σ-Holes and σ-lumps direct the Lewis basic and acidic interactions of noble metal nanoparticles: introducing regium bonds

Abstract

Using local DFT-based probes for electrostatic as well as charge transfer/polarization interactions, we are able to characterize Lewis basic and acidic sites on copper, silver and gold nanoparticles. The predictions obtained using the DFT-probes are compared to the interaction energies of the electron donating (CO, H₂O, NH₃ and H₂S) and the electron accepting (BH₃, BF₃, HCl [H-down] and Na⁺) compounds. The probes include the local electron attachment energy [E(r)], the average local ionization energy [Ī(r)], and the electrostatic potential [V(r)] and are evaluated on isodensity surfaces located at distances corresponding to typical interaction distances. These probes have previously been successful in characterizing molecular interactions. Good correlations are found between Lewis acidity and maxima in V(r), appearing as a consequence of σ-holes, as well as minima in E(r), of the noble metal nanoparticles. Similarly are Lewis basic sites successfully described by surface minima in V(r) and Ī(r); the former are indicative of σ-lumps, i.e. regions of enhanced σ-density. The investigated probes are anticipated to function as reliable tools in nanoparticle reactivity and interaction characterization, and may act as suitable descriptors in large-scale screenings for materials of specific properties, e.g. in heterogeneous catalysis. Because of the similarity between the noble metal nanoparticle's interactions with Lewis bases and the concepts of halogen and hydrogen bonding, a new class of bonds is introduced – regium bonds – taking place between a σ-hole of a Cu, Ag or Au compound and an electron donor.

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1. Introduction

During the last few decades nano-sized transition metal particles and clusters have emerged as versatile groups of materials for a wide range of applications such as heterogeneous catalysis, medical therapy and solar energy harvesting.^1–6 This is largely a reflection of unique properties of the nanoparticles (NPs) in comparison with the corresponding bulk metal, which can be rationalized by perturbations in the electronic structure of the NPs due to atom under-coordination and quantum size effects.⁷ The objective of the current study is to understand the connection between electronic structures of group 11 NPs and their interactions with Lewis bases and Lewis acids. For this purpose, we have employed common DFT-based concepts from molecular theory, and examined their performance in describing the interactions of various Cu, Ag and Au NPs.

We shall first consider the molecular electrostatic potential [V(r)], which has been used extensively in the analysis of chemical reactivity and intermolecular interactions.⁸ The electrostatic potential has mainly been used for main group compounds, but recently the applicability domain has been extended to Cu, Au and other transition metal NPs.^9–13V(r) is rigorously defined by:

where Z_A is the charge on nucleus A, located at R_A, and ρ(r′) is the electron density function. In contrast to other commonly used descriptors, such as partial atomic charges, V(r) is a physical observable that can be determined computationally as well as experimentally.¹⁴ When analyzing the propensity for intermolecular interactions, it is common to compute and depict V(r) on an isodensity surface defined by the 0.001 a.u. contour of the electron density,⁸ denoted V_S(r).

We have recently demonstrated that the size and shape dependent catalytic properties of gold nanoparticles can be explained using the surface electrostatic potentials of the nanoparticles.¹³ The variations in V_S(r) over the nanoparticles were shown to be a consequence of the gold's electron configuration. Here we will use a similar reasoning to explain the interactions of nanoparticles of Cu, Ag, Au with Lewis acids and with Lewis bases.¹³ As a background to the analysis, we will elaborate on the derivation from ref. 13, which starts from an atomic perspective. The electrostatic potential of an atom is spherically symmetric, everywhere positive, and decreases asymptotically toward zero at increasing distances from the nucleus. When atoms are combined to form molecules or particles, the electron density is redistributed toward the more electronegative atoms and regions of negative potential are formed. As an example, V_S(r) of ammonia is negative over the electronegative nitrogen, and the minimum, the V_S,min, is located at the lone pair region (Fig. 1a). The magnitude of V_S,min gives a direct indication of the strength of interactions with Lewis acids or hydrogen bond donors. For the same reason, the hydrogen atoms of ammonia are positive in V_S(r), and the maximum in the surface potential (V_S,max) provides a measure of the hydrogen bond donating strength.

Fig. 1 The electrostatic surface potential [V_S(r)] computed at the 0.001 a.u. isodensity surface for the (a) NH₃, (b) CCl₄, (c) Cl₂, (d) N₂, (e) H₂, (f) Ag₂, and (g) Ag₇ compounds. Coloring from high to low potential, central values of the colors in eV in parentheses: red (0.8) > yellow (0.4) > green (0.0) > cyan (−0.2) > blue (0.4). Selected maxima and minima in the surface potential are marked as V_S,max (σ-hole) and V_S,min (σ-lump) respectively.

An unexpected feature of V_S(r) was first demonstrated by Brinck et al. when analyzing halogenated methanes.¹⁵ They found that the charge distribution within a chlorine or heavier halogen can be polarized to such an extent that the potential at the end of the halogen is positive even when the halogen is bonded to a less electronegative atom. (For future reference, we define the end as the outermost region of the halogen atom, X, close to the point that intersects the C–X axis.) Fig. 1b shows this phenomenon on the molecular surface of CCl₄, which was one of the molecules of the original study. It was further shown that the positive V_S(r) can explain the tendency of halogen compounds to interact with Lewis bases (nucleophiles); this type of interaction is today referred to as halogen bonding.¹⁶ Clark et al. later explained the occurrence of the positive potential by a density depletion at the end of the halogen due to polarization of the σ-orbitals; this was denoted a σ-hole.¹⁷ A similar type of polarization also exists within the homonuclear diatomics. Fig. 1c shows V_S(r) of Cl₂, with the characteristic σ-hole(s) (i.e. positive V_S,max) at the Cl end(s), and a negative potential region around the middle of each Cl atom. The latter can be traced to the combined contributions from the π and π*-orbitals and the polarized σ-orbitals. The L-shaped form of the Cl₂–Cl₂ dimer as well as the crystal structure of Cl₂ are perfectly consistent with the surface potential of Cl₂; the structures are such that V_S,min of one molecule aligns with V_S,max of another.^18–20 Turning toward the N₂ molecule, the picture is largely reversed with the most negative potential at the nitrogen tip and the most positive potential at the middle of the bond (Fig. 1d). The latter observation is remarkable considering that the π*-orbitals are unoccupied and the π-orbitals are polarized towards the bonding region. This shows that the surface electrostatic potential is largely governed by the highest σ-orbital. In N₂ this orbital is polarized towards the end regions, whereas in Cl₂ the orbital is polarized toward the bonding region. Ultimately, this is a reflection of different sp-orbital mixing in the σ-orbitals of the two compounds: in Cl₂ the 3s orbitals mix constructively with the phase of the bonding region of the 3p_z–3p_z σ-orbital, whereas in N₂ the 2s orbitals mix deconstructively with the bonding region and constructively with the 2p_z lobes at the N₂ ends. Figures of the orbitals are provided in the ESI.† Considering the opposite σ-orbital polarization of N₂versus Cl₂, we find it appropriate to introduce the concept of the σ-lump to rationalize the negative end, the lone pair, of each N in N₂. The negative lone pair region of NH₃ can also, for obvious reasons, be described as a σ-lump, albeit a much stronger σ-lump than that in N₂. In other cases, the σ-lump is not equivalent to a lone pair and the introduction of this new concept is therefore warranted.

The influence of orbital polarization on the surface electrostatic potential is much easier to interpret when only the valence s-orbitals contribute to bonding, such as in the H₂ molecule; the bonding σ-orbital is polarized toward the bond, resulting in positive ends and a negative potential in the middle of the bond (Fig. 1e). In other words, there are σ-holes at each end region and a σ-lump in the middle. V_S(r) of Li₂ (not shown) has a similar pattern, but the σ-holes are stronger and the lump is much more diffuse due to the larger polarizability and the much longer bond length of Li₂. The noble metals, Cu, Ag and Au, resemble H and Li in that it is mainly the half-filled s valence orbitals that contribute to chemical bonding. Consequently, the surface electrostatic potential of Ag₂ is similar to that of H₂, but the magnitudes of the V_S,max and V_S,min are larger (Fig. 1f). The relevance of the surface potential for the intermolecular interaction tendencies is demonstrated by the Ag₂'s preference to form T-shaped dimers and to interact end-on with Lewis bases and side-on (middle of the bond) with Lewis acids (see Section S1 of the ESI†).

