σ-Holes on Transition Metal Nanoclusters and Their Influence on the Local Lewis Acidity

Abstract

Understanding the molecular interaction behavior of transition metal nanoclusters lies at the heart of their efficient use in, e.g., heterogeneous catalysis, medical therapy and solar energy harvesting. For this purpose, we have evaluated the applicability of the surface electrostatic potential [V_S(r)] and the local surface electron attachment energy [E_S(r)] properties for characterizing the local Lewis acidity of a series of low-energy TM₁₃ transition metal nanoclusters (TM = Au, Cu, Ru, Rh, Pd, Ir, Pt, Co), including also Pt₇Cu₆. The clusters have been studied using hybrid Kohn–Sham density functional theory (DFT) calculations. The V_S(r) and E_S(r), evaluated at 0.001 a.u. isodensity contours, are used to analyze the interactions with H₂O. We find that the maxima of V_S(r), σ-holes, are either localized or diffuse. This is rationalized in terms of the nanocluster geometry and occupation of the clusters’s, p and d valence orbitals. Our findings motivate a new scheme for characterizing σ-holes as σ_s (diffuse), σ_p (localized) or σ_d (localized) depending on their electronic origin. The positions of the maxima in V_S(r) (and minima in E_S(r)) are found to coincide with O-down adsorption sites of H₂O, whereas minima in V_S(r) leads to H-down adsorption. Linear relationships between V_S,max (and E_S,min) and H₂O interaction energies are further discussed.

1. Introduction

Surface maxima in the molecular electrostatic potential (V_S,max) along the lateral extensions of intramolecular bonds are known as σ-holes [1]. These have been widely used to rationalize molecular interaction behavior and reactivity [2]. In the present contribution we introduce new categories of σ-holes based on the electronic origin of the V_S,max; if the V_S,max arises primarily as a consequence of electron deficiencies in the valence s-orbitals of the compound, we shall denote it an σ_s-hole. Similarly, V_S,max originating from deficiencies in the p- or d-orbitals will be referred to as σ_p- or σ_d-holes. Mixtures of these exist. The new categorization is herein motivated by a detailed analysis of transition metal (TM) nanoclusters, and arises naturally from the occurrence of diffuse (non-directional) or localized (directional) σ-hole on the TM compounds. With some few exceptions, e.g., refs. [3,4,5,6,7], TM compounds have not commonly been characterized by surface electrostatic potential maps. We will here show that σ-holes are useful guides also for TM interactions with clear similarities to halogen or hydrogen bonding.

Representative examples of σ_s- and σ_p-holes can be found on hydrogen and singly coordinated halogen (X = Cl, Br and I) atoms participating in hydrogen and halogen bonding. The V_S,max of the hydrogen atom of e.g., HF (Figure 1) arises because, upon formation of the covalent H–F bond, electron density is relocated from the non-bonding sides of the atoms to the bonding region between them. The large difference in electronegativity between H and F further leads to a strong polarization towards F in the bonding σ-orbital. The σ*-orbital is, on the other hand, highly polarized towards H but because it is unoccupied it will not compensate for the polarization of the σ-orbital. Consequently, the occupation of the σ-orbitals effectively results in a substantial electron deficiency on H. This is manifested by a large positive electrostatic potential at the H atom—an σ_s-hole since it originates in the hydrogen s-orbital occupation. Simultaneously a negative electrostatic potential is built up on F. Due to the spherical symmetry of the H 1s orbital, the corresponding σ_s-hole is diffused over the entire H end of the molecule. This explains the weak directionality of H-bond interactions, which often deviate significantly from 180° [8]. In the following we will show that common features of σ_s-holes are their diffuse and non-directional character. V_S,max have also been used to rationalize the formation of non-covalent bonds between halogen atoms and electron donating compounds, i.e., halogen bonds [9,10]. In contrast to hydrogen bonds, halogen bonds are highly directional. This can be attributed to the partial occupation of the valence p- rather than s-orbitals. Figure 1 includes the example of the I₂ molecule. The intramolecular bonding in I₂ is, by and large, the consequence of the mixing of two I 5p_z orbitals, leading to a σ_pz-bond where the σ_pz-orbital is occupied and the σ*_pz-orbital is unoccupied. Similarly to HF, this gives rise to electron deficiencies in the lateral extensions of the I–I bond, with two corresponding V_S,max at the edges of the σ-framework. Using the principles for σ-hole categorization from above, the V_S,max of halogen atoms should be denoted σ_p-holes due to the p-origin of the hole. Owing to the largely directional character of the p_z orbitals, the σ_p-holes are highly localized in the direction of the σ-bond. Consequently, the halogen bond interactions are, in contrast to hydrogen bonds, found to be directional with A⋯X–R angles close to 180° [8].

For TM compounds, we analogously find that areas of high electrostatic potential are sometimes localized along the extension of TM–TM bonds. In addition, multiple V_S,max may arise on the same atom corresponding to up to one σ-hole per TM–TM bond. This can be traced to the partially occupied d-orbitals of the TM compound and will thus be called σ_d-holes. The square planar Pt₄ cluster of Figure 1 can be taken as an example. Pt has a 5d⁹6s¹ valence configuration and accordingly the local electron deficiencies of Pt₄ are a consequence of the redistribution of both s- and d-orbital densities (the minor mixing of 6p-orbitals is here neglected for the sake of simplicity). The s-deficiencies promote the creation of areas of high electrostatic potential on the corners of Pt₄. The corners do, however, not correspond to maxima as the V_S(r) profile is also affected by the d-occupation. This promotes the formation of V_S,max at the extension of the σ_d-bonds created by the overlap of 5d_z2 and 5d_x2−y2 orbitals, i.e., at each side of the corners of the Pt₄ square structure. These V_S,max are thus best referred to as σ_d-holes. The σ_p- and σ_d-holes have in common that they are highly localized along the direction of the intramolecular bond.

In the present work, we shall investigate the presence and absence of the various kinds of σ-holes on a series of neutral, low-energy TM₁₃ nanoclusters [14,15,16,17,18,19,20,21,22,23]. The nanoclusters comprise the mixed elemental Pt₇Cu₆ nanocluster as well as the Ir₁₃, Pt₁₃, Au₁₃, Pd₁₃, Rh₁₃, Ru₁₃, Cu₁₃ and Co₁₃ clusters. Nanomaterial based on TM elements are essential components in a range of emergent and established applications including heterogeneous catalysis, medical therapy and solar energy harvesting [24,25,26,27,28,29,30,31], and in order to design and efficiently utilize these materials, a thorough understanding of their interaction behavior with ambient molecules is necessary [32]. In line with this, we have recently suggested that the electrostatic potential profile, [V_S(r)], obtained at contours of constant electron density, can be used to rationalize the catalytic activity of Au nanoparticles (NPs) of 0.5–3 nm sizes [3]. We have also found that, besides Au NPs, V_S(r) can be used to characterize the interaction behavior of NPs of the other d¹⁰s¹-valence elements, i.e., Ag and Cu [4,5]. By comparing the adsorption behavior of the H₂O probe molecule with the surface electrostatic potential profiles (computed with hybrid KS–DFT) of the abovementioned group 8–11 TM₁₃ nanoclusters, we will demonstrate that the positions of V_S,max (σ-holes) and V_S,min largely dictate the position and adsorption mode of H₂O. In addition, we will compare the information obtained by the V_S(r) to that of the local electron attachment energy, E(r) (vide infra) [33]. This quantity contains information, not only about the electrostatic contribution to an interaction, but also on the charge-transfer–polarization effects. When evaluated on an isodensity surface, E(r) is denoted E_S(r) and has, similar to V_S(r), been found to be a useful guide in the analysis of Au, Ag and Cu nanocluster interactions, [4] but also for characterizing electrophilicity and the Lewis acidity of organic molecules [33,34].

