Thermodynamic cyclic voltammograms: peak positions and shapes

In this work we concentrate for clarity on the description of metallic electrodes with just a single possible adsorbate species a (generalization to multiple species is conceptually straightforward). In this case, the Gibbs excess energy [13, 16, 50, 64, 65] at applied electrode potential Φ_E is defined as

$$\begin{equation}{G}_{\text{exc}}^{\alpha }={G}_{\text{surf}}^{\alpha }\left({N}_{\text{s}}^{\alpha },{N}_{\text{a}}^{\alpha },{N}_{\text{e}}^{\mathrm{a}\mathrm{b}\mathrm{s},\alpha }\right)-{N}_{\text{s}}^{\alpha }{\mu }_{\text{s}}-{N}_{\text{a}}^{\alpha }{\tilde {\mu }}_{\text{a}}+{N}_{\text{e}}^{\mathrm{a}\mathrm{b}\mathrm{s},\alpha }e{{\Phi}}_{\text{E}},\end{equation} \tag{ 1 }$$

α denotes a certain interface configuration as characterized by its surface geometry (e.g. position of adsorbates) and chemical composition of ${N}_{\text{s}}^{\alpha }$ substrate atoms, ${N}_{\text{a}}^{\alpha }$ adsorbates of type a, and ${N}_{\text{e}}^{\mathrm{a}\mathrm{b}\mathrm{s},\alpha }$ electrons in excess to the charge-neutral pristine electrode surface. Expression (1) in essence determines the free energy cost of creating the interfacial system α with total energy ${G}_{\text{surf}}^{\alpha }\left({N}_{\text{s}}^{\alpha },{N}_{\text{a}}^{\alpha },{N}_{\text{e}}^{\mathrm{a}\mathrm{b}\mathrm{s},\alpha }\right)$ when its constituents are taken from the thermodynamic reservoirs characterized by the (electro-)chemical potentials μ_s, ${\tilde {\mu }}_{\text{a}}$ and −eΦ_E for substrate atoms, adsorbates and electrons, respectively.

With the adsorbates a present in solution as dissolved ions ${\mathrm{A}}^{{\text{q}}_{\text{a}}/\mathrm{e}}\left(\mathrm{a}\mathrm{q}\right)$ of charge q_a, their electrochemical potential ${\tilde {\mu }}_{\text{a}}$ can be determined from tabulated experimental equilibrium potentials ${{\Phi}}_{\text{a,}\text{eq}}^{\mathrm{exp}}$ of the redox reaction $\mathrm{A}{\rightleftharpoons}{\mathrm{A}}^{{q}_{\text{a}}/e}\left(\mathrm{a}\mathrm{q}\right)+\frac{{q}_{\text{a}}}{e}{e}^{-}\left(m\right)$ in combination with theoretical calculations of the chemical potential μ_A in the reference phase. For hydrogen and halides, e.g., μ_A is conventionally determined as half the chemical potential of the biatomic molecules at 298 K and 1 bar (${\mu }_{\text{A}}=\frac{1}{2}{\mu }_{{\text{A}}_{2}\left(\mathrm{g}\right)}$) and ${{\Phi}}_{\text{a,}\text{eq}}^{\mathrm{exp}}$ refers to a 1 M solution [66, 67]. In this case:

$$\begin{equation}{\tilde {\mu }}_{\text{a}}={\mu }_{\text{A}}+{q}_{\text{a}}{{\Phi}}_{\text{a,}\text{eq}}^{\mathrm{exp}}+{\beta }^{-1}\enspace \mathrm{ln}\left({c}_{\text{a}}\right),\end{equation} \tag{ 2 }$$

where c_a is the molar concentration of the ions in solution and β⁻¹ = k_B T.

As noted in the introduction, whenever the interfacial capacitance is independent of the applied potential—as e.g. in many implicit solvent models for high electrolyte concentrations [13, 16, 68] (see also section 1 in the supporting information (SI) (https://stacks.iop. org/JPCM/33/264004/mmedia))—${G}_{\text{surf}}^{\alpha }$ in equation (1) can be described with a second order polynomial in the number of electrons. In this case, the grand canonical, charge-equilibrated excess energy (constant potential conditions) decomposes into an excess energy term valid at the PZC—which is linear in the applied potential Φ_E and identical to the CHE expressions [16, 19, 66]—and an additional, generic DL energy contribution due to capacitive charging [13, 16, 63]. Furthermore, mean-field sampling of configurations α at fixed adsorbate coverages ${\theta }_{\text{a}}={N}_{\text{a}}^{\alpha }/{N}_{\text{sites}}$ can provide an approximate interface free energy landscape as a function of the coverage and the potential, from which any thermodynamic quantity can be inferred, in principle [13].

In the following, we will use expressions that are normalized to the number of surface sites (denoted by lower case letters), such that the mean-field CHE excess energy (per surface site) reads

$$\begin{align}{g}_{\text{exc,}\text{0}}^{{\theta }_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}& ={g}_{\text{exc,}\text{0}}^{\text{clean}}+{\theta }_{\text{a}}{\bar{G}}_{\text{ads,}\text{0}}^{{\theta }_{\text{a}}}-{\theta }_{\text{a}}{\beta }^{-1}\enspace \mathrm{ln}\left({c}_{\text{a}}\right)\\ & \quad +{\theta }_{\text{a}}{q}_{\text{a}}\left({{\Phi}}_{\text{E}}-{{\Phi}}_{\text{a,}\text{eq}}^{\mathrm{exp}}\right),\end{align} \tag{ 3 }$$