We can now ask the question whether the σ-orbital polarization, and the positive and negative potential regions resulting from the σ-holes and σ-lumps, respectively, also govern the interactions of larger metal NPs with Lewis bases and Lewis acids. It is anticipated that metal atoms that are not fully coordinated have associated σ-holes that are potential sites for interactions with Lewis bases, and that the magnitude of V_S,max at these sites reflects the relative interaction energies at each site. Similarly, we expect that exposed bonds, i.e. σ-lumps, are likely to interact with Lewis acids and that the interaction strengths can be predicted from V_S,min (other factors such as charge-transfer and polarization affecting a compound's interaction behavior are discussed below). The surface electrostatic potential of Ag₇ is in agreement with the painted picture (Fig. 1g), i.e. the regions of the most positive potential are found at the exposed atoms and are opposite to the chemical bonds, and the negative potential regions are found at the exposed bonds, as well as in the hollow sites in-between atoms. In the following, we will demonstrate that the positions and magnitudes of V_S,min and V_S,max can be used to identify and rank sites that interact with Lewis acids and Lewis bases, respectively.

It must be emphasized that intermolecular interactions are not solely electrostatic in nature but also have attractive contributions from charge-transfer, polarization and dispersion, as well as repulsive contributions from the Pauli repulsion. Although such a division is artificial, and particularly the differentiation between polarization and charge-transfer is arbitrary, the consideration of different interaction terms can aid the interpretation and prediction of intermolecular interactions. The multifaceted nature of Lewis acid–base interactions was acknowledged already by Pearson who classified Lewis acids and bases as hard or soft.²¹ Interactions between hard acids and hard bases are considered electrostatically controlled, whereas soft–soft interactions have significant contributions from charge-transfer/polarization. While V(r) for obvious reasons is well suited for the characterization of the electrostatic contribution, it does not capture e.g. charge-transfer/polarization effects. Sjoberg et al.²² recognized this when they introduced the average local ionization energy [Ī(r)]. This property gives a measure of the local cost of ionizing a compound, and, in addition to electrostatics, also reflects the local charge-transfer/polarization capacity of a compound. Surface minima in Ī(r) correspond to sites susceptible to interactions with electron-accepting species (Lewis acids), but cannot be used to characterize interactions with electron donors (Lewis bases). Brinck et al.²³ recently introduced the local electron attachment energy, E(r), as a complement to Ī(r). Contrary to Ī(r), surface minima in E(r) correspond to sites where the compound is likely to accept an electron. Hence these sites will, similar to V_S,max, correspond to areas susceptible to interactions with Lewis bases. The Ī(r) and E(r) properties have been proven to be useful tools in the study of molecular interactions including non-covalent bonding as well as local reactivity.^23–26 The performance of Ī(r) and E(r) in the studied NP interactions will here be compared to V(r) in order to better characterize the origin of the interactions.

Owing to the close resemblance between halogen/hydrogen bonding and the interactions of noble metal NP with Lewis basis, we shall also introduce a new class of bonds – regium bonds, reflecting the royal position of Cu, Ag and Au among the elements in the periodic table. In analogy to halogen or hydrogen bonds, regium bonds are controlled by the polarization of σ-orbitals, and takes place between a charge acceptor (in this case a noble metal atom) displaying a high electrostatic potential, i.e. a σ-hole, and a electron donor with a negative electrostatic potential (for instance the N atom of NH₃). Scheme 1 gives a comparison between hydrogen, halogen and regium bonding.

Scheme 1 Comparison between hydrogen, halogen and regium bonds.

2. Methods

2.1. Theory

In this section, the theoretical foundation of the Ī(r) and E(r) properties is provided. An account of their practical evaluation and physiochemical significance is also included. Firstly, the use of isodensity surfaces, such as the 0.001 a.u. isosurface, for the evaluation of the descriptor properties used herein deserves an explanation. Chemical interactions and reactions take place at a certain distance from the nuclear framework that constitutes the molecule or particle. In line with this, it has been found that the evaluation of V_S(r) and Ī_S(r) at a distance from the nuclei corresponding to a constant electron density of 0.001 a.u. [0.004 a.u. for E_S(r)], typically marginally more diffuse [dense] than the van der Waals surface, is often a good choice for both qualitative and quantitative studies.^8,23,25 In addition by performing the evaluation at a constant density, the analysis is simplified since the Pauli repulsion resulting from the intermolecular electron density overlap will be approximately constant when comparing different adsorption sites of the compound.

The average local ionization energy. The average local ionization energy Ī(r) is rigorously defined within the Hartree–Fock (HF) approximation as well as in Kohn–Sham DFT (KS-DFT), and provides the local energy cost for removal of an electron from the studied system. Ī(r) is described by:^22,25where, ε_i is the eigenvalue of the ith spin orbital. The summation runs over all occupied orbitals up to the highest occupied (molecular) orbital, HOMO. Local minima in Ī(r) on an isodensity surface (typically 0.001 a.u.), Ī_S,min, correspond to Lewis basic sites. Ī_S(r) is thus a descriptor of the local electron donation power (i.e. Lewis basicity) of the studied compound; while V_S(r) provides information on both Lewis basicity and acidity. In contrast to V(r), which only contains information about the electrostatic interaction tendencies, Ī(r) also provides information on the charge-transfer/polarization capabilities of a compound. This can be demonstrated by decomposing the Ī(r) property into contributions of the electrostatic potential, the local electron temperature, T(r), and the local exchange–correlation potential, V_xc(r), as:²⁷The local electron temperature T(r) is related to the local kinetic energy density, t_s(r), by 3kT(r)/2 = t_s(r)/ρ(r). Besides the electrostatic potential, V(r), it is mainly t_s(r) that determines the variation of Ī(r) over an isodensity surface, since V_xc(r) is nearly constant at constant density. Surface minima in Ī(r) [Ī_S,min] are typically found in lone pair regions and over π-bonds, and their magnitudes reflect the strength of interactions with Lewis acids. When going down the periodic table for congeners of Lewis bases, Ī_S,min decreases and V_S,min increases indicating that t_s(r) is largely responsible for the soft (charge transfer/polarization) character of I(r) compared to V(r), e.g. comparing H₂O and H₂S, the latter has a much lower Ī_S,min despite its higher V_S,min.²⁸ Hence, Ī(r) and V(r) provide complementary information. This feature will herein be exploited in the evaluation of the electrostatic versus charge-transfer/polarization character of nanoparticle interactions.

The local electron attachment energy. The local electron attachment energy, E(r), was introduced by Brinck et al.²³ as a corresponding property to Ī(r) for the analysis of local Lewis acidity. E(r) can be seen as a modified version of the electron affinity descriptor EA_L(r) of Clark and co-workers^29,30 and sums the contribution of all virtual (unoccupied) orbitals with a negative eigenvalue. E(r) is evaluated by:

E(r) differs from EA_L(r) in the use of a cut-off ε_i < 0 in the selection of virtual orbitals and in the definition of the denominator, where EA_L(r) employs the total virtual orbital density whereas E(r) uses the occupied orbital density.‡ The cut-off is motivated since, within the generalized Kohn–Sham method (GKS-DFT) and given a frozen orbital approximation, only the orbitals with negative eigenvalues will bind a fractional electron, as follows directly from Janak's theorem.^23,31 Analogous to Ī(r), E(r) includes contributions from both the electrostatic potential and V_xc, as well as the kinetic energy densities of the virtual orbitals. E(r) can be decomposed as:²³

In eqn (5) t_i(r) = −½ψ_i*(r)∇²ψ_i(r). As in the case of Ī(r), E(r) can thus be used as a complement to V(r) in the evaluation of the charge-transfer/polarization versus the electrostatic character of a given interaction. E(r) gives the local electron accepting capability (Lewis acidity) of a compound and minima in E(r) on an isodensity surface, E_S,min, correspond to Lewis acidic sites.

