Capacitive Electronic Metal – Support Interactions : Outer Surface Charging of Supported Catalyst Particles

Tobias Binninger,1, ∗ Thomas J. Schmidt,1, 2 and Denis Kramer3, † 1Paul Scherrer Institut, Electrochemistry Laboratory, CH-5232 Villigen PSI, Switzerland 2ETH Zürich, Laboratory of Physical Chemistry, CH-8093 Zürich, Switzerland 3University of Southampton, Engineering Sciences, SO17 1BJ Southampton, United Kingdom Abstract Electronic metal–support interactions (EMSI) in catalysis are commonly rationalized in terms of an electron transfer between support material and supported metal catalyst particles. This general perspective, however, cannot fully explain experimentally observed EMSI for metallic nanoparticulate catalysts, because the strong charge screening of metals should locally confine effects of direct electronic interaction with the support to the catalyst–support interface (CSI), which, apart from the perimeter, is largely inaccessible for catalysis reactants. The concept of capacitive EMSI is proposed here for catalyst particles at the nanometer scale, where electronic equilibration results in a long-range charging of the catalytically active outer surface (CAOS) bypassing the expected strong metallic charge screening, which is confirmed and quantified by electrostatic and density functional theory simulations revealing a strong dependence on the coverage of the support surface with catalyst particles. This long-range charge transfer leads to a shift of the local work function at the CAOS. In order to describe the catalytic consequences, an amendment of d-band theory in terms of ‘d-band + work function’ is proposed. Furthermore, the charging of remote catalytic sites at the CAOS scales with the relative dielectric constant of the surrounding medium and it is concluded that EMSI can have surprisingly strong influence especially in the presence of a strongly polarisable dielectric.


I. INTRODUCTION
Nanoscopic metal catalyst particles supported on metal oxides or carbon materials constitute the majority of heterogeneous catalysts and electrocatalysts used in the chemical industry and studied in catalysis research 1,2 .Furthermore, urgently needed solutions for major global challenges, like global warming and a growing energy demand, will depend on the availability of efficient catalysts.
It was early recognized that the support material can influence the catalytic activity of metal catalyst particles 3 .Besides explanatory schemes based, e.g., on structural or compositional modifications, this 'carrier effect' has been explained by an electron transfer between support material and metal catalyst particles 4,5 .Such 'electronic metal-support interactions' (EMSI) 6 can be rationalized in terms of an electron transfer for metal adatoms and small sub-nanometer sized metal clusters [7][8][9][10][11] due to the formation of polar chemical bonds with more or less ionic character between the support surface atoms and the metal cluster 'adsorbate'.Substantial charge transfer of the order of 0.1-1 |e|/atom can be observed when metal adatoms or sub-nm clusters interact directly with the support surface 12 , especially with oxide support defects or with surface cations of transition metal oxides 11 .This strong electron transfer corresponds to a partial oxidation or reduction of the supported metal atoms with drastic influence on their catalytic properties.
For metallic nanoparticles, a different classification of EMSI emerges: the large number of electronic degrees of freedom of nanometer sized metal particles leads to the formation of a continuum of electronic states with a well-defined Fermi level 13 and an associated work function of the metal nanoparticle.As a consequence, a thermodynamic description of EMSI is justified, where electron transfer is rationalized in terms of electronic equilibration between the support material and the catalyst nanoparticle in analogy to the Schottky theory of metal/semiconductor contacts 3,4,14,15 .At the sub-nanometer scale, the majority of metal cluster atoms are in direct contact with the support surface, especially in the case of monolayer thick two-dimensional clusters.For nanoparticles, where the majority of surface atoms is not in direct contact with the support, the question arises whether EMSI can affect sites at the catalytically active outer surface (CAOS) in addition to how the amount of EMSI-related electron transfer scales with particle size 16 .Due to strong charge screening 17 , the excess charge on metal nanoparticles could be expected to accumulate at the direct contact interface between the metal particle and the support material (the catalystsupport interface, CSI), thus forming a charged double layer at the CSI.In line with this expectation, computational studies on transition metal overlayers fully covering carbon 18 or metal oxide 19 support surfaces came to the conclusion that electronic interactions between metal and support only affect the first two to three metal overlayers.However, as shown in the following, basic electrostatic considerations suggest that, in addition to the sub-nm short-range charge transfer at the CSI, electronic equilibration with the support material leads to a long-range direct charging of the catalyst nanoparticle CAOS, which can affect remote catalytic sites at a distance up to several nanometers away from the CSI.