2. Theory

The electrostatic potential [V(r)] at a position r of a polyatomic system can be obtained by experimental [35,36] as well as computational methods. V(r) is rigorously defined via [2]

$$V\left(r\right)={\displaystyle \sum }_A\frac{{\mathrm{Z}}_A}{\left|{\mathrm{R}}_A-r\right|}-\int \frac{\rho \left(r^{\prime }\right)\mathrm{d}{r}^{\prime }}{\left|r^{\prime }-r\right|}.$$

(1)

Z_A and R_A are the charge and spatial position of A, the nuclei of the system, while ρ(r’) is the total electronic density at position r’. For the study of interaction properties, V(r) is commonly evaluated on an isodensity contour surface of 0.001 a.u. and denoted V_S(r). Maxima in V_S(r), i.e., V_S,max, indicate sites susceptible to interactions with nucleophiles (Lewis basis), e.g., the H₂O molecule when interacting via its oxygen electron lone pairs. Correspondingly, minima in V_S(r), i.e., V_S,min, indicate sites prone to interaction with electrophiles (Lewis acids). V_S(r) has been employed extensively in the study of molecular interactions including hydrogen and halogen bonding, biomolecular recognition interaction, estimations of pK_A and molecular reactivity [2].

Whereas V(r) reflects the electrostatic character of an interaction, the local electron attachment energy, E(r)], is able to capture both the electrostatics and charge-transfer–polarization components of an interaction (see below) and can thus be used complementary to V(r). In the general case, E(r) = E^E0(r) is obtained by summarizing overall virtual (spin) orbitals from LUMO up to a given cut-off level, E₀, via [33]

$$E^{E_0}\left(r\right)=\frac{{\sum }_{i=\mathrm{LUMO}}^{{\epsilon }_i<E_0}\left({\epsilon }_i-E_0\right){\rho }_i\left(r\right)}{\rho \left(r\right)}.$$

(2)

Here ρ_i(r) and ε_i are the local density and eigenvalue of the ith virtual orbital. Within the generalized Kohn–Sham DFT (GKSDFT), it follows directly from Janak’s theorem (dE_tot/dn_i = ε_i, E_tot = total energy) [37], that E₀ = 0 is a sensible choice. This is due to the fact that, if we assume a frozen orbital picture, only orbitals with ε_i < 0 will bind an electron. Hence, we define E(r) = E^E0=0(r) as

$$E\left(r\right)=\frac{{\sum }_{i=\mathrm{LUMO}}^{{\epsilon }_i<0}{\epsilon }_i{\rho }_i\left(r\right)}{\rho \left(r\right)}.$$

(3)

Equation (3) bears resemblance to the local electron affinity property [EA_L(r)] of Clark and co-workers [38,39]; the two quantities differ in the use of the E₀ cut-off in E(r), whereas no cut-off is used in the evaluation of EA_L(r), and in the normalization by the total virtual density in EA_L(r), while E(r) is normalized by the total occupied electron density. Besides the purely physical arguments for only accounting for bound one-electron states (i.e., ε_i < 0) in E(r), the introduction of the energy offset greatly reduces the sensitivity of the property on the basis set size. Furthermore, the use of the unoccupied density in the denominator when evaluating EA_L(r) incorrectly leads to the assignment of high reactivity to regions of low virtual density.

The E(r) quantity can, in analogy to its nucleophilic counterpart Ī(r) (the average local ionization energy [40,41]), be decomposed into contributions from the electrostatic potential, the kinetic energy density t_i(r) and the exchange–correlation potential [V_XC(r)] by [33,42]

$$E\left(r\right)=\frac{1}{\rho \left(r\right)}\left[{\displaystyle \sum }_{i=\mathrm{LUMO}}^{{\epsilon }_i<0}{t}_i\left(r\right)-V\left(r\right){\displaystyle \sum }_{i=\mathrm{LUMO}}^{{\epsilon }_i<0}{\rho }_i\left(r\right)+V_{XC}\left(r\right){\displaystyle \sum }_{i=\mathrm{LUMO}}^{{\epsilon }_i<0}{\rho }_i\left(r\right)\right],$$

(4)

where t_i(r) = −½ψ_i*(r)∇²ψ_i(r). In Equation (4), the t_i(r) term is the only one that directly depends on the functional forms of the virtual orbitals. This term gives the charge-transfer–polarization contribution to the E(r) quantity, whereas, when E(r) is determined on an isodensity contour (i.e., E_S(r)), V_XC(r) is approximately constant [33]. Minima in E_S(r) are denoted E_S,min and can be used to rationalize the electrophilic (Lewis acidic) behavior of Au, Ag, and Cu nanoparticles [4], as well as for characterizing halogen bonding and the local reactivity of aromatic and conjugated electrophilic organic molecules [33,34].

3. Computational Details

The TM nanoclusters were studied in vacuo using hydrid DFT calculations via the PBE0 exchange–correlation functional [11], augmented with Grimme’s D3 dispersion corrections [43], and employing Becke-Johnson damping [44]. The initial geometries for the Au₁₃, Cu₁₃, Pt₁₃, Pt₇Cu₆, Pd₁₃, Co₁₃, Ir₁₃, Rh₁₃, and Ru₁₃ TM nanoclusters were taken from documented low-energy structures (Figure 2) [14,15,16,17,18,19,20,21,22,23]. These were reoptimized under symmetry constraints in the Turbomole 6.4 software package [13] using the Def2–TVZP basis set; the Def2 basis set family of Ahlrichs and co-workers [12] employs effective core potentials (ECPs) for the 4d and 5d, but not for the 3d metals. The optimizations were repeated for all spin-states going from singlet (or doublet) up to tridecatet (or dodecatet). Co₁₃ is an exception where a 2S + 1 = 28 spin-state has been reported as the ground state, which could be corroborated herein [14,18]. The identified ground states (lowest-energy spin-states), reported in

Table 1. TM₁₃ nanocluster Schönflies symmetry group (Sym), ground-state spin multiplicity (2S + 1), and the number of unique atom binding sites (AS), as well as unique V_S,max sites (VS) and E_S,min sites (ES) identified on each cluster. Included are also the atom average valence s-, d- and p-occupations (s-, d- and p-occ) as determined by NBO analysis [51].