In contrast to V_S(r) and Ī_S(r), a small benchmark test on organic molecules, comprising aromatic compounds, unsaturated alkenes and halogen bonding molecules, has previously shown that it is more appropriate to evaluate E_S(r) on the 0.004 a.u. isodensity surface rather than the 0.001 a.u. surface.²³E_S(r) is hence obtained at slightly shorter distances from the nucleus than V_S(r) and Ī_S(r). However, the proper isosurface for interaction analysis on metal NPs remains to be established. In the present study we have thus computed E_S(r) on both the 0.001 and 0.004 a.u. isodensity surfaces.

2.2. Computational details

The Cu₉, Au₉, Ag₉, Ag₁₁, Ag₁₇ and Ag₁₈ nanoparticle structures were studied using the Gaussian 09 program suite.³² The structures were chosen on the basis that they have multiple unique adsorption sites and are of a size that is of technical interest while being computationally manageable. In our previous study we studied Au nanoparticles in the size from Au₂ to Au₅₆₁ and found similar variations in V_S(r) independent of the particle size.¹³ All but the Ag₁₈ particles are doublets in their ground states, while Ag₁₈ is a closed-shell singlet. Adsorption energies of the electron donating H₂O, NH₃ and H₂S molecules and electron accepting Na⁺, HCl (H-down), BH₃ and BF₃ species were determined at the PBE0-D3(BJ)/def2-TVZPP//PBE0-D3(BJ)/def2-TVZP level of theory^33–36 using effective core potentials (ECP)³⁶ for the Ag and Au particles.§ The CO-adsorption energies were obtained from ref. 37. In previous studies it has been found that the use of hybrid functionals, such as PBE0,³³ improves the description of transition metal–adsorbate interactions compared to standard GGA functionals.^38–43 Likewise, one has found that the inclusion of dispersion (via e.g. Grimme's D3 dispersion corrections³⁴ with Becke–Johnson (BJ) damping³⁵) improves the energetics and structures of e.g. Au nanoparticles³⁴ and other transition metal adsorbate systems.^44,45 All structures were characterized as local minima (zero imaginary frequencies) by vibrational analysis employing the harmonic oscillator approximation. Interaction energies, ΔE_int, were determined by:

ΔE_int = E_Ads/NP − (E_Ads + E_NP) (6)

where E_Ads/NP, E_NP and E_Ads are the zero-point corrected Born–Oppenheimer electronic energies of the nanoparticle–adsorbate complex, the bare nanoparticle, and the free adsorbate molecule, respectively.¶ Due to considerable Au₉ structural changes upon H₂S adsorption onto some of the adsorption sites, interaction energies are reported for an optimized structure with the Au atoms fixed in order to facilitate comparison to the descriptor values. Thermal corrections were obtained at the fully relaxed structures.

The modified PBE0³³ exchange–correlation functional with Hartree–Fock exchange (E^HF_x) reduced to 10% was used in the evaluation of the V(r), E(r) and Ī(r) properties. The choice of functional was based on an evaluation of a set of functionals ranging from the pure GGA PBE⁴⁶ to the hybrid PBE0 with different amounts of E^HF_x (up to 66.67%), see Section S4 of the ESI.† Also included in the evaluation were the range separated HSE06,^47,48 the long-range corrected LC-ωPBE⁴⁹ and the TPSSh^50,51 meta hybrid functionals, as well as the hybrid B3LYP^52,53 functional and its long-range corrected version CAM-B3LYP.⁵⁴ The evaluation underlines that a reduced E^HF_x increases the quality of the E(r) and Ī(r) descriptors, especially at the denser isosurfaces (e.g. 0.004 a.u.). In contrast V(r) is relatively insensitive to the amount of E^HF_x. The use of a reduced amount of E^HF_x for DFT calculations of transition-metal compounds is not new, but in-fact a common practice: the so-called B3LYP*⁵⁵ and PBE0*⁵⁶ functionals with 10–15% E^HF_x have for instance been used to describe spin-transitions^55,56 and to study enzymatic processes involving redox active open-shell transition metal ions.^57,58

We have also found that the combination of the LACV3P*//LANL2-DZ basis sets,^59–62 employing the Los Alamos type ECPs for the transition metal atoms, yields results of similar quality for the descriptors as the considerably larger def2 basis set family of Ahlrichs and co-workers³⁶ (i.e. the basis set used for the ΔE_int determination). Hence, in order to show that the descriptors can provide accurate predictions at a low computational cost we have employed the LACV3P*//LANL2-DZ basis set combination for the V_S(r), E_S(r) and Ī_S(r) descriptors throughout this study. The in-house program HS95 (T. Brinck) was used to compute the descriptor values; if not otherwise stated, V_S(r), E_S(r) and Ī_S(r) were evaluated on the 0.001 a.u. isodensity surface, since this isosurface was found to generally give the best correlation with computed ΔE_int (see the results and discussion section, and the ESI†). NBO,⁶³ Mulliken⁶⁴ and Bader⁶⁵ partial charges were determined at the PBE0-D3(BJ)/def2-TVZPP//PBE0-D3(BJ)/def2-TVZP level of sophistication, while the CM5M⁶⁶ charges were obtained from ref. 37.

The cross-correlated R² coefficient of correlation, Q², which was used in the statistical analysis, was evaluated by the leave-one-out procedure:⁶⁷

Above, y_i is the ith value of y in the data series, ȳ is the average value of y over the data series whilst ŷ_i/i is an estimated value of y = y(x_i) determined from a linear regression analysis of the data series upon excluding the ith data point.

3. Results and discussion

Our results are discussed in the following order: first we will give an overview of the general properties of the studied metal nanoparticles before studying the Lewis acidic and Lewis basic characteristics of the Ag₉ nanoparticle in greater detail. This will be compared to the cases of the elementally isomeric Cu₉ and Au₉ nanoparticles. We will then proceed to assess the performance of the V(r), E(r) and Ī(r) descriptors compared to the results from a previous study of Duanmu et al.³⁷ where site-resolved CO adsorption energies on Ag₉, Ag₁₁, Ag₁₇ and Ag₁₈ nanoparticles were correlated with partial atomic charges via the CM5M protocol.⁶⁶

3.1. Geometric and electronic structure of the nanoparticles

The Ag nanoparticle structures were obtained from ref. 37. The Ag₉, Ag₁₁, Ag₁₇ and Ag₁₈ nanoparticles were re-optimized from the reported geometries while the Au₉ and Cu₉ nanoparticles were obtained by structural relaxation starting from the geometry of the Ag₉ particle. The Ag₉ (and hence Cu₉ and Au₉) and Ag₁₁ structures are based on the D_5h Ag₇ structure, and formed by extending Ag₇ with ad-atoms. Ag₁₁ may, alternatively, be seen as two intertwined Ag₇ structures. The Ag₁₇ and Ag₁₈ structures are based on an icosahedral Ag₁₃ substructure. All particles have six or more non-equivalent adsorption sites including sites of three- to six-fold coordination. Over the series of Ag nanoparticles the average coordination number increases with the particle size. As can be seen in Table 1, the average distance between neighboring atoms is quite similar for the Ag particles. There is, however, a slight increase with particle size. For the elemental M₉ (M = Cu, Ag, Au) isomers, the Au₉ and Ag₉ particles are similar in size (similar average bond distances), while the average bond distance in Cu₉ is reduced by approximately 12% compared to Ag₉ and Au₉.