A. Theoretical derivation of catalyst particle outer surface charging
Figure 1 illustrates the electrostatic argument.In general, the isolated catalyst particles will have a work function W c distinct from the work function W s of the bare support surface (cf. Figure 1a).Upon contact, the support material equilibrates with each of the catalyst particles by electron transfer.The resulting polarization generates an electrostatic potential energy step (−e)∆Φ s−c across the CSI that equilibrates the Fermi levels and is equal to the difference of the two work functions W c − W s (cf. Figure 1b).The corresponding polarization double-layer is localized at the CSI due to the strong charge screening inside the metal catalyst particles.This basic reasoning of charge transfer at the direct CSI is well-understood 15,20 .
An additional long-range charging of the CAOS of the catalyst particles must occur if the catalyst particles do not fully cover the support surface.The overlapping dipole fields of each of the supported, polarized catalyst particles generate an overall electrostatic potential step between support and vaccum (dielectric) ∆Φ s−v (cf. Figure 1c) that is proportional to the average surface polarization density p (cf. proof in the Appendix), with the vacuum dielectric constant 0 and, in case the surface is surrounded by a dielectric medium instead of vacuum, the corresponding relative dielectric constant r .Because the average surface polarization density p is a function of the support surface coverage with catalyst particles, the same holds for ∆Φ s−v .Therefore, the overall surface potential step ∆Φ s−v is generally smaller in magnitude than ∆Φ s−c at the direct CSI, which is fixed by W c − W s , for an incomplete coverage of the support surface with catalyst particles.This difference between ∆Φ s−v and ∆Φ s−c is compensated by an additional charge on the CAOS of each catalyst particle.The electrostatic field associated with this additional CAOS charge generates the required additional electrostatic potential step ∆Φ c−v in order to ensure that the value of the electrostatic potential in vacuum is path-independent (cf. Figure 1b): Furthermore, the additional surface charge on the CAOS of the catalyst particle generates a field contribution inside the particle which opposes the dipole field originating from the polarized CSI, thus fulfilling the requirement of zero net electrostatic field inside the bulk of the metal catalyst particle, which has been pointed out earlier in the context of a Schottky model for metal nanoparticles on semiconductor surfaces 15,21,22 .