	Sym	(2S + 1)	AS	VS	ES	s-occ 1	d-occ 1	p-occ 1
Au13	Cs	2	10	10	9	0.73	9.90	0.38
Cu13	C2	2	7	6	6	0.63	9.94	0.44
Pt13	Cs	3	7	15	15	0.61	9.08	0.32
Pt7Cu6	C1	3	13	12	11	0.56 2	9.56 2	0.34 2
Pd13	C2	9	7	7	8	0.37	9.34	0.29
Co13	C3	28	4	4	3	0.65	7.82	0.52
Rh13	Cs	2 3	7	7	14	0.43	8.30	0.27
Ir13	Cs	4	10	17	15	0.67	7.97	0.37
Ru13	Cs	13	7	6	9	0.46	7.22	0.34

¹ In case the total valence occupation does not fully amount to the total number of valence electrons, the remaining electron density is spread over the +1 super-valence s-, p-, d-, and f-orbitals. ² Average valence occupation Pt (s: 0.71, d: 9.25, p: 0.28) and Cu (s: 0.40, d: 9.92, p: 0.41). ³ The 2S + 1 = 10 state is almost degenerate, see supplementary material (Section S3 and Table S5).

, of all clusters were used in the continued study; see the discussion in the supplementary material for details (Section S1 and Table S2). Evaluation of the V_S(r) and E_S(r) quantities were performed at the 0.001 a.u. isodensity contours employing the in-house HS95 program (T. Brinck) and visualized using the UCSF Chimera software [45]. The evaluations were based on Kohn–Sham orbitals obtained from calculations performed in the Gaussian 09 program suite [46] using the Def2–TZVP(−fg) basis set at the optimized geometries. In separate calculations, water molecules were placed O-down at the identified V_S,max sites of the TM nanoclusters. These structures were optimized with the Orca 3.0.2 program [47] using the RIJCOSX-approximation [48] and the Def2–TVZP basis set. The TM nanocluster atoms were constrained at their previously optimized positions. Final energies were determined using the Def2–TVZPP basis set without the RIJCOSX-approximations in both the Gaussian 09 and Orca 3.0.2 programs, yielding essentially identical results. All final electronic configurations were checked for internal (e.g., symmetry reduction) and external instabilities by the methods discussed in the references [49,50]. From the above, site-specific H₂O interaction energies (ΔE_int) could be determined by

ΔE_int = E_TM-H2O − (E_TM + E_H2O),

(5)

where E_TM-H2O, E_TM, and E_H2O correspond to the electronic energies of the H₂O–TM nanocluster adduct, the bare TM nanocluster, and free H₂O molecule, respectively. Note that neither zero-point nor any other thermochemical correction has been included in the interaction energy calculations. The valence s-, p- and d-occupations of the TM nanoclusters were studied by natural bond orbital (NBO) analysis using the NBO version 3.1 implementation in Gaussian 09 [51].

4. Results and Discussion

In this part, we will examine the different kinds of σ-holes that emerge on TM nanoclusters, when to expect them and how these influence the local Lewis acidity (and basicity) of the cluster.

4.1. Exemplifying the Origin of the TM σ_s-Holes and σ_d-Holes—The TM₂ and TM₈ of Ir, Pt and Au

The valence occupation of the neutral compounds of the elements of the Cu-group (group 11) is approximately d¹⁰s¹. Thus, their V_S(r) profiles can be understood in terms of the p-mixed s-occupation [3,4]. For the TM elements to the left of the Cu-group in the periodic table, the occupation of the valence states is changed; especially the d-orbitals are monotonically emptied and consequently a variety of different spin-states become attainable. This affects the electrostatic potential profile of the corresponding TM NPs directly, by changing the electronic configuration, and indirectly by altering the favored geometrical structure [14]. The cubic octamers of Au, Pt and Ir are used as examples below to demonstrate the variations in the V_S(r) profile with the d-occupation (Figure 3).

An NBO analysis of the cubic Au₈ (singlet) cluster indeed suggests that it has approximately fully occupied 5d-orbitals and partially filled 6s-orbitals (NBO valence occupation: 5d^9.926s^0.866p^0.22). As expected, the cubic Au₈ (singlet) display V_S,max at the corner sites. Although the cubic structure is not stable for Au₈, it relaxes to a structure with T_d symmetry (Figure 3) that still display V_S,max at corner sites. It is worth noting at this point that the location of the σ_s-holes of Au₈ at positions deviant from the extensions of the TM–TM bonds suggest that they may not be proper σ-holes; the V_S,max are positioned at angles deviant from straight (180°) or right (90°) with respect to the σ-bonds of the cluster. This may instead motivate the coining of the term pseudo δ-hole (in analogy with the pseudo π-holes of planar structures [52]). However, we argue that the introduction of a pseudo δ-hole is unnecessary in this case and that the proper notation for the V_S,max of Au₈ (and similarly, V_S,max in the continuation of this study) are σ_s-holes, since the positions of the σ-holes are located at extensions of the joint σ_s-orbital overlap of multiple bonds.

The cubic O_h structure is stable for both the Pt₈ (nonet) and Ir₈ (tridecatet) clusters. For Pt₈, V_S,min are created at the interatomic hollow and bridge sites but there are no V_S,max on top of the corner atoms. Instead V_S,max are present at three points at the sides of the corner atoms. These points correspond to the unoccupied positions of an octahedral bonding pattern. The positions of the V_S,max also coincide with the expected extensions of the e_g (d_z2 and d_x2−y2) orbitals of such an octahedral symmetry, and hence to the extensions of the σ-bonds created by the d-orbital overlap in the system. These V_S,max are clear examples of σ_d-holes. We shall find, further on, that the local bonding symmetry is important for the creation of the localized and directional σ_d-holes. For Pt₈, the nonet spin-state means that each atom has a magnetic moment of 1. The partially occupied orbitals of Pt₈ are primarily comprised of mixed d_x2−y2- and d_z2-orbitals (NBO valence occupation: 5d^9.046s^0.756p^0.21 with 5d_xy^1.975d_xz^1.975d_yz^1.975d_x2−y2^1.565d_z2^1.56). The corresponding average atomic magnetic moment of Ir₈ is 1.5 with a clear mixing of 5d orbitals of all different angular momentums, as well as 6s (and 6p) orbitals in the partially occupied orbitals of highest energy. Due to this electronic configuration, the V_S(r) profile is the result of an overlap between, on the one hand, the localized σ_d-holes created by cluster orbitals comprising large d_z2 and d_x2−y2 orbital contributions, and, on the other hand, diffuse σ_d-holes that superficially resemble σ_s-holes. The latter originate from d-orbitals of other angular momenta than d_z2 and d_x2−y2 (NBO valence occupation: 5d^7.976s^0.756p^0.29 with 5d_xy^1.655d_xz^1.655d_yz^1.655d_x2−y2^1.515d_z2^1.51). This gives rise to the triangular shaped V_S,max at the corner atoms of the cubic O_h Ir₈ nanocluster. The resulting σ-holes can be seen as mixed σ_d-holes, or, alternatively, as σ_d-holes masked by pseudo σ_s-holes. We furthermore note that the cubic framework of the TM₈ clusters greatly facilitates the analysis of the d-orbital angular momentum contributions, which will not be possible for the less symmetric TM₁₃ clusters below.