Table 1 Geometrical and electronic data for the Ag, Cu and Au nanoparticles

	d̄ M–M [Å]	Sites^b	N̄ coord	n v-orb	ε LUMO [eV]	ε HOMO(SOMO) [eV]	ΔεHOMO–LUMO [eV]
a Average distance between neighboring atoms. b Number of non-equivalent on top adsorption sites. c Mean coordination number of the particle surface atoms. d Number of virtual spin orbitals below ε = 0.
Cu9	2.48	7	4.67	22	−3.70	−4.46	0.76
Au9	2.81	7	4.67	23	−4.74	−5.42	0.68
Ag9	2.82	7	4.67	23	−3.69	−4.38	0.68
Ag11	2.82	6	4.91	29	−3.72	−4.31	0.59
Ag17	2.86	8	5.88	39	−3.71	−4.28	0.56
Ag18	2.84	11	6.00	40	−3.40	−4.54	1.14

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The structures of all of the studied nanoparticles are displayed in Fig. 2. V_S(r), E_S(r) and Ī_S(r) determined on the 0.001 a.u. isodensity surfaces are also shown, as well as the valence electronic configuration (valence density of states – DOS). Corresponding figures of the HOMO (or, to be specific, the SOMO, singly occupied molecular orbital) and LUMO orbitals of Ag₉ are displayed in Fig. 3. In addition, the HOMO (SOMO) and LUMO orbitals for the other particles are included in the ESI† along with a complete summary of all virtual orbitals used in the evaluation of E_S(r) for Ag₉, i.e. all unoccupied orbitals of energy lower than the E₀ cut-off. The HOMO (SOMO) and LUMO energies can be found in Table 1. An analysis of the frontier orbitals alone is not sufficient to understand the NP interaction behavior (see ESI†). This does not rule out orbital analysis as a tool per se; at the studied level of theory, Ag₉ has 23 virtual spin orbitals below the free electron limit (see Table 1 and ESI†). Many of these have energies similar to the LUMO but predict other preferred sites of interactions. Hence, although the use of the LUMO [and HOMOs(SOMOs)] for predicting interactions is limited for these systems, the general use of orbitals for understanding the reactivity is not; by considering the contributions of all relevant orbitals, a better representation is obtained as shown in the following discussion by the use of the E(r) descriptor for local Lewis acidity, and by the Ī(r) descriptor in the case of Lewis basicity.

Fig. 2 Overview of the Lewis acidic and basic properties of the Cu₉, Au₉, Ag₉, Ag₁₁, Ag₁₇ and Ag₁₈ nanoparticles. The V_S(r), E_S(r) and Ī_S(r) quantities are evaluated on the 0.001 a.u. isodensity surface. Color code for V_S(r) in meV: red > 325; 150 < yellow < 325; −150 < green < 150; −325 < cyan < −150; blue < −325. E_S(r) in eV: red < −8.0 < yellow < −6.0 < green. Ī_S(r) have different color codes for the different metals, in eV; Cu: blue < 5.8 < cyan < 5.9 < green; Ag: blue < 6.1 < cyan < 6.3 < green; Au: blue < 7.2 < cyan < 7.5 < green. The DOS entries show to the electronic valence structure.

Fig. 3 V _S(r), E_S(r), Ī_S(r) at the 0.001 a.u. isodensity surface as well as the LUMO and HOMO (SOMO) orbitals of Ag₉. Color code V_S(r) in meV: red > 325; 150 < yellow < 325; −150 < green < 150; −325 < cyan < −150; blue < −325. E_S(r) in eV: red < −8.0 < yellow < −6.0 < green. Ī_S(r) in eV: blue < 6.1 < cyan < 6.3 < green. The HOMO (SOMO) and LUMO orbitals are differentiated by different spin phases (not indicated in figure) with only minor spatial differences.

The V_S(r) and E_S(r) isodensity maps of Fig. 2 show that Lewis acidic sites (i.e. V_S,max sites = σ-holes) can be identified on top of all surface atoms using the V_S(r) probe, whereas in a few cases there is no local minimum in E_S(r) at the Lewis acidic site. The exceptions include the weak adsorption sites Ag₉(2), Ag₁₇(1), Ag₁₇(5), Ag₁₈(1), Ag₁₈(2), Ag₁₈(6), Ag₁₈(7) and Ag₁₈(8). For these sites, the local value of E_S(r) at the “on top” site that coincides approximately with the intersection between the particle–adsorbate bond and the isosurface has been used in the interaction analysis. From the V_S(r) map we can further identify Lewis basic sites (i.e. V_S,min sites = σ-lumps) at bridge and hollow sites formed in-between the atoms of the particle. A more detailed analysis of the Lewis acidic and basic properties of Ag₉ follows in Section 3.2 below.

3.2. Lewis acidity and basicity of Ag₉

In the following we will elaborate on the details of the local interaction properties of Ag₉. This will serve as a prototypical example of the applicability and conceptual transparency of the considered descriptors. Similar analyses are performed in full or in part for the other particles used in the present study (see discussion further below and in the ESI†). Fig. 3 shows how the Lewis basic and acidic character of Ag₉ varies over the 0.001 a.u. isosurface from the perspective of the V_S(r), E_S(r), Ī_S(r) properties. In addition, the HOMO (SOMO) and LUMO orbitals are included in the figure for comparison. The reader will now recall that the electrostatic potential V_S(r) is able to identify both Lewis acidic (local maxima) and Lewis basic (minima) sites, while minima in E_S(r) mark Lewis acidic sites and minima in Ī_S(r) mark Lewis basic sites, respectively.

General overview. From V_S(r) we find areas of high Lewis acidity (σ-holes) straight above (on top) of all the Ag atoms, indicating that these sites are the favored positions for electrostatically driven interactions with electron donating molecules. Similarly E_S(r) locates Lewis acidic sites at the on top sites, but in some cases, e.g. Ag₉(5), the minima (E_S,min) are not positioned straight above the atoms with respect to the center of gravity of the cluster, but at a slightly tilted angle. For the Ag₉(2) site there is no local minimum in E_S(r). For the comparison with other adsorption sites, the E_S(r) value at the on top site straight above the Ag₉(2) atom will be used in the following (i.e. approximately the site of the intersection of the adsorbate NP bond and the isosurface). V_S(r) ranks the Lewis acidity of the unique, non-equivalent, sites as Ag₉(4) > Ag₉(8) > Ag₉(9) > Ag₉(3) > Ag₉(5) > Ag₉(1) > Ag₉(2), whereas for E_S(r) the order of the Ag₉(8) and Ag₉(9) sites is interchanged. Note that this difference cannot be visualized in Fig. 3, since the red color of the sites only means that, e.g., the potential is larger than 325 meV. As concerning Lewis basicity, areas of low electrostatic potential (indicative of σ-lumps) are found in between the Lewis basic atoms at bridge and hollow sites. The same general areas also correspond to local minima in Ī_S(r), although Ī_S(r) also attributes some Lewis basicity to Ag(9) on top sites. From the analysis of minima in V_S(r) and Ī_S(r) there are two Lewis basic areas of special importance. These are the sites that should be most prone to interactions with electron-acceptors and are located at opposite sides of the b₅₈ bridge formed between Ag₉(5) and Ag₉(8): one area encompasses the two hollow sites h₁₃₅ and h₁₅₈ formed in between the Ag₉(1), Ag₉(3) and Ag₉(5) atoms as well as the Ag₉(1), Ag₉(5) and Ag₉(8) atoms; the other area encompasses the h₂₄₅ and h₂₅₈ hollow sites. Other significant Lewis basic areas are identified at the h₁₇₈, h₃₄₅ and h₇₈₉ sites. The local minima in both V_S(r) and Ī_S(r) rank the Lewis basic sites of Ag₉ as h₁₃₅/h₁₅₈ > h₂₄₅/h₂₅₈ > h₁₇₈ > h₃₄₅ > h₇₈₉. The bridge sites do not correspond to local minima in V_S(r) and Ī_S(r), but are instead Lewis acidic saddle points. These could potentially act as the site of interaction given a suitable adsorbent. We did e.g. find that H⁺ preferentially adsorbs at bridge sites.

