the computational design of new catalysts. In this article, we review first-principles studies of models of real systems of monometallic, bimetallic, and supported nanoparticles. The stability, electronic structure, hydrogen absorption behavior, and small molecule adsorption behavior are reviewed, and advances in first-principles calculations of real system nanoparticles are presented. Further, a combination of machine learning and first-principles studies is also considered. Future perspectives are discussed on the basis of these examples.

1 Introduction

Metal nanoparticles are widely used for various applications in catalysis [1,2,3,4,5]. Various combinations of elements in binary, ternary, and multinary alloys are often investigated to obtain improved catalytic properties [6,7,8,9,10,11]. Recent advances in computer hardware, computer science, and theory are expected to improve the prospects for theoretical catalysis. Further, a data-driven approach is emerging owing to the rapid development of materials informatics [12,13,14,15,16]. Despite these advances, the practical computational design of catalysts is still far from being realized. Dean et al. [17] identified two major reasons. First, catalysts are often designed without considering their stability; consequently, the predicted materials are either difficult to synthesize or unstable under catalytic operation. Second, nanoparticles are catalysts with a high degree of site heterogeneity on surfaces with different planes (e.g., 111, 100), corners, and ridges; thus, their surface properties, such as adsorption and catalytic properties, are also heterogeneous. The big data approach using descriptions, which is widely used in materials informatics, is also affected by these problems, because simplified structure models are typically used in first-principles calculations. Li et al. [18] noted that the discovery of catalysts with improved performance by big data analysis or other descriptor-based approaches is limited when only the bulk structural parameters are considered for data acquisition. Therefore, first-principles calculations of structural models of real systems are necessary. Real system has been an important topic in many disciplines, of which examples can be found in [19,20,21]. Sakaki et al. [21] emphasizes that one of important roles of theoretical and computational chemistry is to present reliable computational results of real systems without any simplification. Challenges toward first-principles calculations of real systems depend on many factors. In the last century or early in this century, an active center, e.g., heme in enzyme and cation substituted site in zeolite frame, is cleaved from real system and often used for first-principles calculation to the huge computational cost to deal with whole of real systems. Along with the development of computers and theory, many of real systems have become computed theoretically, thus the focus of real system shifted to environmental effect such as solvation [19]. However, when we look into the first-principles studies of nanoparticles, we can find many reports adopting small clusters typically consisting of 13 or 55 atoms. There is no doubt that the computationally possibilities forced the early research to those limits and we highly acknowledge pioneer studies such as the works of T. M. Bernhardt, U. Landman, H. Häkkinen and V. Bonačić-Koutecký [22,23,24,25,26,27,28], which set the basics in this field. However, it should be noted that there is an important discipline of cluster and superatom chemistry [29], which have a scope different from nanoparticle chemistry. There is no clear definition of cluster and nanoparticle in size and sometimes used in an interchangeable manner. Though, the former typically consists of a few tens of atoms with the size of 1 nm while the latter is used for systems of a few hundred atoms and more, corresponding to 2 nm and larger. When cluster models consisting of 13 atoms are used to represent a cluster or superatom of the corresponding size, it is real system structure. However, it is obvious that 13 atomic system does not represent the nanoparticle, thus when it is meant to represent nanoparticles, it is not a real system structure, but a simplified structure.

Recent advances in microscopy have enabled the observation of materials at atomic resolution [30, 31], and these experimental observations make it conceptually possible to reconstruct the real system structure and compute its electronic structure. The nanoparticles used in catalysis are typically a few nanometers in size. More than 500 atoms are present in a 3 nm nanoparticle, and more than 1000 atoms are present in a 4 nm nanoparticle. For N computed atoms, the computational cost is proportional to N³ or N⁴ when the density functional theory (DFT) method is used for first-principles calculations [32]. Therefore, calculations of 3–4 nm systems on local computer systems are not practical, and it is necessary to use parallel computing on centralized supercomputer systems. The system size considered in first-principles calculations of nanoparticles in the literature increased dramatically in 2008 when a PtFe nanoparticle with 1289 atoms was reported [33]. Thereafter, first-principles calculations of systems with more than 1000 atoms have been reported [32, 34,35,36,37,38,39]. Nanba et al. [36] realized first-principles calculations of Ru nanoparticles reconstructed from atomic-resolution microscopy observations and discussed the thermodynamic properties of nanoparticles with different shapes to clarify the factors determining the stability of face-centered cubic (fcc) Ru nanoparticles. The stability of Co nanoparticle with up to 1483 atoms was subsequently reported [34], and the stability of a PdPt binary alloy system was also discussed on the basis of 711 atomic models [40]. First-principles calculations of real system nanoparticles are not easy, but they are useful for several purposes. The first is the direct investigation of the thermodynamic stability of nanoparticles, which differs greatly from that of the bulk. Because there is no other effective method of discussing the enthalpy and entropy of nanoparticles, this technique is expected to become a unique and effective approach. Second, the high degree of site heterogeneity is intrinsically incorporated into real system structure models. The calculated electronic structures of vertex and ridge atoms are reportedly much different from those of bulk and low-index surface atoms. The electronic structure and related properties can be obtained by the computation of real system structures. Finally, the calculation results will be indispensable in the data-driven era. As mentioned earlier, data for simplified and ideal models with small sizes are easier to obtain, but they may fail to predict the synthesizability and activity [17]. First-principles calculation data obtained for real system structures will offer a basis for predicting the properties of nanoparticles considering their intrinsic heterogeneous structure. In this article, we discuss the status and prospects of theoretical research based on real system nanoparticles.

2 Stability of Nanoparticles

Owing to their large number of surface atoms, nanoparticles typically have dangling bonds, and thus high surface energy and high reactivity. Although these properties make nanoparticles suitable for use in a wide range of applications, such as electrochemical reactions [41, 42] and heterogeneous catalysis [43, 44], they often result in decreased stability. These properties are clearly size- and structure-dependent; in addition, the size and shape of nanoparticles determine the mixing ratio between surface and core atoms in alloy nanoparticles [45]. Furthermore, to apply the crystalline phase that emerges when a material is nanosized, it is important to understand the thermodynamic phase stability of nanoparticles. All of these factors that affect nanoparticle stability can be captured only by first-principles calculations using real system models. To support this statement, in section 2.1, we discuss in detail the study of Nanba et al. [36], who rigorously showed that the properties of Ru nanoparticles with the most stable structure are still size-dependent.

These observations about the stability of monometallic nanoparticles should also hold for alloy systems. Additionally, the catalytic activity of an alloy depends on its atomic configuration. For example, some elements in an alloy may be vulnerable to dissolution under the operating conditions, and the distribution of elements in the alloy can thus change during operation, degrading the catalytic activity [46, 47]. Depending on the combination of elements, the homogeneous solid-solution (SS) configuration is not necessarily the most stable. Furthermore, nanoalloys resulting from experiments may not necessarily represent the most stable structure, because the more complex energy surface of such alloys results in a slower approach to equilibrium than for monometallic systems [1]. Therefore, it is necessary to study the stability of such systems. In section 2.2, the results of first-principles studies [40, 48] of different binary alloy nanoparticles of PtM, a system used in heterogeneous catalysis, are summarized. These studies provide insights on the stability of a wide range of PtM compositions [42, 49,50,51,52].