where ${g}_{\text{exc,}\text{0}}^{\text{clean}}$ is the excess energy of the clean surface and ${\bar{G}}_{\text{ads,}\text{0}}^{{\theta }_{\text{a}}}$ the average adsorption energy per adsorbate, at coverage θ_a. The subscript 0 denotes values determined at the PZC. Both terms can be approximated e.g. from (charge-neutral) DFT calculations via

$$\begin{equation}{g}_{\text{exc,}\text{0}}^{\text{clean}}\approx \frac{1}{{N}_{\text{sites}}}\left[{E}_{\text{surf,}\text{0}}^{\text{clean,}\text{DFT}}-{N}_{\text{s}}{E}_{\text{s,}\text{bulk}}^{\text{DFT}}\right],\end{equation} \tag{ 4 }$$

$$\begin{align}{\bar{G}}_{\text{ads,}\text{0}}^{{\theta }_{\text{a}}}& \approx \langle \frac{1}{{N}_{\text{a}}^{\alpha }}\left[\right.{E}_{\text{surf,}\text{0}}^{\alpha ,\mathrm{D}\mathrm{F}\mathrm{T}}+{\Delta}{F}_{\text{surf,}\text{vib}}^{\alpha ,\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}\\ & \quad -{{E}_{\text{surf,}\text{0}}^{\text{clean,}\text{DFT}}-{N}_{\text{a}}^{\alpha }{\mu }_{\text{A}}^{\text{DFT}}\left.\right]\rangle }_{\text{MFT}}.\end{align} \tag{ 5 }$$

${E}_{\text{surf,}\text{0}}^{\alpha ,\mathrm{D}\mathrm{F}\mathrm{T}}$ and ${E}_{\text{surf,}\text{0}}^{\text{clean,}\text{DFT}}$ are the DFT 0 K energies of the adsorbate-covered system consisting of ${N}_{\text{a}}^{\alpha }$ adsorbates and N_s substrate atoms and of the pristine substrate system (N_s substrate atoms). ${E}_{\text{s,}\text{bulk}}^{\text{DFT}}$ is the DFT 0 K bulk energy per atom of the substrate material. Here and in the following we assume all DFT surface energetics to be obtained using implicit solvation models. The corresponding terms could also be evaluated with explicit interfacial water to capture specific solvation effects. This can evidently affect the numerical values of both quantities, but would not affect the general thermodynamic description within the present work. The MF average ${\left\langle \cdot \right\rangle }_{\text{MFT}}$ at fixed coverage ${\theta }_{\text{a}}={N}_{\text{a}}^{\alpha }/{N}_{\text{sites}}$ can be efficiently approximated by random sampling or via the use of special quasi random structures [69–71]; for the considered showcase of halides on Ag(111) using only maximally isotropic structures at different coverages yielded already a good agreement between predicted and experimental CV peak shapes [13]. As none of the present discussion depends on the way these MF averages are determined (they might as well be determined by a fit to experimental data) we will refer to these averages at given coverage θ_a simply by a corresponding superscript. In the case of a first-principles parametrization (e.g. based on equations (4) and (5)) inaccuracies due to the use of 0 K energy differences are largely corrected by addition of the term ${\Delta}{F}_{\text{surf,}\text{vib}}^{\alpha ,\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}$—the vibrational free energy contributions of the adsorbates only. As noted previously, within the CHE + DL approximation [13, 16, 63] the DL energy contributions per site due to capacitive charging are simply given by a plate-capacitor-like term

$$\begin{equation}\underline{{g}_{\text{exc}}^{{\theta }_{\text{a}},\mathrm{D}\mathrm{L}}}=\underline{-\frac{1}{2}{A}_{\text{site}}{C}_{0}^{{\theta }_{\text{a}}}{\left({{\Phi}}_{\text{E}}-{{\Phi}}_{0}^{{\theta }_{\text{a}}}\right)}^{2}},\end{equation} \tag{ 6 }$$

with the area per site A_site = A/N_sites. ${{\Phi}}_{0}^{{\theta }_{\text{a}}}$ is the PZC (aka work function of the interface in solution) and ${C}_{0}^{{\theta }_{\text{a}}}$ the area-normalized DL capacitance at the PZC for the coverages θ_a. Both intensive quantities can be determined again from ab initio calculations, e.g. with an implicit solvent model. In the context of the discussion here, they are to be seen as system-specific parameters which can vary for different adsorbates and experimental conditions. Finally, the MF free energy landscape per site as a function of θ_a and Φ_E is given by

$$\begin{equation}{\mathcal{g}}_{\text{exc}}^{{\theta }_{\text{a}},\mathrm{M}\mathrm{F}\mathrm{T}}\approx {g}_{\text{exc,}\text{0}}^{{\theta }_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}+\underline{{g}_{\text{exc}}^{{\theta }_{\text{a}},\mathrm{D}\mathrm{L}}}-T{s}_{\text{conf}}^{{\theta }_{\text{a}}},\end{equation} \tag{ 7 }$$

with ${s}_{\text{conf}}^{{\theta }_{\text{a}}}$ corresponding to the MF configurational entropy term

$$\begin{align}{s}_{\text{conf}}^{{\theta }_{\text{a}}}& =-{\theta }_{\text{a}}^{\mathrm{max}}{k}_{\text{B}}\left[\left(\frac{{\theta }_{\text{a}}}{{\theta }_{\text{a}}^{\mathrm{max}}}\right)\mathrm{ln}\left(\frac{{\theta }_{\text{a}}}{{\theta }_{\text{a}}^{\mathrm{max}}}\right)\right.\\ & \quad +\left.\left(1-\frac{{\theta }_{\text{a}}}{{\theta }_{\text{a}}^{\mathrm{max}}}\right)\mathrm{ln}\left(1-\frac{{\theta }_{\text{a}}}{{\theta }_{\text{a}}^{\mathrm{max}}}\right)\right],\end{align} \tag{ 8 }$$

where ${\theta }_{\text{a}}^{\mathrm{max}}$ is the maximum achievable coverage (per site) for the adsorbates a [13].