Lewis acidity predictions. Now to the critical question: how do the predictions from the descriptors correlate with the chemical interactions of Ag₉? In the lack of high-quality experimental data to compare with, we have created a small computational reference data set comprising interactions with both electron donating and accepting probe molecules as specified in the computational methods. Beginning with the Lewis acidity of Ag₉, we find that the sites identified by V_S(r) and E_S(r) coincide with the predicted adsorption sites of electron donating molecules. These include the H₂O, NH₃, H₂S and CO molecules that have varying Lewis acidic characteristics as shown in Table 2. If we first consider the electrostatic properties of these compounds, the magnitude of the V_S,min (σ-lumps) located at the interacting atom ranks the molecules as NH₃ > H₂O > H₂S > CO. Thus if the Ag₉ interactions are purely electrostatic we expect the interaction strength to be ranked in the same order. However, the calculated average interaction energies are ordered as NH₃ > CO > H₂S > H₂O. In a similar manner, Ī_S,min can be used to estimate the charge-transfer/polarization capacities of the probe molecules. This ranks them as NH₃ > H₂S > H₂O > CO, with the same ordering being expected for the interaction energies if charge-transfer and polarization effects dominate the interactions. Again this does not follow the calculated interaction energy trend. In summary the results suggest that some or all of the interactions have interaction modes that are not purely electrostatically or charge-transfer/polarization controlled. For mixed interaction modes it has previously been shown that combined dual-relationships of V_S,min and Ī_S,min give a good description for trends in the relative interaction strength.^28,68 This approach has been found especially useful in understanding the interaction behavior of molecules spanning over different groups and periods of the periodic table. This will be discussed in the following paragraphs and returned to in Section 3.3 as it turns out that comparison between different substrate NPs benefits the analysis. Table 2 summarizes the geometrical details of the H₂O, NH₃, H₂S and CO adsorption onto Ag₉, and Fig. 6 shows the lowest energy adsorption structures of CO, H₂O, H₂S, NH₃, and CO.

Table 2 Adsorption distances (d) in Å, average interaction energies (ΔĒ_int) in kcal mol⁻¹, and charge-transfer information for the CO, H₂O, NH₃, and H₂S Lewis bases and BH₃, BF₃, HCl (H-down), and Na⁺ Lewis acids onto Ag₉. Additional information is provided in the ESI. V_S,min and V_S,max in kcal mol⁻¹, and Ī_S,min and E_S,min in eV, are given on the 0.001 a.u. isosurface at the site of interactions for the Lewis bases and Lewis acids, respectively

	d Mean	d Max	d Min	r wdW	ΔĒint^d	Δqnbo^b	s Δq	V S,min	V S,max	Ī S,min	E S,min
a van der Waals radius of the interacting atoms, i.e. C, O, S, N, B, B, H, and Na,^75 the Bondi van der Waals radii of Ag is 1.72.^76 b Average charge-transfer in a.u. upon particle–Lewis base interactions evaluated by NBO charges. c Pooled variance in the atomic partial charge for each interaction over the series. d Obtained at the PBE0-D3(BJ)/def2-TVZPP//PBE0-D3(BJ)/def2-TVZP level of theory. e Note that the NBO analysis suggests that BF3 and HCl donate electrons to Ag9, although they interact via a Lewis basic adsorption mode at the hollow sites. f Could not be determined since no virtual orbital with ε < 0.
CO	2.17	2.20	2.14	1.70	8.19	0.12	0.10	−13.3	—	10.91	—
H2O	2.51	2.63	2.46	1.52	5.56	0.05	0.03	−35.1	—	9.58	—
H2S	2.66	2.77	2.61	1.80	7.91	0.14	0.07	−17.7	—	7.99	—
NH3	2.36	2.41	2.33	1.55	10.55	0.09	0.05	−38.9	—	7.79	—
BH3	2.47	2.79	2.42	1.92	24.15	−0.62	0.30	—	40.8	—	−20.18
BF3	3.71	3.75	3.66	1.92	3.99	0.05^e	0.05	—	51.2	—	—^f
HCl	2.80	2.86	2.75	1.10	5.11	0.01^e	0.05	—	44.8	—	—^f
Na^+	3.14	3.18	3.08	2.27	32.78	−0.09	0.21	—	250.1	—	−87.61

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We will begin the analysis by discussing H₂O and NH₃. These are well-behaved probe molecules for evaluation of local Lewis acidity since both H₂O and NH₃ interact via a single atom (O/N-down) by donation of an electron lone pair without altering the probe molecules or the Ag₉ substrate's structure significantly. As described in the Introduction section, this kind of interaction between a noble metal NP σ-hole and an electron donor will henceforth be referred to as a regium bond. The computed regium bond energies of H₂O and NH₃ for each unique site of Ag₉ follow the order predicted by V_S(r) with a coefficient of determination (R²) of 0.971 (H₂O) and 0.961 (NH₃). This is with respect to the local V_S,max values determined on the 0.001 a.u. isodensity surface at the sites of interaction. The corresponding R²-coefficents for E_S,min are slightly higher, 0.989 and 0.987 for H₂O and NH₃, respectively (see Fig. 4).

Fig. 4 V _S,max (top) and E_S,min (bottom) values versus local interaction energies of the H₂O, NH₃, H₂S and CO electron donors at the seven unique sites of Ag₉. V_S,max and E_S,min were evaluated at the 0.001 a.u. isodensity surface.

From this isolated comparison one could conclude that E_S(r) and V_S(r) describe the interactions similarly well and it is difficult to assign the relative charge-transfer/polarization and electrostatic character of the interaction. We can also note that V_S,max and E_S,min mutually correlate with an R² value of 0.984. Upon closer analysis it is, however, found that the positions of V_S,max better coincide with the adsorption geometries of NH₃ and H₂O. This can be observed e.g. for position 5 of Ag₉ (Fig. 3), or from the fact that there is no local minimum in E_S(r) for position 2, which argues for a larger portion of electrostatic control in the interactions. From HSAB theory²¹ it is known that both H₂O and NH₃ are considered hard electron donors (Lewis bases); this means that their interactions are expected to be dominated by electrostatics, and not by charge-transfer/polarization. The H₂S molecule is located on the other (soft) end of the HSAB scale. We hence expect the regium bonds between the H₂S molecule and Ag₉ to include a larger portion of charge-transfer/polarization. However, it turns out that V_S(r) and E_S(r) give similar correlations with the H₂S interaction energy as with the H₂O and NH₃ interaction energies. In addition, V_S(r) shows an equally strong correlation with H₂S adsorption as E_S(r): R² = 0.971 and 0.968, respectively. Hence, at this point we cannot draw any definitive conclusions regarding electrostatics compared to charge-transfer/polarization effects. It can, nevertheless, be concluded that the interactions are not purely electrostatic since the average interaction energies of the probe molecules (including also CO, vide infra) do not follow the ranking of V_S,min of the molecules (Table 2). The comparison between electrostatics and charge-transfer/polarization will be further discussed in connection to the interactions of the Au₉ and Cu₉ NP of the Section 3.3.

A general conclusion from this study is that the correlation, or the lack of correlation, between descriptor values and interaction energies can partly be explained by the amount of charge-transfer; from NBO analysis we find an inverted correlation between the R²-values for the different series and amounts of charge-transfer Δq_nbo (see Table 2 and the ESI†). An even stronger inversed trend is found between R²-vaules for the different series and the variance of the charge-transfer within the series. Taken together, this suggests that the predictive power of the ground-state descriptors V_S(r) and E_S(r) suffers the more the electronic structure of the ground state is altered by the interactions. The implication is that soft interactions are more difficult to describe than hard interactions since these by nature cause a larger deviation from the ground state. Similarly, and also as a general conclusion, we find that the more the geometries of the adsorbate and NP are distorted upon interactions, the weaker are the correlations between interaction energies and the descriptor values (see the ESI†). In addition to the above, some deviation between descriptor predictions and computed interaction energies is expected since the descriptors do not directly reflect dispersion effects or vibrational energy contributions.

The interactions of CO with the Ag₉ have previously been studied by Duanmu et al.³⁷ Although we will here treat CO as a Lewis base, the CO molecule is known to be an ambivalent adsorbate with a mode of interaction that is characterized by both electron donation from CO and back-donation from the substrate as a result of a repulsive σ-interaction and constructive π-orbital overlap,^69–72i.e. CO acts as both a Lewis base and, to some degree, a Lewis acid. Upon interactions with metal surfaces it has, moreover, been found that CO undergoes a substantial valence orbital rehybridization.^73,74 The density difference plots obtained for H₂O, H₂S, NH₃, and CO adsorption are shown in Fig. 5 illustrating the different interaction modes where CO adsorption leads to a constructive orbital overlap leading to a buildup of electron density; this is indicative of a covalent bond, whereas H₂O, H₂S and NH₃ adsorption instead polarizes the charge densities of the interacting species. The latter suggests an electrostatic interaction enhanced by polarization rather than a charge-transfer complex. On the other hand the interactions with all Lewis bases result in adsorption distances considerably shorter than the sum of the van der Waals radii of the interacting atoms, which suggest interactions stronger than van der Waals interactions. Furthermore, as discussed above, there is no correlation between the average interaction energy and V_S,min of the Lewis bases, where, e.g., H₂S binds stronger than H₂O.