2.1 Monometallic Systems

Ru nanoparticles with fcc structure have been reported [30, 53, 54], whereas crystalline Ru is known to have hexagonal close-packed (hcp) structure. Owing to the difference in crystalline structure, the properties of these two materials will differ [30, 55]. Nanba et al. [36] studied the stability of fcc and hcp Ru nanoparticles of various sizes and shapes using models of real system nanoparticles.

Figure 1 shows the relationship between the cohesive energy and the nanoparticle size. As the size of the nanoparticle increases, the cohesive energy decreases, indicating that the system becomes stable. Note that the intercept with the y axis corresponds to the value at an infinite size, i.e., the bulk value. For the bulk structure, hcp Ru has the lowest cohesive energy, indicating that it is the most stable phase of bulk Ru. As the size decreases, the difference in cohesive energy between fcc and hcp Ru nanoparticles also decreases. For small nanoparticles (fewer than 150 atoms), the cohesive energy of icosahedral fcc Ru is the lowest among the investigated nanoparticles. The crossover point for hcp Ru is approximately N = 103. This result can be interpreted as follows. The hcp structure is known to be more stable than fcc structure in the bulk. With decreasing size, the effect of the surface energy becomes significant. The surface of the icosahedral fcc structure consists of the most stable ({111}) facets, with a surface energy of 0.147eV/Ų, whereas that of the hcp structure consists of the most stable ({0001}) and the second-most stable ({$\left\{10\overline{1}0\right\}$}) facets, with surface energies of 0.164 and 0.184eV/Ų, respectively. Thus, the icosahedral fcc structure becomes more stable than the hcp structure as the size decreases. In addition, the presence of a twin boundary in the icosahedral fcc structure also plays a role. A twin boundary generally makes a material less stable. However, the twin boundary energy of the icosahedral fcc structure was calculated to be negative. This result can be understood by considering the local stacking order near the twin boundary, as shown in Figure 2. The stacking of the fcc structure is typically represented as ABCABC, whereas that of the hcp structure is ABAB. The local stacking around the twin boundary is BAB stacking, which is that of the hcp structure. Because Ru with hcp structure is more stable than that with fcc structure, ABAB stacking is more stable than ABCABC stacking; thus, the twin boundary will stabilize the icosahedral fcc structure. Because the twin boundary density increases with decreasing nanoparticle size, the icosahedral fcc structure becomes more stable relative to that of hcp nanoparticles without a twin boundary. Clearly, these effects of surface facets and twin boundaries in nanoparticles can be explored only by first-principles calculations of real system nanoparticles.

Figure 1.

 Size dependence of cohesive energy of Ru nanoparticles. Red squares, blue diamonds, magenta crosses, and green circles represent fivefold twinned decahedral fcc, icosahedral fcc, truncated octahedral fcc, and hcp structures, respectively. The broken lines represent regression lines. The inset shows an enlarged view at N^−1/3 = 0.18−0.24. Note that N represents the number of atoms. [Reprinted with permission from Yusuke Nanba, Takayoshi Ishimoto, Michihisa Koyama. Structural Stability of Ruthenium Nanoparticles: A Density Functional Theory Study. The Journal of Physical Chemistry C 2017, 121 (49), 27445–27452. DOI: 10.1021/acs.jpcc.7b08672. Copyright 2017 American Chemical Society.]

Figure 2.

 Twin boundary models for fcc {111} stacking ∑3. The original fcc {111} stacking model ∑1 is arranged in ABCABC order. [Reprinted with permission from Yusuke Nanba, Takayoshi Ishimoto, Michihisa Koyama. Structural Stability of Ruthenium Nanoparticles: A Density Functional Theory Study. The Journal of Physical Chemistry C 2017, 121 (49), 27445–27452. DOI: 10.1021/acs.jpcc.7b08672. Copyright 2017 American Chemical Society.]

2.2 Binary Alloy Systems

Pd and Pt are known to be immiscible in the bulk phase. Therefore, controlled structure such as core–shell structure is intensively studied to enhance the catalytic activity originating from this unique structure [56,57,58,59,60,61,62]. However, Kobayashi et al. [63] reported that Pd–Pt core–shell nanoparticles form a homogeneous SS structure after hydrogen absorption/desorption. It is thought to be thermodynamically counterintuitive to observe SS structures of immiscible elements. To clarify whether the SS structure is stable or semistable in the nanoparticle phase, Ishimoto and Koyama [40] studied the stability of PdPt nanoparticles using first-principles calculations. Truncated octahedral structure models of particles approximately 3 nm in size with various configurations were prepared, as shown in Figure 3.

Figure 3.

 Side view (top two rows) and cross-sectional view (bottom two rows) of core–shell and SS nanoparticles of Pd₂₀₁Pt₅₁₀, Pt₄₀₅Pd₃₀₆, Pd₄₀₅Pt₃₀₆, and Pt₂₀₁Pd₅₁₀ after geometry optimization. Green and gray balls represent Pd and Pt atoms, respectively. [Reprinted with permission from Takayoshi Ishimoto, Michihisa Koyama. Electronic Structure and Phase Stability of PdPt Nanoparticles. The Journal of Physical Chemistry Letters 2016 7 (5), 736–740. DOI: 10.1021/acs.jpclett.5b02753. Copyright 2016 American Chemical Society.]

The homogeneity of the SS structure was confirmed by determining the short-range order, that is, the Warren–Cowley parameter, $\alpha$ [64].   

$\alpha =1-\frac{P_A^i}{C_A}$(1)

where $C_A$ is the concentration of A atoms, and $P_A^i$ is the conditional probability of an A atom having B atoms as neighbors in the i-th coordination sphere. To discuss the relative stability of nanoparticles, the excess energy, $E_{excess}$, was adopted [65].   

$E_{excess}=\frac{1}{N_{total}}$(2)

(

$E_{AB}-\frac{N_A}{N_{total}}{E}_A-\frac{N_B}{N_{total}}{E}_B$(2)

) where E is the total energy of nanoparticles consisting of N_total atoms. Subscripts AB, A, and B represent the constituent elements of the nanoparticles, and $N_A$ and $N_B$ are the numbers of A and B atoms in the alloy, respectively. Figure 4 (a) shows the excess energies calculated for Pd_711-xPt_x alloy nanoparticles with Pd–Pt core–shell, Pt–Pd core–shell, and SS configurations. The Pt–Pd core–shell configuration is the most stable, followed by the SS and Pd–Pt core–shell configurations. This result can be intuitively interpreted in terms of the surface energy, as the surface energy of Pt is higher than that of Pd. As shown in Figure 4 (a), as the number of Pd atoms at the surface increases, the alloy becomes more stable. To understand the stability at a finite temperature, the temperature dependence of the enthalpy and entropy should be considered. Figure 4 (b) shows the excess free energy at 373 K. When the temperature effect was considered, the stability of the SS configuration became close to or slightly lower than that of the Pt–Pd core–shell configuration. In another work, Ishimoto and Koyama [66] studied the effects of vibration on the enthalpy as well as those of vibration and the configuration on the entropy and found that the configurational entropy is a major factor that should be carefully considered in investigations of the stability of binary alloy nanoparticles.