B. Electrostatic model simulations
The magnitude of the CAOS charging effect can be estimated with a classical electrostatic model.Simulations were performed with COMSOL.The support material and the catalyst particles were modelled without loss of generality as perfect conductors with a fixed electrostatic potential difference of 1 V between catalyst particle and support, corresponding to a work function difference of 1 eV.A catalyst particle was placed above the support surface inside a cuboid supercell with edge length l c-c and periodic boundary conditions in x-and y-direction parallel to the surface.The system is equivalent to a 2-dimensional square array of catalyst particles with inter-particle distance l c-c .Different geometries of the catalyst particle were used: hemispherical, spherical, and cubical.The distance between the support surface and the flat bottom particle surface was fixed at d c-s = 0.3 nm.The numerical convergence of the electrostatic simulations was confirmed by refinement of the 3-dimensional finite element mesh.In order to calculate the net electrostatic potential Φ and the surface charge density σ generated by the electronic equilibration between the support material and the catalyst particle, the support surface was grounded (Φ s = 0 V) and the catalyst particle potential Φ c = 1 V was fixed for a hypothetical work function difference of W c −W s = −1 eV.
Due to the linearity of Poisson's equation, the results for any other work function difference can be obtained by linear scaling.
Figure 2 plots the electrostatic potential Φ and the surface charge density σ on the support surface and the CAOS for hemispherical catalyst particles.It becomes obvious that not only the entire CAOS of the particle carries charge, but also the surrounding empty support surface.The CAOS charge density gradually decreases towards the top of the catalyst particle.However, even the minimum value of σ c (θ = 0) = 0.049 e/nm 2 is substantial (cf.σ c = 0.184 e/nm 2 at the direct CSI), leading to strong electrostatic fields of the order of |E| ≈ 1 V/nm at the CAOS with increasing strength towards the perimeter of catalyst particle and support.
The same model can be used to gauge the dependence of the CAOS charging effect on catalyst loading and particle geometry.Figure 3 shows the surface polarization density per catalyst particle p Ac , averaged over the projected particle area A c , as a function of the support surface coverage with particles γ = A c /A total .Catalyst particle polarization, and thus particle CAOS charging, is largest in the limit of low coverage γ.For increasing cover- age, the CAOS charging decreases.This depolarization results from the mutual interaction between neighboring polarized particles.In the limit of large coverage γ → 1, the CAOS charging converges to zero and p Ac converges towards the fixed polarization p 0 at the direct CSI for a dense catalyst metal layer.Furthermore, the CAOS charging is very sensitive to the particle shape at small γ: the polarization density for γ → 0 of hemispherical catalyst particles is less than half that of spherical and cubical particles.

C. Density functional theory simulations
Predictions of the classical electrostatic model were further investigated by density functional theory (DFT).Platinum nanoparticles supported on a Sb-doped SnO 2 (110) surface were chosen for this purpose, because this system has attracted substantial attention in recent research on electrocatalysts for the oxygen reduction reaction 23,24 .Periodic DFT computations were performed using the Vienna Ab Initio Simulation Package (VASP).The core electrons were taken into account by the projector augmented wave method (PAW) 25,26 .The generalized gradient approximation (GGA) in the PBE form 27 was used for the exchange-   equilibration with the Sb-SnO 2 support was evaluated by Bader charge analysis 28 .Color coded values of δ are presented in Figure 5a; tabulated data can be found in Table I, averaged over different parts of the Pt particle as well as for specific Pt atoms labelled according to Figure 5a.In line with the sign of W c − W s , electrons are transferred on average to the Pt particle on the reduced Sb-SnO 2 surface, whereas electrons are withdrawn from the Pt particle on the stoichiometric support.
The average value of δ across the entire particle depends only weakly on supercell size and is with approximately 0.15 e − /atom in good agreement with the order of magnitude of experimentally observed particle-averaged values reported for different systems in the literature 16,29 .However, the average without CSI is significantly smaller and drops to 0.016-0.028e − /atom.Vice versa, an extremely large transfer of approximately 0.8 e − /atom is found by taking the average only over the CSI.The average value over the entire Pt particle is, therefore, dominated by the electron transfer at the direct CSI, in agreement with experimentally observed increasing EMSI with increasing CSI contact area 30 .
The coverage-dependent charging of the CAOS is weaker than charging at the CSI.Taking the average of δ only over the outer (100) facet of the Pt particle pointing towards vacuum results in a value of 0.023 electrons per Pt atom for the 2 × 2 supercell, which is in good  a The same value is given as for the 1x1 system.W s is independent of the supercell size for periodic boundary conditions.The work function W c of free Pt particles could have a slight dependence on the supercell size due to the different distances between the periodic images of the particles.This dependence, however, is expected to be small.
agreement with the order of magnitude of CAOS charging estimated from the classical electrostatic model assuming a metal surface atom density of approximately 12-15 atoms/nm 2 .
The average δ over the CAOS drops to almost zero for the 1 × 1 supercell confirming the strong dependence of CAOS charging on the coverage γ.
It can be concluded that quantitative analysis of electron transfer in terms of an average value over the entire catalyst particle 4,14,16,31 must be seen in a context of inhomogeneous charging across nanoparticles with strong localisation near the CSI, coverage dependend charging of the CAOS, and different weighting of CSI and CAOS depending on particle size and shape.
Finally, the analysis of individual Pt atoms furthermore confirms that the transferred charge accumulates at the surface of the Pt particle.The center atom is largely screened and the remaining net Bader charges of this atom, which appear uncorrelated with the equilibration-driven charge transfer, both in magnitude and in sign, can be explained with Friedel oscillations 17 not being fully damped at the center atom for the 1 nm sized Pt particle.
The effect of the long-range CAOS charging that circumvents the charge screening of the Pt particle bulk leads to a significant charge especially on edge Pt sites of the vacuum-facing facet, such as Pt atom T2, to which an amount of 0.041 electrons is transferred for the Pt/reduced Sb-SnO 2 system with 2 × 2 supercell.This transfer decreases to almost zero for the same system with 1 × 1 supercell and close proximity of neighboring Pt particles, a result that could indicate a close relationship with experimentally established catalyst particle proximity effects 32 .