Fig. 5 Density difference plots at the 0.00075 a.u. contours for (a) CO, (b) H₂O, (c) H₂S and (d) NH₃ adsorption onto atomic site 4 of Ag₉. Color: purple = density accumulation, blue = density depletion. The figures are obtained at the optimized adsorbate structures by subtraction of the individual densities of Ag₉ and the adsorbate, with frozen coordinates, from the total density of the Ag₉–adsorbate complex.

Fig. 6 The favored adsorption structures for the H₂O, H₂S, NH₃ and CO Lewis bases onto Ag₉.

Given the complex nature of the CO interaction it is expected that neither V_S,max nor E_S,min will fully capture the interaction by itself. Indeed, among the considered adsorbates the CO interaction energy displays the weakest correlation with respect to the descriptor values [R² = 0.918 for V_S(r) and 0.911 for E_S(r)]. Nonetheless, these correlations are significantly better than those obtained by Duanmu et al.³⁷ based on CM5M⁶⁶ partial atomic charges (R² = 0.74). The comparison between the V_S(r) and E_S(r) descriptors and partial charges for CO interactions will be discussed further in Section 3.4.

Lewis basicity. In order to probe the local Lewis basicity of Ag₉ we have studied adsorption of the BH₃, BF₃, Na⁺ and HCl (via the H-down adsorption mode) electron-acceptors. All sites, including the on top, bridge and hollow sites, were considered. The adsorption structures primarily converged to the hollow h₁₃₅, h₁₅₈, h₂₄₅, h₂₅₈, h₁₇₈, h₃₄₅ and h₇₈₉ sites, but also in some few cases to the b₁₃, b₃₄ and b₈₉ sites. In line with the predictions by the descriptors, the h₁₃₅/h₁₅₈ and h₂₄₅/h₂₅₈ regions correspond to the largest interaction energies for all of the considered electron-acceptors. While the qualitative picture obtained by V_S(r) and Ī_S(r) is clearly correct, the correlations between local descriptor values at the 0.001 a.u. isosurface and interaction energies are weaker than those found for local Lewis acidity. For V_S,min we find R²-values of 0.659 (BH₃), 0.922 (BF₃), 0.950 (Na⁺) and 0.821 (HCl) when considering the hollow site adsorption. For Ī_S(r) the corresponding values are 0.768 (BH₃), 0.835 (BF₃), 0.901 (Na⁺) and 0.695 (HCl). The good performance of V_S(r) for Na⁺ indicates primarily electrostatic interactions between Ag₉ and the Na⁺ cation. The weak correlations for the other electron-accepting adsorbents reveal a more complex interaction compared to interactions of the electron-donating molecules (e.g. H₂O). For instance, BH₃ not only interacts via the B atom but also via its H atoms, which are eclipsing adjacent Ag atoms. In addition BH₃ (planar structure as unbound) is bent upon interaction with Ag₉. This adds further complexity to the analysis. However, by forming a linear-combination between Ī_S,min at the site of the B atom interaction and the sum of V_S,max at the H-accepting Ag on top sites, a correlation with an R² value of 0.982 is obtained. This indicates that more information is needed to describe this particular interaction compared to, e.g. the interaction with H₂O.|| Similar arguments can be invoked for BF₃. However, the effects are smaller since BF₃ forms weaker bonds with the Ag₉ nanoparticles than BH₃. The geometrical details for the electron-acceptors, including average adsorption distances and interaction energies, are summarized in Table 2. Fig. 7 shows the lowest energy adsorption structures for the various probe molecules’ interactions with Ag₉.

Fig. 7 The favored adsorption structures for the BH₃, BF₃, Na⁺ and HCl (H-down) electron-acceptors onto Ag₉. Note that BH₃ favors the h₁₅₈ site while the remaining compounds prefer the h₂₃₈ site. The depicted bond distances correspond to those between the central atom of the adsorbent and the adjacent Ag₉ atoms.

Similar to the case of Lewis base adsorption, we can employ local descriptor values (the V_S,max and E_S,min) at the central interacting atom of the probe molecules to better understand the nature of the interaction. We did, for instance, find that E_S,min could not be determined on either HCl or BF₃ since, at the considered level of theory, they do not have virtual orbitals of negative eigenvalues. Consequently the contribution from charge-transfer to the Lewis acidity is expected to be small, which is corroborated by a NBO analysis (Table 2). Moreover, the relative interaction energies do not follow the ranking of V_S,max of the Lewis acids. Again this underlines that the interactions of (some of) the Lewis acids are more complex and not dominated by electrostatics. An interesting observation in connection to this is that in the case of BH₃, NBO analysis shows a large charge-transfer of around 0.6e⁻ from Ag₉ to BH₃. Hence the strong charge-transfer interaction compensates for its comparable small electrostatic driving force giving rise to a comparably strong overall interaction.

3.3. Lewis acidity of Au₉, Ag₉ and Cu₉

As shown above, clearly V_S(r), E_S(r) and Ī_S(r) reflect the interaction behavior of the Ag₉ nanoparticle. However, are these descriptors able to capture trends also for other group 11 NPs? This will be investigated by comparing the Lewis acidity of the three elemental isomers Cu₉, Ag₉ and Au₉ probed by H₂O and H₂S adsorption. Since we are here focusing on Lewis acidity, only the V_S(r) and E_S(r) descriptors will be considered.

Apart from the fact that the Cu particle is considerably smaller in comparison to the other particles, Au₉, Ag₉ and Cu₉ are structurally identical. The similarities are reflected in the V_S(r) and E_S(r) isodensity maps shown in Fig. 2 that display a close resemblance between the particles. The V_S(r) and E_S(r) maps indicate that the on top sites are Lewis acidic centers also for the Au and Cu particles. Accordingly, adsorption of H₂O and H₂S takes place on top of the atomic positions also for the Au and Cu particles. The average binding distances for H₂O are 2.17 ± 0.11 Å for Cu₉, 2.47 ± 0.15 Å for Ag₉, and 2.51 ± 0.12 Å for Au₉. On the average, H₂O adsorbs strongest onto the Cu particle by −7.4 kcal mol⁻¹, by −6.2 kcal mol⁻¹ and 5.6 kcal mol⁻¹ onto Au₉ and Ag₉ respectively. H₂S adsorbs at the same sites but here Au₉ displays the strongest average interactions (−14.8 kcal mol⁻¹) followed by Cu₉ and Ag₉ (−10.2 and −7.9 kcal mol⁻¹). Upon interaction between Au₉ and H₂S, the particle geometry is largely distorted. This is a reflection of the well-known fluxionality (i.e. geometric adaptability) of Au NPs, a feature that is often acknowledged as an important factor for the high catalytic activity of Au NPs.⁷⁷ Due to the large geometric rearrangement upon the H₂S–Au₉ interactions, the Au₉ atoms were constraint at their original positions during optimization of the adsorption structures in order to facilitate the analysis of the correlation between descriptor values and interaction energies.