Above, and in the following, all DL-charging-related terms are underlined to make the impact of the latter more clear.

The equilibrium surface coverage ${\bar{\theta }}_{\text{a}}\left({{\Phi}}_{\text{E}}\right)$ at given applied electrode potential Φ_E minimizes ${\mathcal{g}}_{\text{exc}}^{{\theta }_{\text{a}},\mathrm{M}\mathrm{F}\mathrm{T}}\left({{\Phi}}_{\text{E}}\right)$ (equation (7)) with respect to θ_a. Knowledge of ${\bar{\theta }}_{\text{a}}\left({{\Phi}}_{\text{E}}\right)$ then directly yields the expected number of electrons ${n}_{\text{e}}^{\mathrm{a}\mathrm{b}\mathrm{s}}$ per site as a function of the potential via

$$\begin{equation}{n}_{\text{e}}^{\mathrm{a}\mathrm{b}\mathrm{s},{\bar{\theta }}_{\text{a}}}={\bar{\theta }}_{\text{a}}\frac{{q}_{\text{a}}}{e}\enspace \underline{-\frac{1}{e}{A}_{\text{site}}{C}_{0}^{{\bar{\theta }}_{\text{a}}}\left({{\Phi}}_{\text{E}}-{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}\right)},\end{equation} \tag{ 9 }$$

from which thermodynamic CV currents j can be derived [13]. At a fixed scan rate $v=\frac{\mathrm{d}}{\mathrm{d}t}{{\Phi}}_{\text{E}}$, this current can be written as j = vC_pseudo, where the pseudocapacitance C_pseudo is scan-rate-independent. Within the MF model, ${C}_{\text{pseudo}}^{\text{MFT}}=-e\frac{\mathrm{d}}{\mathrm{d}{{\Phi}}_{\text{E}}}{n}_{\text{e}}^{\text{abs},{\bar{\theta }}_{\text{a}}}$ and thus given by

$$\begin{align}{C}_{\text{pseudo}}^{\text{MFT}}& =\underline{{A}_{\text{site}}{C}_{0}^{{\bar{\theta }}_{\text{a}}}}+{C}_{\text{pseudo}}^{\text{MFT,}\text{sorp}}\\ & =\underline{{A}_{\text{site}}{C}_{0}^{{\bar{\theta }}_{\text{a}}}}-e{l}_{\text{a}}^{\text{MFT}}\frac{\mathrm{d}}{\mathrm{d}{{\Phi}}_{\text{E}}}{\bar{\theta }}_{\text{a}}.\end{align} \tag{ 10 }$$

${l}_{\text{a}}^{\text{MFT}}$ is the electrosorption valency [13–16]—the number of exchanged electrons per adsorbate—which is given by

$$\begin{equation}{l}_{\text{a}}^{\text{MFT}}=\frac{1}{e}\left[{q}_{\text{a}}+\underline{{A}_{\text{site}}{C}_{0}^{{\bar{\theta }}_{\text{a}}}\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}-{A}_{\text{site}}\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{C}_{0}^{{\bar{\theta }}_{\text{a}}}\left({{\Phi}}_{\text{E}}-{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}\right)}\right].\end{equation} \tag{ 11 }$$

Note that expression (10) includes a double-layer charging, baseline contribution ${A}_{\text{site}}{C}_{0}^{{\bar{\theta }}_{\text{a}}}$ plus a contribution due to electrosorption which we refer to as ${C}_{\text{pseudo}}^{\text{MFT,}\text{sorp}}$. The latter is naturally proportional to the change in surface coverage ($\frac{\mathrm{d}}{\mathrm{d}{{\Phi}}_{\text{E}}}{\bar{\theta }}_{\text{a}}$) times the electrosorption valency ${l}_{\text{a}}^{\text{MFT}}$.

${\bar{\theta }}_{\text{a}}$ minimizes ${\mathcal{g}}_{\text{exc}}^{{\bar{\theta }}_{\text{a}},\mathrm{M}\mathrm{F}\mathrm{T}}$ and is thus given by

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{\mathcal{g}}_{\text{exc}}^{{\bar{\theta }}_{\text{a}},\mathrm{M}\mathrm{F}\mathrm{T}}=0={{\epsilon}}^{{\bar{\theta }}_{\text{a}}}+{\beta }^{-1}\enspace \mathrm{ln}\left(\frac{{\bar{\theta }}_{\text{a}}}{{\theta }_{\text{a}}^{\mathrm{max}}}/\left(1-\frac{{\bar{\theta }}_{\text{a}}}{{\theta }_{\text{a}}^{\mathrm{max}}}\right)\right),\end{equation} \tag{ 12 }$$

which can be rewritten into the typical form of an adsorption isotherm as

$$\begin{equation}{\bar{\theta }}_{\text{a}}={\theta }_{\text{a}}^{\mathrm{max}}{\left[1+\mathrm{exp}\left(\beta {{\epsilon}}^{{\bar{\theta }}_{\text{a}}}\right)\right]}^{-1}.\end{equation} \tag{ 13 }$$

