Thermodynamic Cyclic Voltammograms Based on Ab Initio Calculations: Ag(111) in Halide-Containing Solutions

Cyclic voltammograms (CVs) are a central experimental tool for assessing the structure and activity of electrochemical interfaces. Based on a mean-field ansatz for the interface energetics under applied potential conditions, we here derive an ab initio thermodynamics approach to efficiently simulate thermodynamic CVs. All unknown parameters are determined from density functional theory (DFT) calculations coupled to an implicit solvent model. For the showcased CVs of Ag(111) electrodes in halide-anion-containing solutions, these simulations demonstrate the relevance of double-layer contributions to explain experimentally observed differences in peak shapes over the halide series. Only the appropriate account of interfacial charging allows us to capture the differences in equilibrium coverage and total electronic surface charge that cause the varying peak shapes. As a case in point, this analysis demonstrates that prominent features in CVs do not only derive from changes in adsorbate structure or coverage but can also be related to variations of the electrosorption valency. Such double-layer effects are proportional to adsorbate-induced changes in the work function and/or interfacial capacitance. They are thus especially pronounced for electronegative halides and other adsorbates that affect these interface properties. In addition, the analysis allows us to draw conclusions on how the possible inaccuracy of implicit solvation models can indirectly affect the accuracy of other predicted quantities such as CVs.


Digitized Experimental Results and Normalization Proce-
. Figure S2: Digitized full CVs from Foresti et al. S1 . The sharp peaks at the higher potential end are due to formation of silver halide surface layers and not plotted in the main text for reasons of clarity. The orange star marks the reference current j ref before the onset of halide electrosorption, which were used in combination with the capacitance values C ref of the pristine slabs (see Supplementary Fig. S1) to transform experimental CV currents j to scan-rate independent pseudocapacitance curves C exp pseudo reported in the main text. S-3

Additional computational details and results
Implicit solvent model Figure S3: Sketch of the interfacial capacitance in implicit models for high electrolyte concentrations. Under these conditions the potential drop within in the solvent can be neglected as it is small compared to the one that occurs across the vacuum gap between quantummechanical region and the onset of the dielectric in the implicit model with ≈ 78. Thus the interfacial capacitance is dominated by the vacuum-solvent gap and thus the parametrization of the dielectric onset in the implicit model. The sketch allows as well an intuitive understanding of the reduction in the (average) interfacial capacitance for adsorbate-covered surfaces in implicit models, as observed.
As stated in the main text, we chose as an implicit solvation model the SCCS implementation of ENVIRON S2-S5 with optimized interfacial parameters (ρ min = 0.0013, ρ max = 0.01025, α = β = γ = 0) and a Helmholtz-layer representation of the electrolyte via gaussian-shaped planar counter charges (width: 1 bohr) at a distance of 6 Å from the surface. This solvent parametrizations and electrolyte representation yields good agreement in the interfacial capacitances with the experimental system under study and other systems (≈ 40 − 50 µF/cm 2 for clean low index Ag and Pt surfaces, see Refs. S5,S6).
Note that all implicit solvent models behave approximately like two capacitors in series S7,S8 : the first capacitor is characterized by the vacuum gap between the quantummechanical region and the onset of the dielectric, which is where polarization charges screen effectively 99% (1-1/ , with the dielectric constant of water as used in the implicit model, ≈ 78) of the interfacial field. The second capacitor is between the dielectric onset and S-4 the position of the electrolyte counter charges, which are, for reasonable models, confided to the dielectric region. As a result, it is the potential drop between quantum-mechanical region and the onset of the dielectric that puts an upper limit to the total DL capacitance in implicit models, independent on the electrolyte model (cf. also Fig. S3).
Standard parametrizations of SCCS implicit solvent model (e.g. to reproduce the solvation energy of water, or uncharged and positive small molecules) yield interfacial capacitances between 10-20 µF and thus below the values of the experiments studied in this work, independent on electrolyte representation.
Thus it is only possible to approximately reproduce experiments with larger interfacial capacitances by reparametrizing charge-density thresholds for the dielectric onset in the SCCS model. Note that our parameters are in fact approximately the average between parameters used to optimally describe the solvation of anionic, cationic and neutral solvent molecules, thus likey describing all of these good (in average) (cf. Ref. S6).
Turning now to the electrolyte model: A simple Guy-Chapman picture (= solution to the 1D linearized Poisson-Boltzmann model) yields an interfacial capacitance equivalent to a plate capacitor model with a fixed distance between surface and electrolyte screening charges.
This fixed distance is related to the electrolyte concentration and extremely small (O(Å)) for high electrolyte concentrations, which we try to mimic by our chosen planar counter charges at a distance of 6 Å from the surface. In this case the interfacial capacitance does not exhibit a strong dependence on the applied electrode potential (see e.g. Ref. S9 and Fig. S5 here).
For high electrolyte concentrations, also all other reasonable electrolyte models (e.g. modified Poisson-Boltzmann, with or without Stern layer S8,S10 ) that confine the electrolyte charge to the implicit region (dielectric constant ≈ 78), will lead to electrolyte counter charges that are located in close proximity to the interface region (O(Å)). Due to the fact that all fields in this region of space are scaled down by a factor of ≈ 1/78 the total potential drop in this region of space is scaled down by an according factor.
As a result, the total potential drop across the DL in an implicit model at high electrolyte S-5 representations consisting first of the potential drop in the discussed vacuum gap and the consecutive 'solvent' gap up to the electrolyte counter charges, is fully dominated by the former S7 . As a result of the small value of the latter the details of the electrolyte model are unimportant for the observed total capacitance. This picture of the interfacial capacitance also explains the observed reduction when the surface becomes adsorbate-covered and thus there is an effective increase in the vacuum gap (cf. Fig. S3).
It is worth noting, that this picture changes when studying systems at low electrolyte concentrations, where the potential drop within the dielectric region becomes sizable, and thus making the specifics of the electrolyte model visible and significant. For more details, we refer the reader to the discussions in Refs. S5,S11. In general, more complex electrolyte models e.g. based on a modified Poisson-Boltzmann model and with or without Stern layer, lead to a potential dependence in the interfacial capacitance, which however is mostly pronounced for small electrolyte concentrations S5,S11 , and (thus) small capacitances.
Here, we use only a quadratic approximation to the potential dependence for the interfacial energetics, and thus a priori neglect all higher order variations, which is why we also rather want to avoid these in the implicit model calculations.
In summary, our specific choice of implicit model parametrization and electrolyte representation is thus based on the fact that we indeed try to model a situation at high electrolyte concentrations (note the halide concentrations in Ref. S1 are only 0.5 mM, however all experiments are done with 0.15 M NaOH, 0.1 M KPF 6 background electrolyte), the fact that there is no (clear) evidence for a very complex interfacial double layer capacitance (see Figure S1) and the fact that we aim at only discussing the effects of an assumed potential-independent interfacial capacitance.