Figure 4.

 Excess energy of SS and core–shell nanoparticles of Pd₂₀₁Pt₅₁₀, Pt₄₀₅Pd₃₀₆, Pd₄₀₅Pt₃₀₆, and Pt₂₀₁Pd₅₁₀ from Pd₇₁₁ and Pt₇₁₁ nanoparticles (a) without entropy and (b) with entropy correction at 373 K. Calculated values for bulk PdPt SS systems (Pd_0.25Pt_0.75, Pd_0.5Pt_0.5, and Pd_0.75Pt_0.25) are shown as a reference. Cross-sectional views of core–shell nanoparticles are shown. Green and gray balls represent Pd and Pt atoms, respectively. [Reprinted with permission from Takayoshi Ishimoto, Michihisa Koyama. Electronic Structure and Phase Stability of PdPt Nanoparticles. The Journal of Physical Chemistry Letters 2016 7 (5), 736–740. DOI: 10.1021/acs.jpclett.5b02753. Copyright 2016 American Chemical Society.]

As shown in Figure 5, the configurational entropy S_conf is largest for the SS configuration of all Pt₃M nanoparticles, followed by that of the one-skin (S1) and two-skin (S2) configurations. Among the investigated alloy configurations, the S1 configuration is the most stable in all cases. The difference in stability between configurations is the largest for the Pt₃Co nanoparticles, indicating intrinsic stability against structural change during long-term operation. The stability of the S1 configuration is consistent with experimental observations [67]. In summary, first-principles calculations of models of real system nanoparticles can deliver reliable results directly comparable with experimental observations. Incorporating the effect of adsorbates, supports, the external potential, etc., will be essential to further extend the study of real system models.

Figure 5.

 Calculated excess free energy and configurational entropy for different configurations of Pt₃M nanoparticles (M = Co, Ni, and Cu), where SS, S1, and S2 refer to the SS configuration, a one-skin layer configuration, and a two-skin layer configuration, respectively. [Reprinted with permission from David S. Rivera Rocabado, Yusuke Nanba, Michihisa Koyama. Electronic Structure and Phase Stability of Pt₃M (M = Co, Ni, and Cu) Bimetallic Nanoparticles. Computational Material Science 2020 184, 109874-1-9. DOI: 10.1016/j.commatsci.2020.109874. Copyright 2020 Elsevier.]

In addition to alloys of Pt with other metals in the core–shell configuration, other configurations of alloys are being intensively explored with the goal of controlling the electrochemical activity by changing the surface properties [68]. The first-principles study of Rivera Rocabado et al. [48] investigated the stability of bimetallic Pt₃M (M = Co, Ni, and Cu) systems with different configurations by calculating the excess energy and configurational entropy at 353 K for the SS, S1, and S2 configurations (Figure 5).

Finally, supervised learning of the excess energy, E_excess, of nanoparticles with the S1 and S2 configurations revealed that the strain and charge of the Pt atoms in the surface layer are good descriptors. The excess energy values predicted by multiple regression are presented in Figure 6, along with the excess energy values from first-principles calculations. These results would greatly facilitate the computational design of catalysts because they directly indicate the relationships between the electronic properties of a wide range of Pt alloy nanoparticles and their stability, which determines the catalytic activity, as discussed in the next section.

Figure 6.

 Calculated vs. predicted excess energy of Pt₃M nanoparticles with S1 and S2 configurations. [Reprinted with permission from David S. Rivera Rocabado, Yusuke Nanba, Michihisa Koyama. Electronic Structure and Phase Stability of Pt₃M (M = Co, Ni, and Cu) Bimetallic Nanoparticles. Computational Material Science 2020 184, 109874-1-9. DOI: 10.1016/j.commatsci.2020.109874. Copyright 2020 Elsevier.]

3 Electronic Structure

Nanoparticles show unique electronic structure. The activity of metals is often predicted using the d-band center model [69,70,71], which is known as a good descriptor of the catalytic activity associated with the strength of the interaction between the molecules and the catalyst surface. The charge distribution may indicate where reactions tend to occur and provide information about the polarity of the nanoparticles. The dispersion stability of nanoparticles depends strongly on their polarity, which may significantly affect their synthesis from solution [45]. Thus, the electronic structure will also depend on these factors. In this section, the results of first-principles calculations of the electronic structure of monometallic, bimetallic, and supported nanoparticles are presented. Monometallic nanoparticles show potential for a wide range of applications as catalysts [44, 69,70,71,72,73,74]. Among various first-principles studies of the electronic structure of metal nanoparticles such as copper, silver, gold, and platinum nanoparticles [8, 39, 75], here we focus on the studies of Nanba et al. [36] and Rivera Rocabado et al. [35], which are summarized as representative works in section 3.1.

In section 3.2, the electronic structure of PtM nanoparticles (M = Pd, Co, Cu and Ni) is discussed on the basis of the studies of Ishimoto and Koyama [40] and Rivera Rocabado et al. [48]. This type of binary alloy nanoparticle can exhibit ferromagnetic properties [49, 76]; it has been studied to improve the activity of the oxygen reduction reaction [77, 78] and for use as an anode catalyst in direct alcohol fuel cells [41, 79] or as a hydrogen anode [80]. The interested reader is also referred to the first-principles study of transition metal alloy nanoparticles by Roling et al. [81], the combined molecular dynamics/DFT study of the electronic structure of PtPd binary nanoparticles by Chepkasov et al. [82], the combined experimental and first-principles study of real IrCu nanoparticles by Wang et al. [9] and the review article of Ferrando et al. [1] comprising early experimental and theoretical studies for binary clusters and nanoparticles.

Solid catalysts based on nanoparticles are generally embedded in a support or carrier [83]. As mentioned above, the catalytic activity depends on the electronic properties as well as on the shape/size of the nanoparticles. In addition, the support strongly affects the electronic properties and thus the catalytic activity of the nanoparticles [84]. Various first-principles studies have been reported, for example, that of Pt clusters supported on SrTiO₃ by Qureshi et al. [85] and the more recent study of Au nanoparticles supported on ZnO by Hung and McKenna [86]. In section 3.3, we present a first-principles study of the electronic properties of Pt nanoparticles supported on SnO₂ by Rivera Rocabado et al. [35], because this material is a possible candidate to replace carbon as a Pt support in low-temperature fuel cells [87].