4.2. Local Lewis Acidic (and Basic) Characteristics of the TM₁₃ Nanoclusters

The by and large hypothetical TM₈ compounds above are illustrative examples but may or may not be representative of nanoclusters active in real applications. For the purpose of evaluating the expected propensity of σ_s- and σ_d-holes on “real” compounds, we have studied a series of neutrally charged low-energy TM₁₃ nanoclusters, TM = Co, Co, Au, Pt, Pd, Ir, Rh, Ru (Figure 2). In broad terms, the most favorable atomic conformation, magnetic spin-state and electronic valence configuration vary largely for clusters of different TM elements. A common feature is that as the size of the cluster grows, the favorable structure converges towards the compact, close-packed, bulk structure of the corresponding metal [53,54,55,56]. However, at the (sub) nanometer scale, it has been established that clusters of e.g., Rh, Ru, and Ir, adopt an open cubic structure, whereas 3d clusters of Cu, Au and Pd are often found to be compact and possibly disordered [14,57]. There is, furthermore, a preference for 3d clusters to adopt compact atomic configurations while 4d and 5d clusters are generally more open [14]. The TM₁₃ cluster structures of this study will be further discussed below.

Table 1 summarizes the general features of the TM₁₃ nanocluster: their electronic structure, ground-state spin multiplicity, electron valence configuration (as determined by NBO analysis [51]), as well as the number of unique atomic sites, and unique V_S,max and E_S,min. The V_S(r) and E_S(r) profiles of the nanoclusters are displayed at 0.001 a.u. isodensity contours in Figure 4. As can be seen in the figure, both σ_s- and σ_d-holes appear on the TM clusters. In general, the V_S(r) profiles show areas of high electrostatic potential (i.e., V_S,max) on top of atomic sites, whereas low electrostatic potential areas (i.e., V_S,min) are located between atoms at bridge or hollow sites. Thus, at a first approach, it is expected that electron-donating molecules (e.g., H₂O via the O lone-pairs) preferentially adsorb to the atomic on-top sites, whereas electron-accepting molecules (e.g., H₂O via the H atoms) preferentially interact at hollow or bridge sites. V_S,max of large magnitude are often found at atoms of low coordination. The E_S(r) profiles, which can only be used to locate Lewis acidic and not Lewis basic sites, display local minima that largely coincide with the V_S,max. Differences will be discussed in the following sections for each individual nanocluster. The effect of V_S,max and E_S,min on the adsorption behavior of H₂O will also be discussed further below. Figure 5 show the most favorable sites of interaction for all nanoclusters. More details on the H₂O–TM interactions are given further below and in the supplementary material (Table S4).

4.2.1. Au₁₃, Pt₁₃ and Ir₁₃

The low-energy TM₁₃ clusters of Au, Pt and Ir adopt quite dissimilar structures; Ir₁₃ arrange in an open double cubic Ir₁₂ atomic configuration with a 13th capping Ir ad-atom positioned at a bridge site of one of the elongated sides (see Figure 2) [14]. The Pt₁₃ structure cannot be easily categorized, but comprises atoms with a relatively open arrangement [14]. Au₁₃ has a slightly more compact structure based on a prism Au₆ core with capping Au atoms arranged around it [19,21,22,23]. It should, at this point, be noted that 3D Au₁₃ nanoclusters are only stable with respect to 2D clusters if valence electron spin-orbit effects are accounted for [14]. Although such effects are not explicitly included in the present study, we have used the putative global 3D minima structure of Au₁₃ that, presumably, is only metastable at the considered level of theory.

Concerning the electronic structure of Au₁₃, Pt₁₃ and Ir₁₃, NBO analysis [51] suggests that the 6s and 6p occupations vary only slightly over the three nanoclusters. The largest difference instead lies within the 5d orbitals where the d-occupation per atom is approximately d¹⁰, d⁹ and d⁸ for Au₁₃ (doublet), Pt₁₃ (triplet) and Ir₁₃ (quartet), see Table 1. Similarly to the case of the TM₈ example structures in Section 4.1, the Au₁₃, Pt₁₃ and Ir₁₃ display very different V_S(r) profiles (Figure 4). The principles extracted from the cubic examples are, by and large, manifested also on the TM₁₃ clusters; on Au₁₃, dispersed areas of high electrostatic potential are presented at the tips (on-top sites) of the Au atoms, amounting to one V_S,max per atom. These V_S,max are typical σ_s-holes arising mainly from the partially occupied 6s orbitals. On the other hand, both Pt₁₃ and Ir₁₃, favor the formation of localized V_S,max that are positioned along the extension of atomic bonds instead of at the tip of an atomic on-top site. Compared to Au₁₃, this gives rise to a larger number of unique V_S,max on Pt₁₃ and Ir₁₃ with up to three V_S,max per atomic site. These localized V_S,max are to a large extent the product of the unsaturated d-orbitals of the Ir₁₃ and Pt₁₃ and are typical examples of σ_d-holes. Only two clear cases of σ_s-holes are identified on the Pt₁₃ and Ir₁₃ clusters. One is found above the Pt(2) site that is surrounded by a large area of low potential. The other σ_s-hole is found at the tip of the ad-atom site of Ir₁₃. For Pt₁₃, areas that superficially appear similar to σ_s-holes can be identified at the atoms 4, 8, 10 and 13. V_S,max for all of these σ-holes are, however, located at the extension of TM–TM bonds and arranged according to an σ_d-hole pattern.

Analyzing the profile of the E_S(r) in a similar fashion we find that E_S,min sites, i.e., sites susceptible to interactions with electron donors, are located at all the observed V_S,max on Au₁₃, Pt₁₃, and Ir₁₃ except for the Au(12) site where no E_S,min could be identified.

H₂O molecules positioned at all V_S,max (and hence E_S,min) relaxes O-down to sites close to the initial starting positions, i.e., all unique σ-holes correspond to a unique adsorption site (see also discussion under Section 4.3 and reference [3]). For Au₁₃, three adsorption positions deviate slightly from the V_S,max position, namely the atomic sites 7, 10 and 12, where H₂O moves towards a side-on position located in the linear extension of an Au–Au bond. The H₂O adsorption data for Au₁₃ were also discussed in reference [3] and found to correlate closely with CO adsorption. Figure 5 shows the structure of the favored adsorption site for all nanoclusters.