III. DISCUSSION
The long-range effect of EMSI at the nanometer scale can be described in terms of catalysis in a capacitor: The CAOS of the catalyst particle and the surrounding empty support surface correspond to the two charged electrodes of a capacitor at a voltage equal to the "built-in" potential difference ∆Φ s−c between catalyst and support, which is fixed by the corresponding work function difference W c − W s .The influence of such "capacitive EMSI" on the properties of catalytic sites at the CAOS could be classified in terms of electric field effects and surface potential effects on the one hand, and direct charge effects due to the change of electron number at specific CAOS sites on the other hand, as discussed in the following.
According to Sabatier's principle, which is widely accepted in heterogeneous and in electro-catalysis, the binding energies of adsorbing species have a strong influence on the catalytic activity 33,34 .3][44] ).In both cases, a stronger correlation is achieved if the work function is included in the consideration.
The extent to which the catalyst particle outer surface charging influences adsorbate electrodes of a capacitor at a voltage equal to the "built-in" potential difference ∆Φ s−c between catalyst and support, which is fixed by the corresponding work function difference W c − W s .Therefore, enhanced CAOS capacitance is expected in the presence of a dielectric such as water (relative dielectric constant ≈ 80 45 ), and, in a strongly polarisable dielectric, CAOS charging could, in principle, reach values up to the order of 1 |e|/atom.The longrange electron transfer in the presence of a surrounding dielectric could, therefore, become comparable to the short-range electron transfer at the direct CSI, corresponding to a significant valency change of surface atoms with significant consequences for their catalytic properties.However, whereas charging of the CSI affects only catalytic sites that are both in direct contact with the support surface and accessible for the reactants 15,46 , long-range capacitive EMSI can affect all sites at the CAOS of the catalyst nanoparticle as a whole.
Therefore, EMSI can act on catalytic sites at a significant distance from the support surface at the nanometer scale despite the strong charge screening of metals.
As counterpart to the catalyst particle CAOS, the surrounding empty support surface gets charged at the same magnitude but with opposite sign, cf. Figure 2. Corresponding modifications of the co-catalytic properties of oxide supports can be expected in accordance with the electron theory of catalysis on semiconductors 47 .It is, therefore, interesting to contemplate EMSI also in a context of catalytic spillover effects 48 .
Finally, the long-range charge transfer to the CAOS depends on the degree to which the support material can provide mobile electrons to equilibrate with the catalyst particle.
In case of a support material with low density of mobile charge carriers, the formation of a thick depletion layer inside the support decreases the overall capacity of the catalyst CAOSsupport system and results in reduced CAOS charging.Thus, capacitive EMSI are strongest for metallic, weaker for semiconducting, and vanishing for insulating support materials.Introducing free charge carriers into a semiconducting or insulating support therefore increases capacitive EMSI with bearings on the catalytic properties of supported catalyst particles.
This provides motivation to surface treat or dope supports in heterogeneous catalysis applications with a view to provide free charge carriers even though electronic conductivity of the support is not strictly needed.
For a more detailed quantification of the catalytic effects of capacitive EMSI for specific reactions, DFT calculations of binding energies of probe atoms/molecules at the CAOS of supported metal catalyst particles are necessary.In order to observe strong effects due to capacitive EMSI, such computations of adsorbate binding energies must be carried out for systems with very large supercell in the DFT model corresponding to low coverage of the support surface with catalyst particles.Since the required massive computational resources for this purpose exceeded the ones available within the present study, such detailed quantification for specific reactions remains topic of future research on capacitive EMSI.