Vibrational free energy correction
The free energy corrections ∆F α,corr surf,vib due to the vibrations of the adsorbed halides are determined from the concentration-dependent, DFT-derived vibrational spectrum of the S-6 adsorbates only and are plotted in Supplementary Fig. S4.  Figure S4: Numerical values of ∆F α,corr surf,vib for the individual systems. Note, that these are included in the reported adsorption energies in Supplementary Table S1.

Surface charging
All unknown quantities of the derived expressions in the harmonic CHE+DL approximation are properties at the PZC, which are parametrized using first-principles calculations, as dis- (1) The differential interfacial capacitance for given configuration α can be obtained from the work function dependence on the net surface charges. As shown in previous works S9 it is approximately independent on the potential, leading to the employed, harmonic CHE+DL S-7 model of this work S9 (cf. eq. 7 of the main text).
Therefore the interfacial capacitances C α 0 are obtained by polynomial interpolation of Φ α (n net e ) -the work function dependence on the net surface charges -and taking the first order expansion term as estimated capacitance value (cf. Supplementary Fig. S5). The accuracy of the harmonic CHE+DL approximation S9

S-9
Simulated CVs using the revPBE functional Coverage-dependent adsorption energies as well as PZCs were obtained with consistent computational parameters for the revPBE functional (based on in-solvent-relaxed revPBE structures at the revPBE lattice constant). This data in combination with the PBE-determined interfacial capacitances yields the theoretical revPBE CVs as plotted in Supplementary Fig.   S7, which clearly agree less than the PBE results with the experimental CV curves.  Figure S8: Surface charges for Ag(111) electrodes in ion-free (dotted lines) and ion-containing solutions (dashed lines) for the three considered system. Vertical bars indicate the charge difference between the adsorbate-covered and clean surfaces, determined at potentials of full coverage (see text). Top panels: Experimental results from Foresti et al. S1 . Bottom panels: Corresponding theoretical results at the indicated experimental ion concentrations.

Simulated and experimental surface charges
We complement here the results of the main text by a plot of the simulated, equilibrium surface charges σ(Φ E ) = −en abs,θa e /A site and the respective quantity measured in the experiments. and adding also the DL terms (underlined terms in the main text), respectively. Note that surface charges are only defined up to a (arbitrary integration) constant, which is why a comparison of theory and experiment should only be made for the trends and charge differences within the plots.

S-11
In all plots we added as vertical bar the charge difference between the adsorbate-covered (solid lines) and clean surface (dotted lines). In agreement with the approach used in Ref. S1 to estimate the integral electrosorption valencies we determine this charge difference at a potential where the slope of clean and covered surfaces become essentially equal. In the theoretical curves we use the potential where the coverage is 99% the maximum coverage (1/3 ML). Again, the misalignment on the potential scale originates from the accuracy limitations of the predicted adsorption energies by first-principles methods, and will not further be analysed.
As DL charging is not captured within the CHE model, it is not surprising that the  S1 , thus underlining the fact that the CHE+DL results can S-12 correctly reproduce the observed trend across the halide series and yield correct orders of magnitude, while the CHE method is inherently not able to make any such discrimination.
Note also that the total integrated charge in the CHE method (black vertical bars, 71 µC/cm 2 ) -which is directly related to the assumed maximum surface coverage of 1/3 MLis significantly larger than the experimental values (41-62 µC/cm 2 ). This indicates that our setting of θ max a = 1/3 ML can not be far off the experimental values. Indeed a naive (CHElike) translation of surface charge into surface coverage indicates rather lower maximum surface coverages than those studied in our model.
Possible larger values for θ max a (that were not included in the present study) could naturally only lead to a further increase in the theoretical surface charge values. This would then lead to an even larger deviation of the CHE results from the experiments, while improving the CHE+DL predictions This convinces us that an additional inclusion of higher coverages in the theoretical model could only make a stronger case for the CHE+DL method, which our present study did not include due to the strong experimental evidence that this is not needed, for the study of the lower-potential CV peak.

Tabulated Data
All numerical values for adsorption energies, work functions and the interfacial capacitance at PZC as obtained with the PBE functional in the implicit solvent setup are tabulated in table S1. Results for the revPBE functional are in table S2.