Fig. 8 shows that the trend in H₂O interaction energies of the particles closely follows the variations of the relative magnitude of both V_S,max and E_S,min on the 0.001 a.u. isosurface for all the nanoparticles. For H₂O adsorption, E_S,min gives similar correlations for all three particles with R² ≈ 0.985, whereas V_S,max gives slightly weaker correlations for Au and Cu (R² ≈ 0.950) compared to Ag₉ (R² = 0.971). We can, moreover, note that H₂O and Cu₉ form the strongest interaction amongst the nanoparticles. This is well-reflected by the electrostatic potential since Cu₉ has the largest V_S,max of the particles. E_S,min does, on the other hand, predict the Au₉ nanoparticle as the most reactive, which instead is in-line with the H₂S adsorption energies. It is interesting to note that the Cu particle, which is considered a harder metal than Au, forms the strongest bonds to the hard H₂O adsorbate, while the soft Au₉ forms the strongest bonds with the soft H₂S adsorbate. As concerning the H₂S correlations, the Au₉–H₂S interaction gives comparable poor R² values of 0.856 for V_S,max and 0.916 for E_S,min on the 0.001 a.u. isosurfaces, respectively (see Fig. 8). This can be attributed to a large charge-transfer/polarization effect upon H₂S adsorption leading to a redistribution of the electronic structures, which is a reflexing of the large tendency for structural changes of Au₉ upon H₂S adsorption. For Cu₉–H₂S interactions we find correlations of R² = 0.941 for V_S,max and R² = 0.963 for E_S,min, comparable to, but slightly weaker than, those of Ag₉.

Fig. 8 Correlation plots for H₂O (left) and H₂S (right) interaction energies onto the various sites of the geometrical isomers Au₉, Ag₉ and Cu₉versus the site resolved V_S,max (top panel) and E_S,min (middle panel) obtained at the 0.001 a.u. isosurface. The bottommost figures show the calculated interaction energy, ΔE_int, of H₂O and H₂S versus predicted interaction energies, ΔE_pred, from multi-linear combinations of the V_S,max and E_S,min.

In order to find an explanation to the differences between the particles, we turn to an electronic structure analysis. The three particles are approximately isoelectronic in the valence shell and display similar orbital configuration profiles, with Au having unoccupied orbitals of slightly lower energy as well as two more unoccupied orbitals below the zero-energy limit compared to the other particles. Apart from that, all orbitals are isolobal for the Au, Ag and Cu particles, with only slight dissimilarities, as can be seen in for instance the LUMO orbitals of Fig. S4 in the ESI.† Hence the variations in the V_S(r) and E_S(r) amid the particles are due to small differences in their electronic structures, which illustrate the need for a more sophisticated interaction affinity descriptor than e.g. the HOMO and/or LUMO energies. The small deviations between the particles in the correlations of the descriptor values versus adsorption energies can further be explained by geometrical effects upon adsorption, with larger distortions generally giving rise to reduced correlations (see the ESI†).

As mentioned in the introduction, interactions are often rationalized in terms of contributions of different physical characters. While V(r) obviously provides information on the particles’ ability to participate in electrostatic interactions, E(r) also accounts for the ability of the particle to accept electrons in electron-transfer processes taking place upon interactions. The relative influence of electrostatics versus charge-transfer/polarization in the various interactions is attainable by the formation of a multi-variant combination of the V_S,max (electrostatics) and E_S,min (both electrostatics and charge-transfer/polarization) properties of the different sites of the Au₉, Ag₉ and Cu₉ particles. First of all, one should note that, although the NPs display differences in both the descriptor properties and in the interaction energies, the H₂O interaction energies of all the particles put on the same trend line yields a clear correlation with the local values of the individual descriptor, as can be seen in Fig. 8. By this approach, the obtained R² values for V_S,max and E_S,min are 0.790 and 0.823 respectively. This can be compared to the corresponding trends for H₂S adsorption, which display much weaker correlations with R² values of 0.109 (V_S,max) and 0.502 (E_S,min) respectively (see Fig. 8). By the formation of a linear combination between the V_S,max and E_S,min determined at all the unique adsorption sites of the three NPs, the corresponding H₂O interaction energies are predicted with a R² coefficient of 0.851, i.e. only moderately better than the individual descriptors. For the case of H₂S a more pronounced effect is obtained and the correlation is increased to R² = 0.866 (Fig. 8). From the above one could argue that in order to capture the variations of the H₂S interaction over the different particles, one has to account for both their electrostatic and charge-transfer/polarization capabilities. In contrast, the interaction of H₂O seems to be of a simpler nature; based on the hard character of H₂O, and the generally good performance of V_S,max for the estimation of H₂O interaction energies, it is reasonable to assume that the interactions are dominated by electrostatics for all the considered NPs. The difference between H₂O and H₂S can be further demonstrated by assessing the weighting coefficients in their respective multi-linear combinations: for H₂O the weighting coefficients of V_S,max and E_S,min are −0.134 and 0.495, whereas for H₂S the coefficients are 0.838 and 3.301 (V_S,max in kcal mol⁻¹ and E_S,min in eV). Note here that the sign of the V_S,max coefficient is changed going from H₂O to H₂S. Since E_S(r) consists of both electrostatic and charge-transfer/polarization contributions, this indicate that, in order to be able to compare the H₂S interaction of the different particles, we have to remove a portion of the electrostatic component. This is in contrast with H₂O, where the electrostatic component of E(r) instead has to be reinforced.

3.4. CO adsorption onto Ag₉–Ag₁₈ particles

In the study of Duanmu et al.,³⁷ CO adsorption energies onto each unique Ag site were determined for the Ag₉, Ag₁₁, Ag₁₇ and Ag₁₈ nanoparticles using the N12⁷⁸ DFT exchange–correlation functional in combination with the 6-311+G(2df,2p) basis set for carbon and oxygen and cc-pVDZ-PP for Ag. As described in Section 3.3, CO interacts C-down and adsorbs at on top sites of the Ag-atoms. The interaction energies range from −3.5 to −16.1 kcal mol⁻¹. By re-optimization of the adsorption structures on the Ag₉ particle, we find binding distances of 2.14 ± 0.03 Å.

Duanmu et al.³⁷ used partial charges to rationalize the variations of the CO interaction energies at different sites of the Ag nanoparticles. As shown in Table 3, the V_S,max descriptor performs better than the CM5M partial atomic charges presented in the work of Duanmu et al., both overall and for each of the particles separately. The same is true when comparing V_S,max to atomic charges obtained by other protocols, including Mulliken, Bader and NBO charges. The success of V_S,max is not surprising in this context. In contrast to partial charges, the electrostatic potential is a rigorously defined property and provides an unbiased understanding of the interaction. In addition, the local electrostatic potential is a physical observable and its spatial variations are known to be influential in directing chemical interactions.⁸

Table 3 Squared coefficients of correlation (R²) for the site resolved CO adsorption energies^a onto Ag_n, n = 9, 11, 17 and 18, compared to local descriptor values of V_S(r), E_S(r) and Ī_S(r) on both the 0.004 a.u. and 0.001 a.u. isosurfaces as well as CM5M, Mulliken, Bader and NBO partial charges. Cross-correlated R² = Q², and standard errors (SE) for the total series are included for comparison

	V 0.004	V 0.001	E 0.004	E 0.001	V + E^d	Ī S,ave	E + I^e	CM5M^a	Mulliken	Bader	NBO
a From ref. 37. b The Ag9(2), Ag17(1), Ag17(5), Ag18(1), Ag18(6), Ag18(7) and Ag18(8) sites of the 0.004 a.u. isodensity surface have no true minimum in E(r). c The Ag9(2), Ag11(4), Ag17(1), Ag17(5), Ag18(1), Ag18(3), Ag18(6), Ag18(7) and Ag18(8) sites of the 0.001 a.u. isodensity surface have no true minimum in E(r). Instead the E(r) values on the isodensity surface along the Ag-nucleophile bond were used. d Estimated interaction energy obtained by a linear combination of V = V0.001(r) and E = E0.001(r): ΔEint(estimated) = 0.256V − 0.034E − 0.048 [kcal mol^−1]. e Estimated interaction energy obtained by a linear combination of E = E0.001(r) and I = ĪS,ave: ΔEint(estimated) = −0.047E + 0.257I − 37.420 [kcal mol^−1]. f The corresponding R^2 with H2O adsorption energies at the PBE0-D3(BJ)/def2-TVZPP//PBE0-D3(BJ)/def2-TVZP level of theory is V0.004 = 0.950 and V0.001 = 0.938 as well as E0.004 = 0.919 and E0.001 = 0.922. g A similar linear combination of V + I gives a R^2 of 0.830.
Ag9	0.934	0.918	0.887	0.911	0.918	0.505	0.881	0.74	0.021	0.007	0.751
Ag11	0.907^f	0.868^f	0.743^f	0.761^f	0.832	0.516	0.794	0.79	0.135	0.074	0.614
Ag17	0.828	0.838	0.942	0.941	0.901	0.938	0.960	0.84	0.072	0.001	0.440
Ag18	0.781	0.812	0.859	0.834	0.827	0.823	0.835	0.78	0.037	0.097	0.341
Total	0.817	0.819	0.835	0.819	0.839	0.731	0.850^g	0.73	0.047	0.030	0.427
Q ^2	0.779	0.786	0.804	0.782	0.808	0.690	0.822	0.68	−0.111	−0.072	0.357
SE	1.393	1.385	1.322	1.387	1.307	1.689	1.263	1.687	3.180	3.209	2.467