While equation (13) defines ${\bar{\theta }}_{\text{a}}\left({{\Phi}}_{\text{E}}\right)$ in general only implicitly, it is possible to derive an analytic expression for the inverse function ${{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}\enspace {:=}\enspace {{\Phi}}_{\;\text{E}}\left({\bar{\theta }}_{\text{a}}\right)$, which is given as the solution to the quadratic equation (in Φ_E) defined by equation (12) and with

$$\begin{align}{{\epsilon}}^{{\bar{\theta }}_{\text{a}}}& =\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}\left({g}_{\text{exc,}\text{0}}^{{\bar{\theta }}_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}+\underline{{g}_{\text{exc}}^{{\bar{\theta }}_{\text{a}},\mathrm{D}\mathrm{L}}}\right)\\ & =a{\left({{\Phi}}_{\text{E}}-{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}\right)}^{2}+b\left({{\Phi}}_{\text{E}}-{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}\right)+\mathcal{E},\end{align} \tag{ 14 }$$

$$\begin{equation}a=\underline{\left[-\frac{1}{2}{A}_{\text{site}}\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{C}_{0}^{{\bar{\theta }}_{\text{a}}}\right]},\end{equation} \tag{ 15 }$$

$$\begin{equation}b={q}_{\text{a}}+\underline{\left[{A}_{\text{site}}{C}_{0}^{{\bar{\theta }}_{\text{a}}}\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}\right]}={q}_{\text{a}}\left(1+{\underline{\delta }}_{\text{a}}\right),\end{equation} \tag{ 16 }$$

$$\begin{align}\mathcal{E}& ={\bar{G}}_{\text{ads,}\text{0}}^{{\bar{\theta }}_{\text{a}}}+{\bar{\theta }}_{\text{a}}\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{\bar{G}}_{\text{ads,}\text{0}}^{{\bar{\theta }}_{\text{a}}}-{\beta }^{-1}\enspace \mathrm{ln}\left({c}_{\text{a}}\right)\\ & \quad +{q}_{\text{a}}\left({{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}-{{\Phi}}_{\text{a,}\text{eq}}^{\mathrm{exp}}\right).\end{align} \tag{ 17 }$$

The general solution to this second order equation in Φ_E (equations (12) and (14)) however hampers meaningful further analysis and we restrict ourselves in the following to the case where a = 0 in equation (14). This approximation implicitly assumes $\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{C}_{0}^{{\bar{\theta }}_{\text{a}}}=0$ and thus refers to systems, where the interfacial capacitance is independent on the surface coverage (${C}_{0}^{{\bar{\theta }}_{\text{a}}}={C}_{0}$) or where this is a good approximation. In this case

$$\begin{equation}{{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}=-\frac{1}{{q}_{\text{a}}}\left[\mathcal{G}+{q}_{\text{a}}{\underline{{\Delta}}}_{\text{a}}\right]={{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}-{\underline{{\Delta}}}_{\text{a}},\end{equation} \tag{ 18 }$$

$$\begin{equation} \mathcal{G}=\mathcal{E}-{q}_{\text{a}}{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}+{\beta }^{-1}\enspace \mathrm{ln}\left(\frac{{\bar{\theta }}_{\text{a}}}{{\theta }_{\text{a}}^{\mathrm{max}}}/\left(1-\frac{{\bar{\theta }}_{\text{a}}}{{\theta }_{\text{a}}^{\mathrm{max}}}\right)\right),\end{equation} \tag{ 19 }$$

$$\begin{equation}{\underline{{\Delta}}}_{\text{a}}={\underline{\delta }}_{\text{a}}{\Delta}{{\Phi}}_{\text{E}}\quad \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\quad {\Delta}{{\Phi}}_{\text{E}}={{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}-{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}},\end{equation} \tag{ 20 }$$

where ${\underline{\delta }}_{\text{a}}$ is a unitless measure for DL effects as defined in equation (16). Note, that the CHE result is given by the first term in equation (18) with ${{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}=-\frac{\mathcal{G}}{{q}_{\text{a}}}$, where $\mathcal{G}$ is the differential, adsorption-energy-like per adsorbate quantity defined in equation (19). Inclusion of the (underlined) DL terms shifts ${{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}$ thus by an amount of $-{\underline{{\Delta}}}_{\text{a}}=-{\underline{\delta }}_{\text{a}}{\Delta}{{\Phi}}_{\text{E}}$ at given coverages ${\bar{\theta }}_{\text{a}}$ with ${\Delta}{{\Phi}}_{\text{E}}={{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}-{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}$ the difference between the applied potential and the PZC ${{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}$. Note that ${\underline{\delta }}_{\text{a}}={D}_{\text{a}}{C}_{0}$ is the product of an adsorbate-dipole-related quantity ${D}_{\text{a}}=\frac{{A}_{\text{site}}}{{q}_{\text{a}}}\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}$ and the DL capacitance C₀ (assumed independent on coverage already before). Thus the DL-induced shifts in the electrosorption potential can be rationalized by a dipole field interaction with the field in the DL ∝C₀ΔΦ_E. This term corresponds to the interface charge at applied potential that is used as relevant variable in other works [18]. Furthermore, ${q}_{\text{a}}{\underline{\delta }}_{\text{a}}$ relates directly to the electrosorption valency via $e{l}_{\text{a}}^{\text{MFT}}={q}_{\text{a}}\left(1+{\underline{\delta }}_{\text{a}}\right)$ (cf equation (11) for $\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{C}_{0}^{{\bar{\theta }}_{\text{a}}}=0$ and equation (16)). The close connection between electrosorption valency and dipole has been clarified previously [15, 16]. Although adsorbate dipoles and interfacial fields are more common descriptors when discussing adsorbates at electrochemical interfaces, we use here mostly the unitless quantity ${\underline{\delta }}_{\text{a}}$, as it represents a direct measure of the size of dipole-field interactions (at given DL capacitance) relative to the work by the electron transfer process upon adsorption, and it relates in a straightforward way to ${l}_{\text{a}}^{\text{MFT}}$.