Table 2. H₂O adsorption position and the corresponding V_S,max (kcal/mol) and E_S,min (eV) for the TM₁₃ nanocluster. Interaction energies (ΔE_int in eV) and distances (d_TM-O in Å) are also included, as are the σ-hole character of the V_S,max (σ_type) and distance from the V_S,max to the optimized position of the O-atom of H₂O upon adsorption (d_σ–O in Å).

	ASfav	VS,max	σtype	ES,min	ΔEint	dTM-O	dTM-σ	dσ-O
Au13	13	16.80	σs	−8.88	−0.37	2.44	2.20	0.48
Cu13	1(2)	19.55	σs	−6.53	−0.54	2.14	2.02	0.56
Pt13	12	20.12	σd	−11.93	−0.80	2.22	2.13	0.13
Pt7Cu6	1 1	35.38	σs	−10.88	−0.73	2.09	1.91	0.23
Pd13	8(11) 2	13.31	σs	−5.14	−0.59	2.28	2.16	1.35
Co13	8 3	21.65	σs	−5.64	−0.48	2.13	2.06	0.90
Rh13	1(7)	20.43	σd	−8.74	−0.81	2.23	2.13	0.27
Ir13	6	33.75	σs	−12.86	−1.00	2.17	2.18	0.33
Ru13	3(1) 4	29.16	σs	−12.11	−0.92	2.22	2.07	0.96

¹ Cu atom. ² Fifth highest V_S,max of series (AS 13 (two sites), 9 and 4 higher). ³ Same as position 12 and 13. ⁴ Second highest V_S,max (AS 9 [10] higher).

gives the corresponding adsorption energies, distances and magnitude of the local V_S,max and E_S,min. We find that there is a clear tendency for H₂O to prefer the sites with larger magnitudes of V_S,max (and E_S,min). Linear relationships between V_S,max (and E_S,min) and H₂O interaction energies are discussed further under Section 4.3.

We also investigated H₂O adsorption towards the areas of low electrostatic potential. V_S,min sites are generally found at bridge or hollow sites between atoms. For Au₁₃, the most significant V_S,min is located at the three-fold hollow site between atoms 1, 2 and 3 (h_1-2-3, equivalent to the h_5-5-6 position) above the central prism motif. Pt₁₃ has its lowest V_S,min in the four-fold hollow h_1-7-9-12 site, and Ir₁₃ at the three-fold hollow h_2-6-7 site in proximity to the Ir(6) ad-atom. For both Au₁₃ and Pt₁₃, H₂O preferentially adsorbs H-down (ΔE_int = −0.23 and −0.29 eV for Au₁₃ and Pt₁₃) towards the abovementioned sites, whereas for Ir₁₃, no H-down adsorption mode could be identified but instead H₂O migrated to an adjacent V_S,max and adsorbed O-down. A possible explanation for this is the on average stronger O-down interaction between Ir and H₂O compared to e.g., Au; on Ir₁₃ the flip from H-down to O-down would thus be more beneficial than for Au₁₃.

4.2.2. Rh₁₃ and Ru₁₃

The Rh₁₃ and Ru₁₃ clusters’ structures arrange in a double cubic TM₁₂ atomic configuration similar to that of Ir₁₃. The 13th TM ad-atom (the capping atom) is, however, positioned at a four-fold hollow site at the extended side for Ru₁₃ and Rh₁₃, instead of at the bridge site position of Ir₁₃ [14]. For Ru₁₃, the cube motif underneath the ad-atom is distorted to form a parallelepiped structure (Figure 4). Rh₁₃ (doublet) belongs to the same group of the periodic table (group 9) as Ir₁₃, but its electronic valence structure is clearly redistributed as compared to Ir₁₃: NBO analysis suggests a much lower s- and p-occupation and an increased d-occupation of d^8.3 (Ir₁₃: d^7.97). Ru₁₃ (tridecatet) shows a similar electronic valence distribution as Rh₁₃, but with a d^7.22-occupation. An analogous analysis for Rh₁₃ in the decatet spin-state (almost degenerate to the doublet state) is provided in the supplementary material (Section S3, Figure S1, and Tables S3, S5 and S6). This also includes a discussion of the possibly multi-configurational character of Rh₁₃ (and Ir₁₃), and the usage of spin-contamination correction schemes.

The electronic structures described above affect the electrostatic potential of Rh₁₃ and Ru₁₃ compounds compared to the closely related Ir₁₃ cluster. From Figure 4 we can conclude that the V_S(r) profiles of the Rh₁₃ and Ru₁₃ clusters have both similarities and differences compared to the Ir₁₃ cluster; for Rh₁₃, the overall appearance of the V_S(r) profile suggest that σ_d-holes are located at the extensions of all Rh–Rh bonds, similar to Ir₁₃. However, true V_S,max could only be identified at the capping atom, as well as on all six sites of the extended cubic side perpendicular to the position of the capping atom with respect to the cubic structure. Could this be an effect of the larger d-occupation of Rh₁₃ compared to Ir₁₃, or is it a geometrical effect of the position of the capping atom? While the position of the capping atom could certainly induce some electronic redistribution, the largest effect of its position is probably the split of the σ-hole of the capping atom into two σ_d-holes (σ_s-hole on Ir₁₃). The relatively large d-occupation of Rh₁₃ is, on the other hand likely to affect the shape of the σ-holes. At full d¹⁰-occupation (e.g., Au), as outlined above, we only expect σ_s-holes. Hence the increased d-occupation and reduced s-occupation of Rh₁₃ is likely to partly quench the formation of σ_d-holes.

As concerning the E_S(r) profile of Rh₁₃ it contains σ_d-holes to a larger degree than V_S(r) with E_S,min located at the extension of all but the 6–3 and 12–9 bonds. There are in fact twice as many E_S,min as V_S,max, suggesting that there remains a driving force (including also charge-transfer–polarization effects) for interactions along the extension of all Rh–Rh bonds.

For Ru₁₃ the positions of the identified V_S,max are overlapped almost perfectly by E_S,min sites. However, and similarly to the case of Rh₁₃, Ru₁₃ displays a larger number of E_S,min compared to V_S,max. A distinct difference compared to Rh₁₃ is, however, that there is a larger amount of σ_s-holes on Ru₁₃. For instance, the σ-hole located at the capping atom is a σ_s-hole for Ru₁₃, whereas it is a σ_d-hole for Rh₁₃. Characteristic σ_s-holes are also found at atoms 5 and 7. In addition, the σ_d-holes of the atoms 1, 3, 9, 10, 11 and 12 are clearly distorted from the ideal Ru–Ru bond extension, suggesting a mixed σ_s/d-hole, or alternatively, that the σ_d-holes are to some degree masked by overlapping σ_s-holes. In comparison to Rh₁₃, Ru₁₃ displays σ-holes in all directions.