3.1 Monometals

Nanba et al. [36] discussed the dependence of the electronic properties of Ru nanoparticles on their size and structure. To perform a detailed analysis, the atomic domain of the nanoparticles is divided into sets of atoms on the surface (vertex, ridge, or facet), within the subsurface, and in the interior. Figure 7 shows the values of the d-band center calculated by Nanba et al. [36] for those regions and the nanoparticle structure as a function of nanoparticle size. The calculated values for the interior of the nanoparticles are similar to the bulk values; thus, the electronic structure may also be similar. By contrast, the atoms within the subsurface show a lower d-band center compared to the atoms in the interior, and the atoms at all the surfaces show a higher d-band center (the highest value is found for atoms on the vertex, followed by those for atoms on the ridge and on the facet). This order reflects the fact that the coordination number increases in the order vertex, ridge, facet. The differences could result in different catalytic properties depending on the local position of a nanoparticle. All of the Ru nanoparticle structures exhibit these trends. Moreover, the icosahedral fcc {111} facet has less-negative values of the d-band center than the other facets. This finding can be explained by the long interatomic distance in this surface layer. Only a real system calculation can capture these effects, which affect the catalytic properties of nanoparticles but cannot be explored using simulations based on bulk or slab models without considering the local geometry.

Figure 7.

 Results of first-principles calculations of d-band centers in different Ru nanoparticle structures: (a) decahedral fcc, (b) icosahedral fcc, (c) truncated octahedral fcc, and (d) hcp) and for different numbers of atoms (given as $N^{-1/3}$). [Reprinted with permission from Yusuke Nanba, Takayoshi Ishimoto, Michihisa Koyama. Structural Stability of Ruthenium Nanoparticles: A Density Functional Theory Study. The Journal of Physical Chemistry C 2017, 121 (49), 27445–27452. DOI: 10.1021/acs.jpcc.7b08672. Copyright 2017 American Chemical Society.]

Furthermore, the results suggest that local charge transfer from the subsurface layer (positively charged atoms) to the surface layer (negatively charged atoms) occurs in Ru nanoparticles. The charge was more negative for atoms with lower coordination numbers. The increase in charge transfer compensates for the lower coordination number at the surface [36]. These characteristics were also observed in Pd and Pt nanoparticles by Ishimoto and Koyama [40].

Rivera Rocabado et al. [35] studied the size dependence of the density of states (DOS) of Pt nanoparticles. As shown in Figure 8(a), small Pt nanoparticles such as Pt₄ or Pt₁₃ exhibit molecular characteristics; i.e., their DOSs show more discrete energy levels owing to the limited number of atoms. As the number of atoms in the nanoparticles increases, the DOS becomes more continuous owing to orbital overlapping; for large particles ($N > 711$), the shape of the DOS does not change substantially with the number of atoms, as shown in Figure 8(b). Finally, Figure 8(c) shows the DOS profiles of the outer-shell atoms of different Pt nanoparticles. For smaller nanoparticles, the DOS profiles of the outer-shell atoms resemble that of the entire nanoparticle, but they converge to the Pt (111) profile for larger nanoparticles because the {111} site is dominant.

Figure 8.

 DOS of Pt nanoparticles. (a) Partial DOS, (b) difference in partial DOS between contiguous nanoparticles, and (c) partial DOS of atoms in the outer shell of Pt nanoparticles. [Reprinted with permission from David S. Rivera Rocabado, Takayoshi Ishimoto, Michihisa Koyama. The Effect of SnO₂(110) Supports on the Geometrical and Electronic Properties of Platinum Nanoparticles. SN Applied Sciences 2019 1, 1485. DOI: 10.1007/s42452-019-1478-0. Copyright 2019 Springer. CC by 4.0.]

The d-band center of the entire nanoparticle and that of the outer shell as a function of nanoparticle size was also studied (Figure 9). The results shows that with increasing nanoparticle size, the d-band center decreases in both cases, converging to the d-band center of bulk Pt for the entire nanoparticle and to the d-band center of Pt(111) for the outer shell, as expected from the discussion of the DOS. Only the d-band centers of the outer shells of Pt₂₀₁ and Pt₄₀₅ were slightly lower than the values for Pt(111). Much more detailed investigations are necessary. As observed for Ru nanoparticles, Pt nanoparticles also showed negatively charged atoms in the outer shell, and the charges were more negative when the d-band center was lower.

Figure 9.

 (a) Size dependence of d-band center of Pt nanoparticles (gray circles) and of the d-band center of only the outer-shell atoms (blue squares). The values for bulk Pt (dashed purple line) and Pt (111) (dashed orange line) are shown for reference. (b) Relationship between the d-band center and the charge per atom of the outer-shell atoms of the nanoparticles. The linear regression line (dashed black line) and coefficient of determination are also shown. [Reprinted with permission from David S. Rivera Rocabado, Takayoshi Ishimoto, Michihisa Koyama. The Effect of SnO₂(110) Supports on the Geometrical and Electronic Properties of Platinum Nanoparticles. SN Applied Sciences 2019 1, 1485. DOI: 10.1007/s42452-019-1478-0. Copyright 2019 Springer. CC by 4.0.]

3.2 Binary Alloys

Ishimoto and Koyama [40] analyzed the electronic structure of PdPt nanoparticles. The d-band centers obtained using the calculated total DOS and partial DOS of Pd and Pt are shown in Figure 10 for Pt–Pd core shell and SS nanoparticles (see Figure 3 in section 2.2.) The d-band center of the total DOS of all types of nanoparticles has a linear relationship with the Pd concentration; the d-band center increases from pure Pt₇₁₁ to pure Pd₇₁₁ nanoparticles. A strong dependence is not observed for the d-band centers of Pd and Pt in the SS nanoparticles, but it is observed in the core–shell nanoparticles, where the d-band centers of the Pt₄₀₅Pd₃₀₆ and Pt₂₀₁Pd₅₀₁ nanoparticles are deeper than those of pure Pt and Pd nanoparticles.

Figure 10.

 Calculated d-band centers from total and partial DOS in core–shell and SS PdPt nanoparticles. [Reprinted with permission from Takayoshi Ishimoto, Michihisa Koyama. Electronic Structure and Phase Stability of PdPt Nanoparticles. The Journal of Physical Chemistry Letters 2016 7 (5), 736–740. DOI: 10.1021/acs.jpclett.5b02753. Copyright 2016 American Chemical Society.]

Figure 11 shows the average charge in each layer of various PdPt nanoparticles. The atomic charges in the inner layers are approximately zero for the SS nanoparticles and thus are similar to those in the bulk. Furthermore, the atoms in the subsurface (layer 2) are positively charged, and the surface atoms (layer 1) are negatively charged. This charge transfer from subsurface to surface atoms was also observed in monometallic nanoparticles (section 3.1). However, the charge transfer behavior is different in core–shell nanoparticles. For SS Pd₄₀₅Pt₃₀₆ nanoparticles, the absolute values of the positive charge at the subsurface and the negative charge at the surface are larger than those for monometallic nanoparticles. By contrast, the atomic charges on the subsurface and surface are almost neutral for core–shell Pt₄₀₅Pd₃₀₆. The mechanism of charge transfer from the subsurface to the surface owing to the lower coordination number at the surface (section 3.1) is strengthened (or weakened) by charge transfer from Pd to Pt, as expected from the work function values. Because Pd has a smaller work function (5.12 eV) than Pt (5.65 eV) [88], charge transfer may occur from Pd to Pt. These findings reveal one of the most remarkable differences between monometallic and bimetallic nanoparticles.