Evidently, ${{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}$ (equation (18)) can be approximated as a perturbation to the CHE result ${{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}$ via

$$\begin{equation}{{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}\approx {{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}-{\underline{\delta }}_{\text{a}}{\Delta}{{\Phi}}_{\text{E}}^{\mathrm{C}\mathrm{H}\mathrm{E}},\end{equation} \tag{ 21 }$$

where ${\Delta}{{\Phi}}_{\text{E}}^{\mathrm{C}\mathrm{H}\mathrm{E}}={{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}-{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}$ is the potential drop in the DL, as estimated from the CHE electrosorption potentials ${{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}$. With Cl@Ag(111) having electrosorption valencies of ∼−0.5 [16] we can directly derive $\vert {\underline{\delta }}_{\text{a}}\vert \approx 50\%$ and thus estimate the DL-charging-related shift of the CHE peak position as $\approx 0.5\vert {\Delta}{{\Phi}}_{\text{E}}^{\mathrm{C}\mathrm{H}\mathrm{E}}\vert$, which is easily in the range of hundreds of meV, and thus far from negligible ³ .

3.1.1. Peak position

As discussed above, CVs are typically dominated by the electrosorption peak ${C}_{\text{pseudo}}^{\text{MFT,}\text{sorp}}$ (equation (10)). Whenever ${l}_{\text{a}}^{\text{MFT}}$ depends only weakly on the coverage, the position and width of the CV peak are hardly affected by its neglect, as it enters only as a multiplicative constant in equation (10) ⁴ . Thus, we focus in the following on the quantity $\mathcal{C}=\sigma \frac{\mathrm{d}}{\mathrm{d}{{\Phi}}_{\text{E}}}{\bar{\theta }}_{\text{a}}$, with $\sigma =\frac{-{q}_{\text{a}}}{\vert {q}_{\text{a}}\vert }$. This expression yields positive electrosorption peaks, independent on the sign of the adsorbate charge q_a for physically sound, i.e. convex free energy landscapes ⁵ .

Thus, CV peak position and shape can be evaluated from the moments of $\mathcal{C}$ via the integrals

$$\begin{equation}\langle {{\Phi}}_{\text{E}}^{n}\rangle =\frac{1}{N}{\int }_{-\infty }^{\infty }\mathcal{C}\cdot {{\Phi}}_{\text{E}}^{n}\mathrm{d}{{\Phi}}_{\text{E}},\end{equation} \tag{ 22 }$$

and the normalization $N={\int }_{-\infty }^{\infty }\mathcal{C}\cdot \mathrm{d}{{\Phi}}_{\text{E}}$. These can be rewritten as integrals over the coverage ${\bar{\theta }}_{\text{a}}$ (see section 2 in the SI for more details), as

$$\begin{equation}\langle {{\Phi}}_{\text{E}}^{n}\rangle =\frac{1}{{\theta }_{\text{a}}^{\mathrm{max}}}\enspace {\int }_{\enspace 0}^{{\theta }_{\text{a}}^{\mathrm{max}}}{\left({{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}\right)}^{n}\mathrm{d}{\bar{\theta }}_{\text{a}}.\end{equation} \tag{ 23 }$$

The peak position P on an absolute potential scale is given by the first moment

$$\begin{equation}P=\langle {{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}\rangle \approx \langle {{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}\rangle -\langle {\underline{{\Delta}}}_{\text{a}}\rangle ,\end{equation} \tag{ 24 }$$

which can be readily computed from the coverage-dependent adsorption energetics (cf equations (18) and (23)). This result clarifies the above mentioned influence of the dipole-field interaction ${\underline{{\Delta}}}_{\text{a}}$ on the expected peak position relative to the CHE estimate. Furthermore, it is straightforward to derive peak shifts induced by a variation of the concentration c_a via the identities

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\enspace \mathrm{log}\left({c}_{\text{a}}\right)}\langle {{\Phi}}_{\text{E}}^{n}\rangle =\langle {n{\Phi}}_{\text{E}}^{n-1}\frac{\partial {{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}}{\partial \enspace \mathrm{log}\left({c}_{\text{a}}\right)}\rangle ,\end{equation} \tag{ 25 }$$

$$\begin{equation}\frac{\partial {{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}}{\partial \enspace \mathrm{log}\left({c}_{\text{a}}\right)}\approx \frac{{\beta }^{-1}\enspace \mathrm{ln}\left(10\right)}{{q}_{\text{a}}}\left(1-{\underline{\delta }}_{\text{a}}\right).\end{equation} \tag{ 26 }$$

The latter approximation is valid for small ${\underline{\delta }}_{\text{a}}$ (cf equation (21) and section 2 in the SI). Evidently, the first term in equation (26) describes the perfect Nernstian behaviour of the CHE energetics, whereas the DL terms induce non-Nernstian shifts according to

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\enspace \mathrm{log}\left({c}_{\text{a}}\right)}P=\frac{\mathrm{d}}{\mathrm{d}\enspace \mathrm{log}\left({c}_{\text{a}}\right)}\langle {{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}\rangle \approx \frac{{\beta }^{-1}\enspace \mathrm{ln}\left(10\right)}{{q}_{\text{a}}}\left(1-\langle {\underline{\delta }}_{\text{a}}\rangle \right),\end{equation} \tag{ 27 }$$

which can be directly evaluated from the coverage dependence of the PZC and knowledge of the DL capacitance C₀ (cf equation (16)). Therefore, non-Nernstian peak shifts are directly related to non-ideal electrosorption valencies $e{l}_{\text{a}}^{\text{MFT}}={q}_{\text{a}}+{q}_{\text{a}}{\underline{\delta }}_{\text{a}}\ne {q}_{\text{a}}$, which depend on the adsorbate dipole and the interfacial capacitance, not however on the magnitude or direction of the field in the DL (within the present approximations).