H₂O adsorption sites (O-down) were found on the TM₁₃ clusters with only moderate deviations from the positions of the σ-holes for both Rh₁₃ and Ru₁₃ (see further discussion under Section 4.3). Two noteworthy exceptions are adsorption to the 5(7) and 9(10) positions of Ru₁₃ where H₂O moves from the σ_d position along the TM–TM bond extensions and instead resides at a typical σ_s site at an Ru–Ru–O angle significantly deviating from 90° or 180°. For Rh₁₃, the most favorable position of adsorption corresponds to the site with the highest V_S,max (atom 1 and 7), whereas for Ru₁₃ the strongest interaction is at position 1 (and 3), which correspond to the second highest V_S,max. The most favored adsorption sites are thus not the capping atoms, which can be explained by a relatively large coordination of these sites, with four neighbors compared to only three for the cubic corner sites (cf. Ir₁₃ where the capping atom only has a coordination number of two and is the favored site of H₂O adsorption). The global V_S,min of Ru₁₃ is found at the four-fold hollow h_2-4-9-10 site, and for Rh₁₃ at the three-fold hollow h_5-10-13 site, which alternatively can be described as a low electrostatic potential ring around the capping atom. Similarly to the case of Ir₁₃, H-down adsorption of H₂O to Rh₁₃ and Ru₁₃ converges to the O-down adsorption modes.

4.2.3. Cu₁₃ and Pt₇Cu₆

Cu₁₃ and Au₁₃ have many characteristics in common; both clusters adopt doublet spin-states, approximately fully occupied d-orbitals, and are organized in compact atomic configurations. The structure of Cu₁₃ resembles a distorted biplanar hexagonal structure (see also Co₁₃ below) [14]. Analogous to Au₁₃, and in line with the conclusions of ref. [4], the V_S(r) profile of Cu₁₃ displays a delocalized σ_s-hole on top of each atomic site, with areas of low V_S(r) located in between the atoms. This profile reflects the dominance of the partially occupied s-orbitals in the V_S(r). As for Au₁₃, the positions of the E_S,min sites of Cu₁₃ coincide well with the V_S,max. H₂O adsorbs O-down to the V_S,max (and E_S,min) sites with the optimized position of the interacting H₂O deviating little from the V_S,max (and E_S,min) positions (see also Section 4.3). H-down adsorption (ΔE_int = −0.27 eV) takes place at the V_S,min sites, where the most prominent V_S,min corresponds to the three-fold hollow h_5-10-12 site—this adsorption mode is in line with previous reports of H₂O interactions with Cu nanoclusters [5,58].

The low-energy structure of Pt₇Cu₆ (triplet) is compact and disordered [15]. The comparison between Cu₁₃, Pt₁₃ and Pt₇Cu₆ gives rise to several questions with regards to the V_S(r). For instance: will σ-holes occur on both Pt and Cu atoms, or will the electrostatic potential be polarized to one atomic type; and if both atom types have associated σ-holes, will the preference of σ_d-holes for Pt and σ_s-holes for Cu be mirrored in the alloy nanocluster?

The average electronic valence configuration of Pt₇Cu₆ is approximately a mean value of that of the Cu₁₃ and Pt₁₃ clusters. Notably however, the NBO analysis implies that some electron density has been transferred from the s(p)-orbitals to the d-orbitals (Table 1). In particular, the analysis suggests that the occupation of the d-orbitals of Pt is increased at the expense of, primarily, a reduced Cu s-occupation. This is reflected in the detailed resolution of the V_S(r) of Pt₇Cu₆. Whereas all Cu sites still display one single σ_s-hole on top of each atomic site—expected since the s-occupation still dominates the bonding of the Cu atoms—the characteristics of the Pt atoms are slightly modified compared to the pure Pt cluster. For Pt, and despite the average increase of the d-occupation, the Pt atoms do maintain their general preference for formation of σ_d-holes along the extension of Pt–Pt bonds. However, when there are no Pt neighbors, as is the case for e.g., the Pt(8) atom, σ_s-holes are formed also on the Pt atoms. The Pt(8) is in addition the Pt atom that possesses the highest d-occupation (d^9.55), and hence electronically shows closest resemblance to Cu. Another notable feature is that only one σ-hole is identified per Pt atom, despite the fact that there is a larger number of possible sites along Pt–Pt extensions. This is partly attributed to the local bonding symmetry of the Pt atoms; in e.g., Pt₈ or Pt₁₃ the symmetric arrangement of atoms with (close to) linear or perpendicular angels allows for the distinction of different σ_d-holes. In Pt₇Cu₆, on the other hand, the disordered cluster structure prevents formation of V_S,max along the extension of all Pt–Pt bonds. The overlap of different contributions to the V_S,max further leads to the σ_d-holes being slightly distorted from the linear Pt–Pt…σ_d angle. In total, the Pt₇Cu₆ cluster has 12 σ-holes, one per atom (regardless if it is Cu or Pt). The exception is Pt(7), which does not have any V_S,max. Instead, Pt(7) is surrounded by a large area of low electrostatic potential. One can further conclude that it is both the atomic arrangement and the atom type that determine the magnitude of the local V_S(r); although the Pt atoms of the Pt₇Cu₆ cluster in general are associated with larger V_S,max than Cu, the overall V_S(r) maximum is located at a Cu atom.

Regarding the E_S(r), we observe eleven E_S,min but only seven coincide with V_S,max. The location of the E_S,min and V_S,max overlap for all Pt atoms (except for the Pt(7) atom, vide supra), as well as for the Cu(1) site that display the E_S,min and V_S,max with the largest amplitude. For the remaining Cu atoms, E_S,min are identified at bridge sites, whereas V_S,max are located at atomic on-top sites. Upon adsorption of H₂O to the atomic sites, it converges O-down to the σ-holes and not to the E_S,min bridge sites of Cu. For about half of the sites, including both Cu and Pt atoms, the H₂O does, however, show some preference for positions further along the extension of TM–TM bonds—i.e., potential σ_d-hole sites in close proximity to the original σ_s-hole. This includes both Cu and Pt sites.

H₂O adsorbs H-down in proximity to the Pt(7) site with a ΔE_int of −0.30 eV.

4.2.4. Pd₁₃ and Co₁₃

The low-energy structures of Co₁₃ and Pd₁₃ are closely related [14]; Co₁₃ arranges in a C₃ hexagonal bilayer structure that is constructed from two overlapping, close-packed hexagonal layers bearing resemblance to the bulk hexagonal close-packing (hcp) of Co. One of the overlapping layers has a triangular shape, while the other is arranged in a honeycomb structure. Pd₁₃ adapts to a distorted hexagonal bilayer structure, similar to that of Cu₁₃. On the average, Co₁₃ has a somewhat decreased d⁸ occupation (NBO: d^7.82), whereas for Pd₁₃ a large redistribution from the s(p) orbitals leads to a notably large d-occupation of d^9.34. Compared to Pt₁₃, where Pt is placed just below Pd in the periodic table, the Pd₁₃ cluster has a significantly different V_S(r) profile with one V_S,max per atom placed on top of the atomic sites. The exception is the 10(13) site that has a split V_S,max resembling two σ_d-holes. Overall Pd₁₃ has an σ_s-hole-dominated V_S(r), whereas the V_S(r) of Pt₁₃ is dominated by σ_d-holes. This can be understood by the increased d-occupation of Pd₁₃ compared to Pt₁₃. V_S,max and E_S,min of Pd₁₃ coincide well for the five sites with highest V_S,max, but E_S,min have a larger tendency to form σ_d-hole equivalents along the extensions of Pd–Pd bonds. V_S,max and E_S,min are located at different positions for the 6(12) site, and no E_S,min could be identified at atom 3.