Figure 11.

 Calculated average atomic charge of each layer in SS and core–shell Pd₂₀₁Pt₅₁₀, Pt₄₀₅Pd₃₀₆, Pd₄₀₅Pt₃₀₆, and Pt₂₀₁Pd₅₁₀ nanoparticles. Pd and Pt appear as gray and green, respectively in the cross-sectional view of each nanoparticle. [Reprinted with permission from Takayoshi Ishimoto, Michihisa Koyama. Electronic Structure and Phase Stability of PdPt Nanoparticles. The Journal of Physical Chemistry Letters 2016 7 (5), 736–740. DOI: 10.1021/acs.jpclett.5b02753. Copyright 2016 American Chemical Society.]

It is also worth discussing the electronic structure of a wider range of bimetallic Pt₃M (M = Cu, Ni, and Co) nanoparticles with the S1, S2, and SS configurations, which was studied by Rivera Rocabado et al. [48] (section 2.2, Figure 5). Here, the d-band center of surface Pt atoms calculated from the DOS is lower than that of monometallic Pt nanoparticles, except for Pt₃Cu nanoparticles with the SS configuration (Figure 12).

Figure 12.

 Calculated d-band centers of surface Pt atoms in monometallic Pt nanoparticles and in S1 (diamonds), S2 (crosses), and SS (circles) Pt₃M nanoparticles. [Reprinted with permission from David S. Rivera Rocabado, Yusuke Nanba, Michihisa Koyama. Electronic Structure and Phase Stability of Pt₃M (M = Co, Ni, and Cu) Bimetallic Nanoparticles. Computational Material Science 2020 184, 109874-1-9. DOI: 10.1016/j.commatsci.2020.109874. Copyright 2020 Elsevier.)

For the S1 configuration (the most stable configuration), the atoms on the surface of the Pt₃Co nanoparticles have the lowest d-band centers, followed by those on the Pt₃Ni and Pt₃Cu nanoparticles. This tendency is consistent with other theoretical studies of Pt₃Co and Pt₃Ni [89, 90], whereas the calculated values are different from those in previous studies because a different modeling approach (slab vs. isolated nanoparticle) was used. Moreover, the experimental values of the d-band center [91] of Pt₃Co and Pt₃Ni with the S1 configuration are in good agreement with the values mentioned here.

An analysis of the surface atom charges [48] revealed that all bimetallic Pt nanoparticles show lower charges than monometallic Pt nanoparticles. These results are consistent with the work function difference, where charge transfer occurs from Co, Ni, or Cu (lower values) to Pt (higher values). Moreover, the SS nanoparticles show the lowest surface atom charge, followed by the S1 and S2 nanoparticles. For the S1 and SS configurations, the order of the atomic charges is Pt₃Co, Pt₃Ni, and Pt₃Cu (from lowest to highest). For the S2 configuration, all the values are similar, and they are only slightly lower than the surface atom charge of monometallic Pt nanoparticles.

From the analysis of the electronic structure, one can argue that Pt₃M nanoparticles with the S1 configuration are more active toward the oxygen reduction reaction than Pt nanoparticles because of the lower surface d-band center and more negatively charged surface atoms. Both mechanisms together weaken the interaction of nucleophilic species and thus improve the oxygen reduction reaction kinetics of the nanoparticles. However, Pt₃Cu nanoparticles with the SS configuration are more likely to interact with nucleophilic species than Pt nanoparticles owing to the upshift in the d-band center of the outer-shell Pt atoms.

3.3 Supported Systems

Rivera Rocabado et al. [35] modeled SnO₂-supported Pt nanoparticles on the basis of the experimentally observed configurations of Takasaki et al. [87], although the Pt(111) plane in the nanoparticles acts parallel to the SnO₂(110) plane (Figure 13). The size of the nanoparticles ranges from Pt₄ to Pt₂₃₃.

Figure 13.

 Models of supported Pt nanoparticles. [Reprinted with permission from David S. Rivera Rocabado, Takayoshi Ishimoto, Michihisa Koyama. The Effect of SnO₂(110) Supports on the Geometrical and Electronic Properties of Platinum Nanoparticles. SN Applied Sciences 2019 1, 1485. DOI: 10.1007/s42452-019-1478-0. Copyright 2019 Springer. CC by 4.0.]

The electron distributions within supported Pt nanoparticles differ from those in isolated Pt nanoparticles. Owing to the work function difference between the support and the nanoparticles, electron transfer occurs from Pt to SnO₂. Furthermore, most of the electrons are redistributed at the nanoparticle–support interface, as indicated by the charge distribution in Figure 14. The charge per atom in the outer shell becomes moderate as the number of atoms in the nanoparticles increases. The top atomic layer is negatively charged, except in Pt₁₃. Moreover, the average charge per atom of the atoms in the outer shell ranges from 0.19 e/atom (for Pt₄) to −0.008 e/atom (for Pt₂₃₃). The charge of the outer-shell atoms of SnO₂-supported nanoparticles decreases with increasing nanoparticle size. However, it does not become as negative as that of the outer-shell atoms of isolated Pt nanoparticles. This behavior may increase the ability to interact with nucleophilic species and thus result in slower oxygen reduction reaction kinetics. This low activity was observed experimentally in SnO₂-supported Pt nanoparticles 2 nm in size [92].

Figure 14.

 Top: charge distribution of Pt atoms within nanoparticles. Bottom: average charge per atom for the outer-shell atoms in different layers within isolated Pt nanoparticles and SnO₂-supported Pt nanoparticles. [Reprinted with permission from David S. Rivera Rocabado, Takayoshi Ishimoto, Michihisa Koyama. The Effect of SnO₂(110) Supports on the Geometrical and Electronic Properties of Platinum Nanoparticles. SN Applied Sciences 2019 1, 1485. DOI: 10.1007/s42452-019-1478-0. Copyright 2019 Springer. CC by 4.0.]

The d-band centers of all the outer-shell Pt atoms in the supported nanoparticles are downshifted compared to those of the outer-shell atoms of isolated Pt nanoparticles, where the highest downshift (0.305 eV) appears in supported Pt₁₃. The downshift of the d-band center decreases with increasing particle size, with a minimum of only 0.007 eV for Pt₂₃₃. At the interface, the d-band center increases with increasing number of atoms, from 0.672 eV for Pt₁₃ to 0.083 eV for Pt₁₁₉ and Pt₂₃₃. By contrast, the first value for Pt₁₃ is downshifted, and the values for Pt₁₁₉ and Pt₂₃₃ are upshifted with respect to those of the outer-shell atoms of isolated Pt nanoparticles. The downshift changes to an upshift between Pt₁₃ and Pt₃₇. These results are inconsistent with the expected correlation between the atomic charges and the d-band center according to the Hammer–Norskov model [93, 94]. Although the outer-shell atoms of the supported Pt₁₃ nanoparticles are positively charged with respect to those of isolated Pt nanoparticles, Pt₃₇ to Pt₂₃₃ nanoparticles are more likely to interact with nucleophilic species than isolated Pt nanoparticle, as indicated by the charge distribution, even though the d-band centers of the atoms at the surface and the layers between the top and the Pt/SnO₂ interface are downshifted.