3.1.2. Proxies for peak position and peak shape

In analogy to the previous analysis, peak widths can be analyzed via the second central moment, as done in the SI (sections 2 and 3), however, the intricate averages over the coverage hamper further insightful, analytical analysis. Therefore, we follow a simpler route to derive proxies for the position and peak shape of CVs by estimating above averages over the coverage ⟨⋅⟩ by simple evaluation of the coverage-dependent terms at an intermediate coverage, e.g. the half maximum coverage. This procedure allows to derive in full rigour general and more intelligible proxies for CV peak properties, as elaborated in the SI. We refer the interested reader to the respective section 4 in the SI, also to understand better the conditions under which the various approximations made are expected to remain valid. All derived expressions are reported in a self-contained way for CV peak position and shape (table 1) as well as for their variation when changing the ion concentration c_a and the DL capacitance C₀ (table 2).

Table 1. Approximate expressions for the CV peak position P, the potential drop in the DL ΔΦ_E and the CV peak shape as defined by its width W and height H. While all other variables are as introduced in the main text, the variables $\sigma =-\frac{{q}_{\text{a}}}{\vert {q}_{\text{a}}\vert }={\pm}1$ and $K{ >}0=\mathcal{O}\left(1\right)$ can be understood from the detailed derivations in section 4 of the SI. For halides on Ag(111) K ≈ 0.27. All coverage-dependent terms are to be evaluated at half maximum coverage (${\bar{\theta }}_{\text{a}}=\frac{1}{2}{\theta }_{\text{a}}^{\mathrm{max}}$)

Property	Proxy
CHE peak position	$$\begin{equation}{P}^{\text{CHE}} =-\frac{\mathcal{G}}{{q}_{\text{a}}}\\ \mathcal{G} ={\bar{G}}_{\text{ads,}\text{0}}^{{\bar{\theta }}_{\text{a}}}\hspace{-1pt}+\hspace{-1pt}{\bar{\theta }}_{\text{a}}\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{\bar{G}}_{\text{ads,}\text{0}}^{{\bar{\theta }}_{\text{a}}}\hspace{-1pt}-\hspace{-1pt}{q}_{\text{a}}{{\Phi}}_{\text{a,}\text{eq}}^{\mathrm{exp}}\\ \quad -{\beta }^{-1}\mathrm{ln}\left({c}_{\text{a}}\right)\\ \quad +{\beta }^{-1}\mathrm{ln}\hspace{-2pt}\left(\hspace{-1pt}\frac{{\bar{\theta }}_{\text{a}}}{{\theta }_{\text{a}}^{\mathrm{max}}}/\hspace{-4pt}\left(\hspace{-2pt}1\hspace{-1pt}-\hspace{-1pt}\frac{{\bar{\theta }}_{\text{a}}}{{\theta }_{\text{a}}^{\mathrm{max}}}\hspace{-1pt}\right)\hspace{-3pt}\right)\hspace{-2pt},\end{equation} \tag{ 28 }$$
CHE + DL peak position	$$\begin{equation}P={P}^{\text{CHE}}-{\underline{\delta }}_{\text{a}}{\Delta}{{\Phi}}_{\text{E}},\end{equation} \tag{ 29 }$$
DL potential drop	$$\begin{equation}\begin{aligned}{\Delta}{{\Phi}}_{\text{E}}& ={{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}}}-{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}\\ & =P-{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}\approx {\Delta}{{\Phi}}_{\text{E}}^{\text{CHE}}\\ & ={P}^{\text{CHE}}-{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}},\end{aligned}\end{equation} \tag{ 30 }$$
	CHE + DL width	$$\begin{equation}W={W}^{\text{CHE}}+{W}^{\text{DL}}\end{equation} \tag{ 31 }$$ $$\begin{equation}{W}^{\text{CHE}}=\sigma K\left(\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{{\Phi}}_{\text{E}}^{{\bar{\theta }}_{\text{a}},\mathrm{C}\mathrm{H}\mathrm{E}}\right),\end{equation} \tag{ 32 }$$ $$\begin{equation}\begin{aligned}{W}^{\text{DL}}& =-\sigma K\cdot \frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{\underline{{\Delta}}}_{\text{a}}\\ & =-{W}^{\text{CHE}}\cdot {\underline{\delta }}_{\text{a}}-\frac{\vert {q}_{\text{a}}\vert K}{{A}_{\text{site}}{C}_{0}}\cdot {\underline{\delta }}_{\text{a}}^{2}\\ & -\sigma K\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{\underline{\delta }}_{\text{a}}\cdot {\Delta}{{\Phi}}_{\text{E}},\end{aligned}\end{equation} \tag{ 33 }$$
		CHE + DL height	$$\begin{equation}H=\vert {q}_{\text{a}}\left(1+{\underline{\delta }}_{\text{a}}\right)\vert \frac{K}{W}.\end{equation} \tag{ 34 }$$

Table 2. Proxies for the concentration dependence $\frac{\mathrm{d}}{\mathrm{d}\enspace \mathrm{log}\left({c}_{\text{a}}\right)}$ and the capacitance dependence $\frac{\mathrm{d}}{\mathrm{d}{C}_{0}}$ for the CV peak position P, the width W and the height H.