IV. CONCLUSIONS
Electronic equilibration between metallic catalyst particles and support material results in long-range charging of the catalytically active outer surface of catalyst particles at the nanometer scale bypassing the strong charge screening of the catalyst particle bulk.In this way, electronic metal-support interactions can affect remote catalytic sites at a significant distance from the direct catalyst-support interface.
where ρ is the net electrical charge density resulting from the electronic equilibration between catalyst particles and support material.This surface multipole expansion consists in a Taylor expansion of the factor The first order term Φ (1) (x, y, z) is generated by the surface polarization density p(x , y ) := dz z ρ(x , y , z ).Under the general assumption that the surface polarization density fulfills the limiting property lim A→∞ 1 A A p dA = p with a well-defined area averaged surface polarization density p , the following limiting behavior of Φ (1) holds: This can be derived directly from equation (C4) with the use of Lemma 1 presented below.
Since the first order surface multipole term is antisymmetric in z, the limiting value for z → −∞ is given by the negative of this expression.Proof.For any given > 0 we have to find a z 0 so that ∀z > z 0 the following holds:

Figure 1 .
Figure 1.Schematic of energy levels, work functions, and electrostatic potentials.a) Work functions of the isolated support and catalyst particles; b) illustration of the additional net electrostatic energy contribution (−e)Φ resulting from electronic equilibration.The zero potential level is chosen to correspond to the potential level in vacuum.An outer surface charge on the catalyst particle is necessary to account for the difference ∆Φ c−v of the net electrostatic potential levels in vacuum and inside the catalyst particle, respectively; c) the equilibrated system is characterized by a common Fermi level and work function W c|s .

Figure 2 .
Figure 2. Electrostatic model results.Surface charge density σ on the outer catalyst surface and the support surface and the electrostatic potential Φ for a hemispherical catalyst particle with a diameter of 3 nm.Inset: Angular dependence of the outer catalyst surface charge density.

Figure 3 .
Figure 3. Electrostatic model results.Average surface polarization density per catalyst particle as a function of the support surface coverage with catalyst particles γ for different catalyst particle geometries.

Figure 4 Figure 4 .
Figure 4 plots the net electrostatic potential energy (−e)Φ due to electronic equilibration across the surface for the three different systems.Figure 4a visualises 2D slices through