As mentioned above, the key factor affecting the charge distribution and charge transfer between the nanoparticles and the support is the work function. The work function of Pt is5.65 eV [88], and that of SnO₂ is 7.08 eV. To support this claim, the calculated average charge of the outer-shell Pt atoms in a graphene-supported Pt₃₇ nanoparticle was compared to the average charges of the outer-shell atoms of a SnO₂-supported Pt₃₇ nanoparticle and an isolated Pt₃₇ nanoparticle, as shown in Figure 15(b). A graphene support was chosen for comparison [95]. The charges of the Pt outer-shell atoms in the graphene-supported nanoparticle at the triple-phase boundary are negative, indicating that charge is transferred from the support to the nanoparticle, which differs from the charge transfer in the SnO₂-supported nanoparticle.

Figure 15.

 (a) Charge variation for a supported single Pt atom and Pt atoms in a supported Pt₄ nanoparticle on different support materials (SnO₂, SnO_2-δ, and Sn_0.98Nb_0.02O₂). (b) Average charges of outer-shell Pt atoms in SnO₂-supported Pt₃₇ nanoparticle, graphene-supported Pt₃₇ nanoparticle, and isolated Pt₃₇ nanoparticle. [Reprinted with permission from David S. Rivera Rocabado, Takayoshi Ishimoto, Michihisa Koyama. The Effect of SnO₂(110) Supports on the Geometrical and Electronic Properties of Platinum Nanoparticles. SN Applied Sciences 2019 1, 1485. DOI: 10.1007/s42452-019-1478-0. Copyright 2019 Springer. CC by 4.0.]

It has been shown experimentally that Pt nanoparticles supported on reduced SnO₂ [92] or Nb-doped SnO₂ [87] have higher activity than Pt/SnO₂. This finding is consistent with the work function values, which range from 6.14 eV for SnO_1.98 to 4.58 eV for SnO_1.85 and 6.39 eV for Sn_0.98Nb_0.02O₂; these values are lower than that of SnO₂ (7.08 eV) [35]. Figure 15(a) shows the charges for a single Pt atom and for the atoms of a Pt₄ nanoparticle supported on SnO₂, reduced SnO₂, and Sn_0.98Nb_0.02O₂. Note that reduced and Nb-doped SnO₂ models are nonstoichiometric; i.e. the models are prepared by introducing oxygen vacancy by removing oxygen from SnO₂ and Sn_0.98Nb_0.02O_2.01, respectively. The charges of Pt and Pt₄ on the SnO₂ and Nb-doped SnO₂ are positive, whereas that of Pt on Sn_0.98Nb_0.02O₂ is lower than those on SnO₂; this result can be understood in terms of the work functions of the support material. As the number of oxygen vacancies in the reduced SnO₂ support increases, the Pt charge becomes smaller until it becomes highly negative, which is consistent with the work function of SnO_2-δ. The high activity of Nb-doped SnO₂ may result from the presence of oxygen vacancies, which may reduce the work function and improve charge transfer from the support to Pt nanoparticles. Similar results were obtained by Binninger et al. [96] using DFT calculations for Sb-doped SnO₂-supported Pt nanoparticles, where only the presence of oxygen vacancies enables charge transfer from the support to Pt.

Finally, Rivera Rocabado et al. [35] evaluated the results of studies based on real system models by listing many other findings that are consistent with reported experimental observations [87, 92, 97,98,99,100].

4 Properties of Nanoparticles

In this article, we emphasize the importance of studying the stability and electronic structure of nanoparticles. The interactions of nanoparticles with small molecules also make important contributions to their catalytic properties. In this section, first-principles studies of hydrogen absorption in real system nanoparticles are first discussed. As stated by Greeley and Mavrikakis [101], hydrogen absorption is crucial to energy storage and production; in addition, hydrogen is a common element in the molecules used for all possible catalytic applications. For this purpose, the study of Ishimoto and Koyama [102] is summarized as a representative example in section 4.1.

The catalytic activity of nanoparticles is strongly related to the adsorption of small molecules, as shown by many first-principles studies [35, 103,104,105,106]. We are aware of various rigorous first-principles studies of the adsorption of small molecules such as CO or NO_x on real system nanoparticles [2, 6, 7, 10]. However, we believe that the study of Nanba and Koyama [107] is representative of such studies because the authors thoroughly analyze the adsorption of NO on different sites on Rh₄₀₅, Pd₄₀₅, Pt₄₀₅, Ag_405, and Ir₄₀₅ nanoparticles. This study is discussed in detail in section 4.2.

4.1 Hydrogen Absorption

Ishimoto and Koyama [102] investigated H absorption at the octahedral or tetrahedral site of a Pd₄₀₅ nanoparticle with truncated octahedral structure (Figure 16) considering the unique nanoparticle properties observed experimentally [108,109,110,111]. After geometrical optimization, a volume expansion of 16% at the tetrahedral sites and 5% at the octahedral sites was observed, whereas the sites near the (111) surface showed greater expansion because of hydrogen absorption. Figure 17 shows the absorption energy of hydrogen on different octahedral and tetrahedral sites. The absorption energy is calculated from the total energy $E\left({\text{Pd}}_{405}\text{H}\right)$ and the energies of the Pd₄₀₅ nanoparticle $E\left({\text{Pd}}_{405}\right)$ and a hydrogen molecule H₂$E\left({\text{H}}_2\right)$ as follows:   

$E_{abs}=E\left({\text{Pd}}_{405}\text{H}\right)-\left(E\left({\text{Pd}}_{405}\right)+\frac{1}{2}E\left({\text{H}}_2\right)\right)$(3)

Figure 16.

 Distribution of nearest-neighbor interatomic distance in a real Pd₄₀₅ model (yellow: distribution on the surface, blue: distribution in the interlayer between surface and subsurface layers). (Reproduced with the permission of AIP Publishing from Takayoshi Ishimoto, Michihisa Koyama. Theoretical Study of Tetrahedral Site Occupation by Hydrogen in Pd Nanoparticles. The Journal of Chemical Physics 2018 148, 034705. DOI: 10.1063/1.5005976.)

Figure 17.

 Absorption energy of hydrogen on (a) octahedral and (b) tetrahedral sites in a Pd₄₀₅ nanoparticle with and without ZPE correction. (Reproduced with the permission of AIP Publishing from Takayoshi Ishimoto, Michihisa Koyama. Theoretical Study of Tetrahedral Site Occupation by Hydrogen in Pd Nanoparticles. The Journal of Chemical Physics 2018 148, 034705. DOI: 10.1063/1.5005976.)

The energy was corrected using only the zero-point energy (ZPE) correction of hydrogen atoms embedded in tetrahedral or octahedral sites consisting of four or six Pd atoms by partial vibrational frequency analysis (Figure 17).