Property	Proxy
Change with ion concentration  log(ca)
CHE + DL peak position	$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\enspace \mathrm{log}\left({c}_{\text{a}}\right)}P=\frac{{\beta }^{-1}\enspace \mathrm{ln}\left(10\right)}{{q}_{\text{a}}}\left(1-{\underline{\delta }}_{\text{a}}\right),\end{equation} \tag{ 35 }$$
CHE + DL width	$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\enspace \mathrm{log}\left({c}_{\text{a}}\right)}W=K\cdot \frac{{\beta }^{-1}\enspace \mathrm{ln}\left(10\right)}{\vert {q}_{\text{a}}\vert }\cdot \frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{\underline{\delta }}_{\text{a}},\end{equation} \tag{ 36 }$$
CHE + DL height	$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\enspace \mathrm{log}\left({c}_{\text{a}}\right)}H=-\frac{H}{W}\cdot \frac{{\beta }^{-1}\enspace \mathrm{ln}\left(10\right)}{\vert {q}_{\text{a}}\vert }\cdot \frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{\underline{\delta }}_{\text{a}},\end{equation} \tag{ 37 }$$
Change with DL capacitance C0 (${C}_{0}={C}_{0}^{{\bar{\theta }}_{\text{a}}}$, coverage-independent; $\frac{{\underline{\delta }}_{\text{a}}}{{C}_{0}}={D}_{\text{a}}$, independent on C0)
CHE + DL peak position	$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}{C}_{0}}P=-{\Delta}{{\Phi}}_{\text{E}}\frac{{\underline{\delta }}_{\text{a}}}{{C}_{0}},\end{equation} \tag{ 38 }$$
CHE + DL width	$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}{C}_{0}}W=\left[-{W}^{\text{CHE}}-\frac{2\vert {q}_{\text{a}}\vert K{\underline{\delta }}_{\text{a}}}{{A}_{\text{site}}{C}_{0}}-\sigma K\frac{\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{\underline{\delta }}_{\text{a}}}{{\underline{\delta }}_{\text{a}}}{\Delta}{{\Phi}}_{\text{E}}\right]\cdot \frac{{\underline{\delta }}_{\text{a}}}{{C}_{0}},\end{equation} \tag{ 39 }$$
CHE + DL height	$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}{C}_{0}}H=\frac{\vert {q}_{\text{a}}\vert }{\vert {q}_{\text{a}}\left(1+{\underline{\delta }}_{\text{a}}\right)\vert }H\cdot M\cdot \frac{{\underline{\delta }}_{\text{a}}}{{C}_{0}},\end{equation} \tag{ 40 }$$
	$$\begin{equation}M=2+\left(1+\frac{2H}{{A}_{\text{site}}{C}_{0}}\right){\underline{\delta }}_{\text{a}}+\sigma \frac{\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{\underline{\delta }}_{\text{a}}}{{\underline{\delta }}_{\text{a}}}\frac{H{\Delta}{{\Phi}}_{\text{E}}}{\vert {q}_{\text{a}}\vert }.\end{equation} \tag{ 41 }$$

Indeed, the respective equations show that changing the ion concentration c_a can affect CV peak positions and shapes, and these changes can be understood from the changes in dipole-field-like interactions which rationalize the respective descriptors (see more detailed discussion later on).

In the same way, a change of the DL capacitance C₀ can introduce changes in peak position and shape. As C₀ changes with electrolyte concentrations but also with electrolyte ion type [17] due to variations in ionic solvation shell sizes [18], the derived dependencies on C₀ are highly important to rationalize non-trivial changes under different experimental conditions.

We will provide a more detailed discussion and evaluation of these proxies in tables 1 and 2 later on.

DL effects were found relevant for understanding thermodynamic CVs of Ag(111) in halide containing solutions [13], as evidenced by comparing experimental results of Foresti et al [72] with theoretical predictions of the present mean-field description at the CHE and CHE + DL level of theory, parametrized from DFT calculations in an implicit solvent model (see figure 1 and reference [13]). The inability of the pure CHE description to replicate the experimental variations in the peak shapes, hints at the importance of capacitive DL charging for these systems.

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This becomes even more clear, when studying the changes in the peak shapes with changing the ion concentrations c_a for the according CHE + DL mean field models, as reported in figure 2. ⁶ Results at the experimental conditions (c_a = 5 × 10⁻⁴ M [72], as in figure 1) are colored and bold, while results for selected alternative concentrations are in grey (molar concentrations as indicated, partly unphysical). The plot demonstrates the strong concentration-dependence of the according CVs and clarifies that all halide peak shapes are expected in essence identical, given that their position P on the potential axis is the same. This can be easily understood from the previous discussions and the fact that all adsorbate-related descriptors but the adsorption energy are essentially the same for Cl, Br and I (see reference [13] and figure SF1 in the SI).