H₂O adsorbs on top and O-down to all V_S,max of Pd₁₃ with a tendency to move towards the E_S,min position of atomic site 4(5). The average deviation of the O positing of H₂O compared to the V_S,max is small (d_σ-O,ave = 0.67Å, see Section 4.3). The most significant V_S,min of Pd₁₃ is located at the hollow site of atoms 2(7), 1(9) and 10(13), at which H₂O adsorbs H-down with an ΔE_int of −0.29 eV.

For Co₁₃, the C_s symmetry results in only four unique atomic on-top adsorption sites: the triangular and honeycomb corner atoms, as well as the triangular edges and the hcp site on top of the honeycomb. The latter is associated with negative electrostatic potential and H₂O is repelled from this site. Overall, all attempts to identify H-down structures converged to O-down adsorption to the corner and edge sites. The V_S,max are all of σ_s-type and H₂O adsorbs O-down to the three unique, positive σ_s-holes with small deviation of the optimized H₂O position from the location of the σ_s-hole site (see Section 4.3). E_S,min are identified at the corner and edge σ_s-hole sites of the triangular side, but for the honeycomb side the E_S,min are located at the bridge sites.

4.3. General Discussion

From the above discussion, we can conclude on some general features of σ_s- and σ_d-holes. Diffuse and weakly directional σ_s-holes are created when there is an electron deficiency in the s-orbitals. Such deficiencies are present on all the TM nanoclusters, but may in some cases be weak in comparison to the effects of electron deficiencies in the d-orbitals. In such cases, the electron polarization within the d-orbitals leads to localized and directional σ_d-holes if certain criteria are fulfilled:

• Partially occupied d-orbitals
• Favorable electronic configuration (e.g., spin-state with low degree of spd-hybridization)
• Locally symmetric bonding arrangement (e.g., cubic)

The above should be seen in the light of the limited study in the current contribution and future work will be directed towards further understanding of the principles behind the creation of σ_s- and σ_d-holes. We found, however, that in cases where any one of the above criteria is not fulfilled, other types of σ-holes dominate over the prototypical σ_d-hole. If the d-orbitals are fully occupied (e.g., Au₁₃ or Cu₁₃), no σ_d-holes will be present but instead σ_s-holes will appear at the atomic on-top sites located at the corners of the nanoclusters. If the clusters have unsaturated d-orbitals but the overall d-occupation is large compared to the ideal occupation (e.g., d⁹ in Pt or d⁸ in Ir) ―for instance due to a large redistribution of s-electrons via s(p)d-hybridization―the σ_d-holes will be fully or partially quenched resulting in σ_s-holes or mixed σ_s/d-holes. Examples of this are Ru₁₃, Rh₁₃, and Pd₁₃. In addition, if the local bonding symmetry is low with TM–TM bond angles deviating significantly from straight or right angles, σ_d-holes may, to a varying degree, be masked by the overlap of the diffuse σ_s-holes or by other σ_d-holes. This effectively yields (pseudo) σ_s-hole or mixed σ_s/d-hole dominated V_S(r) profiles, e.g., Pt₇Cu₆ or the capping atoms of Ru₁₃ and Ir₁₃.

Overall, the locations of the various σ-holes coincide well with the sites of O-down H₂O interactions. As a measure of this,

Table 3. Averaged data for the TM₁₃ series: average O-down interaction energies (ΔE_int in eV) and adsorption distances (d_TM-O,ave in Å), as well as average distances from the nearby TM atom to the closest σ-hole upon H₂O interaction (d_TM-σ,ave), and the distance between the σ-hole and the O atom of the adsorbed H₂O (d_σ-O,ave). Included are also the coefficient of determination (R²) for the correlation between the H₂O interaction energies and the local V_S,max (R²_V) or E_S,min (R²_E).

	ΔEint,ave	dTM-O,ave	dTM-σ,ave	dσ-O,ave	R2V	R2E
Au13	−0.29	2.57	2.21	0.83	0.84	0.81
Cu13	−0.47	2.19	2.06	0.43	0.85	0.95
Pt13	−0.47	2.38	2.20	0.46	0.67 1	0.82 1
Pt7Cu6	−0.47 2	2.30 2	2.09	0.57 2	0.92 2	0.65 2
Pd13	−0.45	2.34	2.15	0.82	0.58	0.25
Co13	−0.470	2.15	2.10	0.61	n.a. 3	n.a. 3
Rh13	−0.64	2.29	2.15	0.37	0.19	0.18
Ir13	−0.59	2.41	2.28	0.45	0.28 (0.90) 4	0.09 (0.75) 4
Ru13	−0.64	2.25	2.17	0.91	0.03	0.10

¹ Including only the sites with identified V_S,max. ² Values resolved for the Cu (Pt) atoms are, in the order of the table columns: −0.46 (−0.45), 2.22 (2.39), 2.00 (2.17), 0.62 (0.52), 0.99 (0.85), and 0.98 (0.81). ³ Not applicable since there are only three unique adsorption sites. ⁴ Including only the eight strongest interaction sites.

includes the average distance (d_σ-O,ave) from the σ-holes to the optimized position of the O-atom of H₂O for all of the TM₁₃ nanoclusters. The d_σ-O,ave are generally small and in the order of 0.37–0.91 Å. Some few exceptions are outlined in Section 4.2.2, Section 4.2.3 and Section 4.2.4. Additionally, it should be noted that the d_σ-O,ave distance includes the difference between the position of the 0.001 isodensity surface relative to the TM atom centers (approximately 2.0–2.3 Å, Table 3) and the binding distance of the H₂O molecule that adsorbs at slightly larger distances (2.2–2.6 Å). In general, there is a trend that the d_σ-O,ave deviations for the σ_d-holes are smaller than for the σ_s-holes, in line with the proposed larger directionality of the σ_d-holes compared to the σ_s-holes. The σ_d-holes display, on the average, larger ΔE_int (−0.54 eV) but reduced V_S,max (11.5 kcal/mol) compared to the σ_s-holes (average ΔE_int = −0.45 eV and V_S,max = 13.0 kcal/mol). In addition, the maximum V_S,max and strongest ΔE_int are clearly larger for σ_s-holes (V_S,max = 35.4 kcal/mol and ΔE_int = −1.00 eV) compared to σ_d-holes (V_S,max = 25.4 kcal/mol and ΔE_int = −0.81 eV).