At the octahedral sites, the hydrogen absorption energy in the core region (2 to 5 Å) is −0.12 eV, which is close to that of bulk Pd. Beyond the core region, near the (100) surface (6 to 7 Å), the absorption energy becomes less negative; i.e., the absorption becomes less stable. The hydrogen absorption becomes much more stable at 8 to 10 Å, where the ZPE correction is less significant than that in other regions. The local differences in absorption stability originate from the hydrogen position. In the core region, hydrogen is located at the center of mass of the octahedral site, but it is displaced toward the surface because of the difference in atomic charge distribution between the surface and subsurface, which results in an asymmetric potential at the octahedral and tetrahedral sites. This change in position also affects the ZPE.

At the tetrahedral sites, hydrogen absorption is unstable from the core region to the (100) surface region, i.e., from 2 to 7 Å, and is most unstable in the (100) surface region. Furthermore, hydrogen absorption is stable only near the (111) surface (8 to 10 Å). This stability can be explained by the volume expansion of the (111) surface in Pd₄₀₅ and the distance of the hydrogen position from the center of mass of the site. Stable absorption states for hydrogen near the (111) surface of Pd nanoparticles are supported by experimental observations [110].

These results indicate that the activation barrier to hydrogen diffusion from octahedral to tetrahedral sites is larger than that from tetrahedral to octahedral sites. Kofu et al. [111] experimentally observed only hydrogen diffusion from octahedral to octahedral sites via tetrahedral sites in bulk Pd (Figure 18). The calculated hydrogen absorption energies near the core region at the octahedral and tetrahedral sites agree well with this experimental observation. Furthermore, the activation energy of hydrogen diffusion from octahedral to tetrahedral sites is expected to be lower near the subsurface region of the Pd nanoparticles than in the core region, as shown in Figure 18; thus, the diffusion mechanism is expected to be different. This experimental observation [112] is consistent with the hydrogen absorption energy on Pd₄₀₅, whereas shorter distances between stably adsorbed hydrogen atoms in neighboring octahedral and tetrahedral sites are reported, which may result in a lower activation energy. Nevertheless, more rigorous studies are needed that take account of, e.g., surface hydrogen adsorption, multiple hydrogen absorption, and transition states.

Figure 18.

 Hydrogen diffusion mechanism in core and subsurface regions in Pd nanoparticles. (Reproduced with the permission of AIP Publishing from Takayoshi Ishimoto, Michihisa Koyama. Theoretical Study of Tetrahedral Site Occupation by Hydrogen in Pd Nanoparticles. The Journal of Chemical Physics 2018 148, 034705. DOI: 10.1063/1.5005976.)

4.2 Adsorption of Small Molecules

Nanba and Koyama [107] analyzed the adsorption of NO on M₄₀₅ (M = Rh, Pd, Ag, Ir, and Pt) real system nanoparticles. The adsorption sites are defined as the spatial adsorption locations on the nanoparticle surface. In addition, the adsorption configuration is distinguished from the adsorption sites using the coordination number of the adsorbed NO molecule. Considering the symmetry, 33 of the 1094 adsorption sites in the M₄₀₅ nanoparticles were investigated (Figure 19).

Figure 19.

 (a) Top, (b) bridge, and (c) hollow adsorption sites on M₄₀₅ nanoparticles. The (100) and (111) facets are represented by red squares and green hexagons, respectively. Orange symbols: adsorption sites on ridge (T_1x, B_1x); red symbols: adsorption sites on (100) facet (T_2x, B_2x, H_2x); green symbols: adsorption sites on (100) facet (T_2x, B_2x, H_2x). [Reprinted with permission from Yusuke Nanba, Michihisa Koyama. NO Adsorption on 4d and 5d Transition-Metal (Rh, Pd, Ag, Ir, and Pt) Nanoparticles: Density Functional Theory Study and Supervised Learning. The Journal of Physical Chemistry C 2019 123 (46), 28114–28122. DOI: 10.1021/acs.jpcc.9b05748). Copyright 2019 American Chemical Society.]

The adsorption energy on different metal nanoparticles was found to exhibit different features; the common characteristics are described as follows. The adsorption energy on the facet in the nanoparticle model agrees with that of slab models. However, the adsorption energy on the ridge generally differs from that on the facet.

The correlation between the d-band center and the adsorption energy of the top sites in Rh₄₀₅ nanoparticles reportedly has a coefficient of determination of 0.959. However, it is not possible to correlate the NO adsorption energies of different sites with the d-band centers of all the M₄₀₅ nanoparticles, as shown in Figure 20. Therefore, a multiple regression analysis was conducted. To obtain the highest coefficient of determination, four descriptors were defined on the basis of the average d-band center, the standard deviation of the d-band center, the number of metal atoms coordinated to the adsorbed NO molecule, the average interatomic distance, and the generalized coordination number (GCN). Figure 21 shows the adsorption energy predicted by multiple linear regression. All the values have a predictive squared correlation coefficient $R_{LOO}^2$ higher than 0.7, except for that of Pt_405.

Figure 20.

 NO adsorption energy at different sites for different facets of (a) Rh₄₀₅, (b) Pd₄₀₅, (c) Ag₄₀₅, (d) Ir₄₀₅, and (e) Pt₄₀₅ nanoparticles. The coefficient of determination $R^2$ and mean absolute error $\sigma$ are given for a linear regression of the dependence on the d-band center (black lines). [Reprinted with permission from Yusuke Nanba, Michihisa Koyama. NO Adsorption on 4d and 5d Transition-Metal (Rh, Pd, Ag, Ir, and Pt) Nanoparticles: Density Functional Theory Study and Supervised Learning. The Journal of Physical Chemistry C 2019 123 (46), 28114–28122. DOI: 10.1021/acs.jpcc.9b05748. Copyright 2019 American Chemical Society.]

Figure 21.

 Comparison of NO adsorption energies predicted by multiple regression analysis (SL) and those calculated by DFT for M₄₀₅ nanoparticles, where M = (a) Rh, (b) Pd, (c) Ag, (d) Ir, and (e) Pt. $R_{LOO}^2$ is the predictive squared correlation coefficient, and $\sigma$ is the mean absolute error. [Reprinted with permission from Yusuke Nanba, Michihisa Koyama. NO Adsorption on 4d and 5d Transition-Metal (Rh, Pd, Ag, Ir, and Pt) Nanoparticles: Density Functional Theory Study and Supervised Learning. The Journal of Physical Chemistry C 2019 123 (46), 28114–28122. DOI: 10.1021/acs.jpcc.9b05748. Copyright 2019 American Chemical Society.]

The first descriptor is given by the average of the d-band center, and it shows the highest standard partial regression coefficient, indicating that the d-band center is important for describing the NO adsorption. However, the standard partial regression coefficient of the third descriptor, which is related to the average interatomic distance and the number of atoms, is as high as 70% of that of the first descriptor. This descriptor may capture the effects of direct orbital interactions, which depend on the interatomic distance. The fourth descriptor, which is given by the d-band center divided by the GCN, captures the effect of adsorption strength on metal–metal interaction within the nanoparticle. These findings could be applicable to NO adsorption on other monometallic nanoparticles [107].