FigureFigure 5 .
Figure 5b plots the net electron density change for the Pt/reduced Sb-SnO 2 system with 2x2 supercell.The Pt particle CAOS charging is clearly visible in terms of a halo around the CAOS corresponding to an increased electron density in agreement with the direction of electron transfer expected from the work function difference W c − W s .The change of electron number δ for each individual Pt atom of the nanoparticle as a result of electronic Table I.DFT results.Work function difference W c − W s between the individual systems; Net potential energy step (−e)∆Φ s−c across the direct CSI; Number of electrons transferred per Pt atom: Average (AVG) over the total Pt cluster, AVG without the first Pt layer in direct contact with the support (w/o CSI), AVG only over the first Pt layer in direct contact with the support (CSI), AVG only over the outer (100) facet towards vacuum; Specific Pt atoms: Atom C (cluster center), Atom S1 (side (100) facet center), Atom S2 (side corner), Atom T1 (outer (100) facet center), Atom T2 (outer edge), Atom T3 (outer corner), cf. Figure 5a.Values in e − /atom with positive values corresponding to an increased electron number.
. a. Electric field effects of capacitive EMSI.The binding energies of adsorbates with polar bonds are influenced by the dipole-field interaction energy E = −d • E between the bond dipole d and the electrostatic field E at the CAOS.Typical dipole moments associated with polar adsorbate bonds are of the order of 0.01 |e|nm 36 yielding maximum changes of adsorbate binding energies of the order of 0.01 eV for field strengths of approx. 1 V/nm at the CAOS.Thus, electric field effects of capacitive EMSI via dipole-field interactions appear to be comparably small.b.Surface potential/work function effects of capacitive EMSI.Stronger catalytic effects due to capacitive EMSI can be expected from changes of the local work function 37 , i.e. the surface potential at the CAOS 38 , which arise from the strong electrostatic fields at the charged CAOS.Since the local work function influences the relative alignment of electronic adsorbate states and metallic surface states, strong implications for the catalytic properties can be expected 37,39 .These implications are discussed in more detail in the following proposing an amendment of d-band theory to include effects of the work function.According to d-band theory, molecular adsorption energies are correlated with the position of the electronic d-band center d with respect to the Fermi level f of transition metal catalysts, because the relative positions of catalyst d-states, catalyst Fermi level f , and HOMO/LUMO states E i of the adsorbing molecule determine the strength of the splitting between bonding and anti-bonding adsorbate states and the degree of filling of the latter 40-42 .However, Figure 6 illustrates that the relative alignment of the interacting electronic states is determined not only by the position of the d-band center relative to the Fermi level of the transition metal, f − d , but also by the position of the Fermi level relative to the vacuum level, i.e. the work function W .In case that the adsorbate state is located entirely outside the surface potential step, it is the sum of work function and d-band center (relative to f ) that determines the energetic position of the metal d-states relative to the original molecular orbitals.Changes in the work function shift the metal d-states relative to the molecular states thereby altering the energetic resonance between these interacting states.This has a threefold effect on the binding energy of the adsorbate by changing (i) the absolute positions of bonding/anti-bonding adsorbate states with respect to the unperturbed states, (ii) the splitting between bonding and anti-bonding adsorbate states, and (iii) the degree of filling of anti-bonding adsorbate states.This reasoning led us to revisit previously established correlations between catalytic activities and metal d-band centers.Figure 7 plots oxygen and nitrogen adsorption energies, as proxies of catalytic activity towards oxygen reduction and ammonia synthesis, as a function of d-band center (relative to f ) alone and d-band center plus work function, respectively

Figure 6 . 3 )Figure 8 Figure 7 .Figure 8 .
Figure 6.Illustration of the relevance of the work function in d-band theory for the relative alignment of energy levels and filling of anti-bonding states.