The E_S(r) and V_S(r) profiles show a strong mutual correlation where E_S,min and V_S,max sites generally overlap well, with some notable exceptions (see the discussion in Section 4.2.2, Section 4.2.3 and Section 4.2.4). The mutual correlation is not surprising bearing in mind that the E_S(r) property includes a contribution from V_S(r) (cf. Equation (4)). From a computational point of view this is an attractive feature of E_S(r), because the computation of V_S(r) is significantly more time-consuming than that of E_S(r). The two quantities do, however, to a certain extent yield different and complementary information; E_S(r) provides, besides information on electrostatics, a measure of the charge-transfer and polarization effects. Table 3 includes R² (coefficient of determination) values for the correlation between the site resolved H₂O interaction energies and the local E_S,min and V_S,max. As can be seen, there are clear correlations for both E_S,min and V_S,max, especially for the Au, Cu and Pt compounds. Pd₁₃ shows weak correlations, while for Ru₁₃, Ir₁₃ and Rh₁₃ the correlations are poor. Some insight into why certain trends are strong and others weak can be gained from the Ir₁₃ case. Here there are 17 unique adsorption sites. By including only the eight sites with the strongest V_S,max, a correlation of R² = 0.90 can be found between V_S,max and ΔE_int. For the sites with weaker V_S,max one can thus argue that factors other than electrostatics become dominant for the interaction.

The difficulties for V_S,max and E_S,min for ranking reactivity are most pronounced for the particles that have stronger interactions as well as for the particles that have partially occupied d-states (with the possible exception of Pt). Since E_S(r) and V_S(r) are ground-state properties, it is reasonable that a strong, partly covalent interaction (e.g., Ru₁₃ or Rh₁₃) that is associated with a large redistribution of the electronic configuration is more difficult to describe compared to a weak, non-covalent interaction (e.g., Cu₁₃). The DFT methods are, in addition, known to exhibit difficulties in describing the multi-configurational character of certain TM elements, the exceptions being e.g., Au, Ag and Cu, whereas, as discussed further in the supplementary material (Sections S1 and S3), large spin contaminations are found for Ir₁₃ and, especially, Rh₁₃, indicating possible multi-configurational states. It can, moreover, be noted that the d-band model has also been reported to fail to reproduce adsorption trends for e.g., the Rh₁₃, Ir₁₃ and Pt₁₃ nanoclusters [16].

As discussed above, both E_S,min and V_S,max can successfully be used to rank the adsorption trends of Au₁₃, Pt₁₃, Cu₁₃ and Pt₇Cu₆. Figure 6 shows as an example the ΔE_int of H₂O versus E_S,min and V_S,max resolved for all the combined Cu and Pt atoms of the Pt₁₃, Cu₁₃ and Pt₇Cu₆ compounds. It illustrates that the adsorption characteristics can be ranked also when including sites from multiple clusters. The trends are strong for Cu and weaker, but still clear, for Pt.

We lastly note that the localized V_S,max (σ_d-holes) of the TM₁₃ nanoclusters bear some resemblance to the concept of the π-holes [59], i.e., V_S,max positioned perpendicular to the molecular (here particle) framework. In addition, and in analogy to the σ- and π-hole concept, one could also claim that the reported localized V_S,max for some of the nanoclusters are in-fact δ–holes since these appear primarily as a consequence of electron deficiencies in the d-states of the clusters. However, based on the positions of the V_S,max along the lateral extensions of the metal–metal bonds, i.e., in the direction of the nanoclusters’ σ-bonding system?either along the extensions of TM–TM σ_d-bonds or at the focal points (often at atomic on-top sites) of the joint s-orbital states of the cluster?we conclude that the V_S,max identified on the TM nanoclusters are best described as σ-holes and that these can be divided into σ_s- and σ_d-holes.

5. Conclusions

We have herein discussed different types of maxima in the surface electrostatic potential, V_S,max, also known as σ-holes, and their influence on the local Lewis acidity of TM nanoclusters. A new categorization is introduced where σ-holes are denoted σ_s, σ_p or σ_d depending on whether their origin can primarily be traced to the s-, p- or d-occupation. It is demonstrated that highly localized and directional σ_d-holes appear at the extension of TM–TM bonds when there is partially occupied d-orbitals, a relatively weak s(p)d-hybridization, and locally beneficial bonding symmetry. Under other circumstances, diffuse and weakly directional σ_s-holes that correspond to the half-filled s-orbitals—or similarly diffuse and mixed σ_s/d-holes (from the summarized contribution of the valence orbitals formed by the hybridization of s-, [p-] and d-orbitals)—dominate the surface electrostatic potential profile. In addition, the localized σ_d-holes can be fully or partly masked by the overlap of the diffuse σ_s-holes if such are present.

By studying a series of different TM₁₃ nanoclusters, we have found, for example, that low-energy clusters of Ir₁₃, Pt₁₃, Rh₁₃, Ru₁₃ and Pt₇Cu₆ exhibit σ_d-holes. In contrast, only σ_s-holes and σ_s/d-holes are found on the Au₁₃, Cu₁₃, Pd₁₃, and Co₁₃ nanoclusters. No σ_p-holes were identified on the TM₁₃ structures, but we argue that such holes are found on e.g., singly coordinated halogen atoms.

Regardless of the character of the V_S,max (be it σ_s or σ_d), the interactions of the TM₁₃ nanoclusters with water are found to be directed by the position of the V_S,max with H₂O adsorbing O-down at the site of the V_S,max. Similarly, we found that areas of low surface electrostatic potential (V_S,min) on all but the Ir₁₃, Rh₁₃ and Ru₁₃ nanoclusters gave rise to a H-down adsorption mode of H₂O. We furthermore note that the magnitude of the V_S,max can be used to rank the interaction energies of H₂O O-down adsorption with particularly strong correlations for the d¹⁰s¹ TM elements as well as for Pt. Similarly, the local electron attachment energy quantity [E_S(r)] can be used to identify adsorption sites, minima in E_S(r) coincide well with the identified V_S,max and H₂O adsorption positions with some few exceptions. E_S(r) can also be used to rank the H₂O interaction energies for especially the d¹⁰s¹ TM elements, as well as Pt nanoclusters.

Based on the findings of the present study, we predict that both V_S(r) and E_S(r) quantities will find general use in the study and rationalization of the interaction behavior of TM nanomaterials. We envisage applications in e.g., heterogeneous catalysis, medical therapy, drug delivery systems, nanotoxicity, sorption studies, dissolution and nucleation, as well as in nanoparticle transportation. Future studies should evaluate the applicability of V_S(r) and E_S(r) to e.g., larger TM and oxide particles, and to surfaces. Furthermore, they should include a thorough benchmark of the applicability of different DFT methods and basis sets.