5 Future Perspectives

We have summarized current progress in first-principles calculations of real system nanoparticles. Owing to their structures, nanoparticles show high reactivity, but this reactivity can decrease their stability. We showed throughout this article that first-principles calculations based on real systems can provide insights into these peculiarities. We discussed the current literature on Ru nanoparticles, where the most stable structure changes from hcp to icosahedral fcc as the particle size decreases owing to the impact of surface facets and twin boundaries. In addition to discussing monometallic particles, we also provided representative results from the literature regarding the stability of bimetallic particles, especially PtM compounds (M = Pd, Co, Cu, and Ni). We also discussed the correlation between the nanoparticle stability and electronic properties, which may open a path to the computational design of catalysts.

If nanoparticles are to be used as catalysts, the DOS and charge distribution should be studied, because these properties affect the reactivity of nanoparticles. We presented results from studies of Ru nanoparticles showing that the catalytic properties may depend on the nanoparticle shape. Furthermore, we discussed findings that show a correlation between the d-band center of monometallic nanoparticles and the charge distribution in the outer shell. We also summarized results for bimetallic Pt nanoparticles; notably, it has been found that Pt₃Cu nanoparticles are more likely to interact with nucleophilic species than with Pt nanoparticles. It was also shown that real system calculations for supported nanoparticles deliver results that are in good agreement with experimental observations.

Recent studies of hydrogen absorption on Pd₄₀₅ nanoparticles were also discussed, where it has been shown that the activation barrier for hydrogen diffusion can depend on the diffusion path within the nanoparticle. Finally, we discussed results showing that the adsorption energy of small molecules can be correlated with various structural and electronic properties of the nanoparticles via multiple linear regression analysis, for example, reports on the NO adsorption by M₄₀₅ nanoparticles (M = Rh, Pd, Ag, Ir, and Pt). All of these literature findings demonstrate the advantages of first-principles calculations of nanoparticles using real system models over conventional bulk or slab calculations.

There are many further possibilities for research in this young field. The study of the size and structure dependence of the properties of monometallic nanoparticles can be expanded to the study of other relevant properties such as the ferromagnetic properties (for example, the Curie temperature) or thermal properties (for example, the heat conductivity or heat capacity), which are important for the application of nanoparticles beyond the use of their catalytic properties, for example, their use in magnetic materials [113] or nanofluids [114, 115].

Further study of metallic nanoparticles with more than two components is necessary. As we discussed in section 2.2, it has not always been possible to identify trends in the stability of different PtM nanoparticle configurations. The analysis of trimetallic Pt nanoparticles may address this limitation. In addition, the investigation of metallic nanoparticles with more than two components is of the highest interest because such alloy nanoparticles may show other thermophysical properties that would be suitable for further study. These statements also apply to supported nanoparticles, where many combinations have not yet been investigated.

In this context, high-entropy alloy (HEA) PtM nanoparticles are very promising owing to their high thermal and chemical stability [116]. According to the stability studies of PtM nanoparticles reported in this article, the contribution of the free excess energy is much higher than that of the entropy; thus, the free excess energy has been used to quantify the nanoparticle stability. However, HEA nanoparticles consisting of five or more elements mixed almost equiatomically from 5 to 35 mol%, depending on the number of constituents, exhibit a configurational entropy that may make a larger contribution than the free excess energy [117]. Moreover, Kusada et al. [118] pointed out the importance of calculating the DOS profile of such new compounds, especially the local DOS on the surface, to study the catalytic activity. The calculation of the DOS would make it possible to conduct DOS engineering [119], where the shape of the DOS and the position of the d-band center can be precisely controlled by combining different elements. The experimental studies of Wu et al. [116, 117], who provided scanning tunneling electron microscopy images of synthetized HEA nanoparticles, are an invitation to develop real system models that can be used in first-principles calculations to study their stability and electronic structure.

Beyond metallic nanoparticles, it may also be worth extending these studies to carbon-based, ceramic, semiconductor, polymeric, and lipid-based nanoparticles owing to their wide range of applicability [120]. For example, the features of metal oxide nanoparticles may differ from those of metallic nanoparticles. ZnO nanoparticles can have hexagonal wurtzite structure with polar atom sites having one to three dangling bonds [121], which would endow it with distinctive catalytic properties that, to our knowledge, have not yet been studied using first-principles calculations with real system models.

The study of the adsorption properties can be extended to account for more than one adsorbed molecule. The difference in the adsorption energies in different adsorption configurations may provide insights about the effects of other fluid molecules on the diffusion path along the surface, which, as discussed in section 4.1, is poorly known for hydrogen adsorption on Pd nanoparticles. Larger molecules can also be investigated, as Šulce et al. [122] did for β-keto esters on Pt nanoparticles. The results of this study were in excellent agreement with experimental observations overall, even though they used a surface to model the Pt nanoparticles. Such rigorous investigations can be complemented/extended by using real system models of (supported) nanoparticles.

In this article, we presented literature correlations that predict the stability and adsorption energy using supervised learning, where both geometrical and electronic properties are used as descriptors. Other machine learning methods, such as artificial neural networks [123, 124] or gradient-boosting regression [125], could be tested alongside other descriptor combinations. Jinnouchi and Asahi [126] showed that descriptors should be defined as local properties, especially in studies of catalytic properties. These types of approaches can be extended to study hydrogen absorption and small molecule adsorption. Finally, it will be of great interest to examine HEA nanoparticles using such machine learning approaches, for example, finding appropriate descriptors to identify correlations with the configurational entropy. In this case, genetic algorithms could also be used to identify the minimum-energy structures [125,126,127]

Beyond first-principles calculations, it is also desirable to extract information from the results to build the force field potential for use in molecular dynamics or Monte Carlo simulations of systems with large numbers of atoms. These simulations are necessary to determine the reaction and diffusion paths of molecules during catalysis or other processes. Despite advances in the development of interfacial force fields [128, 129], most of these fields are based on surfaces and not on real system nanoparticles. On the basis of the findings summarized here, a new type of interfacial force field for the description of solvents and nanoparticles should be developed to account for the different atomic sites originating from the nanosizing of materials. This can be done using the results of first-principles calculations for real systems that distinguish the different atomic sites in the force field model. Machine-learning potentials based on these first-principles calculations may also be helpful to model and accurately predict nanoparticle growth in solution.

Advances in quantum computing will provide an alternative method of performing more demanding calculations, which would be advantageous for the computation of real systems of larger nanoparticles. As is the case for all first-principles calculations, considerable work is still needed to develop computational methods that may be applicable to quantum computers. However, these computational methods should also be developed toward the computational design of materials, which may involve new machine learning algorithms that seek the most stable nanoparticle configuration with specific electronic, mechanical, or thermodynamic properties.

Acknowledgments

Part of the research was supported by ACCEL, JST (JPMJAC1501) and JSPS KAKENHI Grant Number 20H05623. The computation was performed using ITO at the facilities of the Research Institute for Information Technology, Kyushu University and MASAMUNE-IMR at Center for Computational Materials Science, Institute for Materials Research, Tohoku University.

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