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Regarding the E_S(r) descriptor, it gives the best overall correlation amongst all individual descriptors if evaluated at the 0.004 a.u. isosurface, but performs poorly for the Ag₁₁ particle. This can be traced to an underestimation of the interaction strength at the Ag₁₁(7) site, which is also manifested, but to a smaller degree, for the V_S(r) descriptor. On the other hand, the E_S(r) correlation for the CO adsorption onto the Ag₁₇ particle is very good with a coefficient of determination (R²) of 0.942 on the 0.004 a.u. isodensity surface. Moving on over the series of Ag particles, we find that E_S(r) indeed performs well as an indicator of the CO-affinity. The performances of both E_S(r) and V_S(r) for CO adsorption are, however, slightly inferior to that of e.g. H₂O adsorption, and could be linked to the mixed donor–acceptor interaction character of CO, as pointed out previously in Section 3.2, whereas H₂O is purely an electron donor. Fig. 9 gives an overview of the performances of V_S(r) and E_S(r) for the estimations of the local CO-affinity evaluated on the 0.001 a.u. isodensity surfaces.

Fig. 9 Plots presenting the correlation between site-resolved CO adsorption energies and descriptor values at the adsorption sites for the Ag₉, Ag₁₁, Ag₁₇ and Ag₁₈ nanoparticles separately and combined (total). V_S,max (top) and E_S,min (bottom) were obtained at the 0.001 a.u. isodensity surface.

In addition to the analysis on the 0.004 a.u. isodensity surface, V_S,max and E_S,min values obtained on the 0.001 a.u. isosurface are also included in Table 3. Although the differences are small, the comparison suggest that the 0.004 a.u. isosurface is slightly better overall for E_S(r) while for V_S(r) the 0.001 and 0.004 a.u. isosurfaces give similar predictions of the Ag NP CO affinity. However, for the interactions with other adsorbates and including the Au and Cu particles, the general conclusion is that the 0.001 a.u. isosurface gives slightly more accurate results than the 0.004 a.u. isosurface for the group 11 metal NPs.

Since E(r) and V(r) are known to provide complementary information to some degree,²³ it is not surprising that a multi-linear combination of E(r) and V(r), as seen in Table 3, increases the correlation with the interaction energies – this is because we are now able to describe a larger portion of the interaction energy. We, furthermore, find that the dual Lewis acid/base character of CO is reflected by the finding that the Ī_S(r) Lewis basicity descriptor correlates fairly well with the CO adsorption energies [Ī_S(r) is here in the form of the mean average local surface ionization potential at the 0.001 a.u. isosurface, Ī_S,ave, of the atom of interaction since minima in Ī_S(r) could in general not be identified at the atomic sites]. The mixed electron donor/acceptor character of CO is further supported by the fact that a multi-linear combination of Ī_S,ave and E_S,min gives the best overall correlation with the CO–Ag-particle interactions (R² = 0.850).

4. Conclusion

In the current work we have evaluated the performance of a series of isodensity surface based DFT-descriptors for analyzing and predicting local Lewis acidity and basicity. The evaluation was carried out on a selection of Ag, Au and Cu nanoparticles, by correlating descriptor values with site-resolved interaction energies of small probe molecules. The descriptors include the density dependent surface electrostatic potential [V_S(r)] as well as two multi-orbital descriptors: the surface average local ionization potential [Ī_S(r)] that characterizes the local electron donating capacity of the particles, and the local surface electron attachment energy [E_S(r)] that depicts the local electron acceptor capacity of the particle. Qualitatively, both electron donating and accepting properties are well described. From adsorption studies of small probe molecules, we conclude that positions of high susceptibility towards electron-accepting compounds are found at bridge and hollow sites of the noble metal nanoparticles. Upon interactions with electron-donors, the atomic on top sites are instead favored. These findings coincide with areas of high Lewis acidity and basicity defined by the descriptors, e.g., Lewis acidic areas of high V_S(r), i.e. σ-holes, correspond to adsorption sites for electron donors, whereas Lewis basic sites of low V_S(r), i.e. σ-lumps, adsorb electron acceptors.

We find that the interaction energies for the adsorption of CO, H₂O, NH₃ and H₂S Lewis bases can be successfully ranked, both by local maxima in V_S(r), σ-holes, as well as minima in E_S(r) at the on top sites of the noble metal nanoparticles. The descriptors readily outperform the comparatively simple FMO concept and have a much larger predictive power than partial atomic charges obtained by various protocols. Due to the similarity between halogen and hydrogen bonding and the interaction of the NPs with electron donors (e.g. H₂O) via sites of high V_S(r), i.e. σ-holes, we have introduced a new class of bonds – regium bonds – where the name reflects the noble character of the Cu, Ag, and Au metals.

As concerning the interactions with the BH₃, BF₃, HCl, and Na⁺ electron accepting probe compounds, the same quality of correlation cannot be obtained as for the interactions with electron donors. This is attributed to an increased level of complexity upon interactions, including adsorbate bending, multi-atom interaction and rearrangement of the substrate nanoparticle.

We have further described that the Ī_S(r) and E_S(r) descriptors provide complementary information to V_S(r) with regard to the different contributions to the interaction energy: V_S(r) reflects the electrostatic part of the interactions while Ī_S(r) and E_S(r) also quantify the metal particles’ charge transfer/polarization abilities. Although E_S(r) and Ī_S(r) display excellent capabilities to identify and rank nanoparticle adsorption sites, we find from our analysis that the adsorption sites are best identified by V_S,max and V_S,min. Accordingly, electrostatics is expected to be the dominating factor for interactions with hard species. Nevertheless, we also find that in order to reflect trends in the interaction energies of the Cu, Ag and Au particles with e.g. the soft H₂S molecule, both electrostatic and charge-transfer/polarization effects have to be included.

As has previously been demonstrated within the field of molecular chemistry, we anticipate that the tested descriptors will be broadly applicable in the nanoparticle and materials sciences. In the near future we will show this for systems ranging from nanoparticles of various sizes and composition to semi-infinite materials and surfaces including metals and metal oxides. The findings of the present study are envisaged to have implications in areas as diverse as heterogeneous catalysis, nucleation/dissolution processes, particle transportation, chromatography, corrosion and nanotoxicology.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors gratefully acknowledge the financial support from the Swedish Nuclear Fuel and Waste Management Company (SKB) and the School of Chemical Science and Engineering at KTH (via its Excellence award to J. H. S.). The calculations were performed at resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre (NSC) in Linköping University as well as at the PDC Center for High Performance Computing (PDC-HPC).

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp06259a

‡ Note that in KS-DFT most neutral compounds have one or more virtual orbital(s) with negative eigenvalue(s), exceptions are e.g. small and hard molecules such as H₂O. The NPs considered in this study have 22–40 virtual orbitals of negative eigenvalue (see Table 1).

§ Relativistic effects are important, in particular for Au compounds, and affect the geometry and chemical properties significantly. By the use of ECPs scalar relativistic effects are implicitly accounted for. We have previously showed that ECP and relativistic all-electron calculations yield similar results.¹³

¶ At the considered level of theory basis set superposition errors (BSSEs) are small. Using a similar computational set up for H₂O adsorption onto the Cu₇ nanoparticle we have previously found a BSSE of 0.06 eV.⁹

|| It should, nevertheless, be noted that only five points are used in the multi-linear regression, which in-fact is not a sufficient number for a proper analysis.

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