Appendix A :
Computational details for DFTPeriodic DFT computations were performed using the Vienna Ab Initio Simulation Package (VASP).The core electrons were taken into account by the projector augmented wave method (PAW)25,26 .The generalized gradient approximation (GGA) in the PBE form27 was used for the exchange-correlation functional.The simulated system consisted of cuboctahedral Pt 55 clusters supported on Sb-doped SnO 2 slabs comprising nine atomic layers with the most stable (110) surface orientation of the rutile crystal structure.Both stoichiometric (oxidized) and reduced (110) surfaces of the oxide support were studied.In the latter case, the topmost row of oxygen atoms was removed from the (110) surface.The detailed atomic coordinates for all investigated systems are specified in the POSCAR input files for VASP which are provided as Supplemental Material49 .The Pt 55 clusters were placed symmetrically on both support slab surfaces in order to avoid problems with long-range dipole interactions between adjacent images of the catalyst decorated slabs generated by the periodic boundary conditions.The energy cutoff of the plane wave basis set for the wave functions was chosen at E cutoff = 500 eV.The extension of the periodic supercells in z-direction perpendicular to the surface was chosen large enough to accommodate a vacuum region of ≈ 1.6 nm in between adjacent images.In x-and y-direction, the smaller supercells (1x1) measured ≈ 1.3 nm containing eight unit cells of the SnO 2 (110) surface.Due to the diameter of the Pt 55 cluster of ≈ 1.0 nm, the spacing between adjacent Pt 55 clusters was only ≈ 0.3 nm resulting in a high support surface coverage γ = A Pt /A total ≈ 0.6.The Pt/reduced Sb-SnO 2 system was furthermore investigated with a large supercell comprising 2x2 the dimension of the small supercell in x-and y-direction, yielding a coverage γ ≈ 0.15.Only the Γ-point in reciprocal k-space was sampled due to the large size of the system.Computations were performed both on the IRIDIS High Performance Computing Facility, University of Southampton, UK, and on the ARCHER UK National Supercomputing Service.The lattices of the Pt cluster and of the SnO 2 slabs were fixed for the individual systems and atomic relaxation of the combined system was switched off in the calculations.In this way, structural interactions between catalyst and support were excluded.Only the distance between the Pt cluster and the support (110) surface was relaxed until an energy minimum was reached.The lattice constant of the fcc Pt unit cell was determined in a fully-relaxed Pt bulk calculation to a Pt = 3.967 Å.A first attempt of a fully-relaxed bulk SnO 2 calculation yielded a band gap of ≈ 0.4 eV, which is much smaller than experimentally observed values > 3 eV.Therefore, in order to obtain an SnO 2 band gap of E g = 2.3 eV, it was decided to fix the lattice constants of the SnO 2 body centered tetragonal unit cell at a SnO 2 = b SnO 2 = 4.5 Å and c SnO 2 = 3.1 Å, which still agrees well with experimentally determined unit cell parameters.The Sb-doped support slabs were constructed by replacing five evenly distributed Sn atoms by Sb atoms (per smaller supercell), corresponding to a support composition of Sb 0.035 Sn 0.965 O 2 .Work functions were determined as the difference between the plateau value of the electrostatic Hartree potential in the vacuum region and the Fermi level of the system, W = Φ Hartree (vac) − E Fermi .The superposition of the electrostatic Hartree potentials of the free Pt clusters and of the bare Sb-doped SnO 2 slabs was subtracted from the electrostatic Hartree potential of the combined system in order to obtain the net electrostatic Hartree potential Φ(r) due to equilibration.The same procedure was performed with the corresponding electron density distributions in order to obtain the net electron density change due to equilibration.In addition, for an analysis of the net electron transfer between support and Pt particles, Bader charges 28 of each Pt atom of the free Pt clusters were subtracted from those of the combined system.electrostatic potential

All terms of order > 1 0 .Lemma 1 .
of the surface multipole expansion do not contribute to the overall potential step and thus the work function: Their respective contribution to Φ gains one effective z-factor in the denominator with each increase of the multipole order, thus, yielding a zero limiting value for |z| → ∞.The overall potential step across the surface, therefore, is entirely determined by the average surface polarization density p and given by the expression∆Φ := lim z→∞ Φ (1) (x, y, z) − lim z→−∞ Φ (1) (x, y, z) =p Given a function p : R 2 → R on the 2-dimensional plane that fulfills the limitlim A→∞ 1 A A p dA = p (C7)with a well-defined area average p , then the following relation holdslim z→∞ ∞ 0 dr r p(zr) [1 + r 2 ] 3/2 = p ,(C8)where p(r ) := 1 2π 2π 0 p(r , φ) dφ and p is written in terms of polar coordinates (r , φ) on the 2-dimensional plane.The existence of the integral on the left-hand-side is implicitly assumed.
r 2 ] 3/2 − p < .(C9)By definition of the integral with upper bound ∞, we know that ∃R > 0 so that∞ 0 dr r p(zr) [1 + r 2 ] 3/2 − R 0 dr r p(zr) [1 + r 2 ] 3/2 < 5 .(C10) SystemW c − W s (−e)∆Φ s−c AVG AVG AVG AVG Atom Atom Atom Atom Atom Atom This long-range charge transfer scales with the work function difference between catalyst and support material, and it depends on the size, shape, and proximity of the catalyst nanoparticles, which suggests a close connection between such capacitive EMSI and well-established catalyst particle size and proximity effects.Furthermore, strongest catalytic effects are predicted in the presence of a dielectric due to the scaling of CAOS charging with the relative dielectric constant of the surrounding medium.