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In order to assess the accuracy of the derived proxies, we use these results for the CV peaks for Ag(111) in Cl, Br and I solutions, to test in how far the differences between the CHE and the CHE + DL results (figure 1) and the relevant changes with the concentration (figure 2) can be described and understood from the derived proxies. Unfortunately, the non-negligible degree of asymmetry in the peak shapes (see figures 1 and 2) hampers an unambiguous definition of position P, width W and height H. Therefore, we restrict ourselves to an analysis of P and W, for which the above moments-based definition can provide unambiguous and well-defined results. Verification of the proxies for P and W will also validate the proxies for H as it is the former two that enter the derivation of the peak-height-related proxies (see section 4 in the SI).

Figure 3(a) reports peak positions P and peak widths W for the experiments (encircled stars), as well as for the ab initio parametrized mean-field models (CHE: crosses, CHE + DL: squares) with quadratic coverage dependencies (see reference [13] and section 6 in the SI), at an ion concentration of c_a = 0.5 mM. P and W are computed from the mean and 2nd central moment of the experimental and theoretical curves of ${C}_{\text{pseudo}}^{\text{sorp}}\left({{\Phi}}_{\text{E}}\right)$ according to

$$\begin{equation}P=\frac{\int {C}_{\text{pseudo}}^{\text{sorp}}\left({{\Phi}}_{\text{E}}\right)\cdot {{\Phi}}_{\text{E}}\mathrm{d}{{\Phi}}_{\text{E}}}{\int {C}_{\text{pseudo}}^{\text{sorp}}\left({{\Phi}}_{\text{E}}\right)\mathrm{d}{{\Phi}}_{\text{E}}},\end{equation} \tag{ 42 }$$

$$\begin{equation}W=2\sqrt{\frac{\int {C}_{\text{pseudo}}^{\text{sorp}}\left({{\Phi}}_{\text{E}}\right)\cdot {\left({{\Phi}}_{\text{E}}-P\right)}^{2}\mathrm{d}{{\Phi}}_{\text{E}}}{\int {C}_{\text{pseudo}}^{\text{sorp}}\left({{\Phi}}_{\text{E}}\right)\mathrm{d}{{\Phi}}_{\text{E}}}}.\end{equation} \tag{ 43 }$$

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We then evaluate the accuracy of the proxies by adding to each CHE data point (P^CHE, W^CHE) the approximate expressions for the DL-induced peak shift (equation (24), horizontal vectors in figure 3(a) and the width change (equation (33), vertical vectors). The individual contributions to the DL-induced width change (first, second and third term in equation (33)) are illustrated as dashed, dotted and solid vertical arrows, respectively. All estimates are evaluated at half coverage ($0.5{\theta }_{\text{a}}^{\mathrm{max}}$) and with K = 0.27 in equation (33) (see section 4 in the SI for the detailed derivation of this value). All approximate results that derive from adding to each CHE data point the respective, perturbative shifts are plotted as filled circles and denoted as 'CHE + ↑'. Evidently, these estimates fall close to the CHE + DL solutions (squares) which provides confidence in the accuracy of the derived proxies.

Shaded regions enclosing experimental and theoretical data points across the halide series support a strong overall dependence of peak widths W on the peak position P for the experiments, which is only fully captured by theory when including the DL energetics. Note, that the inclusion of only the first two terms of equation (33) (dashed and dotted vertical arrows) leads to identical widths across the halide series. Thus only the latter term, which varies with the potential drop in the DL (ΔΦ_E), seems responsible for the variation of the halide peak shapes, in line with the results in figure 2 and the conclusions of reference [13].

Figure 3(b) plots the theoretical values for P and W when changing the ion concentration by two orders of magnitude around the experimental value of c_a = 0.5 mM (c_a = 0.5 × 10^±2 mM). While the CHE method only yields trivial Nernstian shifts in the peak position, the CHE + DL theory and their approximation (CHE + ↑) lead to non-trivial position and width changes (cf figure 2). As analysed subsequently, the former is related to the fact that halides on Ag(111) exhibit a normal work function change (see figure SF1 in the SI). Furthermore, as the work function change per adsorbate increases with the coverage for these systems (see figure SF1 in the SI) equation (36) results in a decreasing peak width with increasing c_a as $\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{\underline{\delta }}_{\text{a}}{< }0$, which explains the behaviour observed in figure 3(b).

Figure 3(b) includes as well the PZC of Ag(111) at half maximum coverage. At these potentials most DL-induced alterations to the CHE results are expected to vanish as there are nearly no interfacial fields present. Indeed, the concentration-dependencies of the CHE + DL and CHE + ↑ methods extrapolate nicely to the CHE results in this potential regime, which underlines again that the CHE results are intrinsically only valid at the PZC (cf figures 1 and 2).

Halides on Ag(111) exhibit significant coverage dependencies (e.g. for ${C}_{0}^{\bar{{\theta }_{\text{a}}}}$ and ${\underline{\delta }}_{\text{a}}$, see figure SF1 in the SI), however, their approximate treatment by evaluation at half maximum coverage and their complete neglect ($a=0;{C}_{0}^{\bar{{\theta }_{\text{a}}}}={C}_{0}$ in equation (14)) does not prevent the proxies from reproducing the CHE + DL results even in a semi-quantitative way. This is even more so surprising, as $\vert {\underline{\delta }}_{\text{a}}\vert \in 0.2\dots 0.5$ for halides on Ag(111) and thus not small compared to 1, which is assumed at certain steps of the derivations (see SI).

Therefore, we expect these proxies to remain valid also for other systems, where the approximations are typically less problematic than for the here discussed showcase system.

An insightful and accessible interpretation of the physical meaning and origin of the derived proxies in tables 1 and 2 can be achieved by analyzing a Frumkin model, where the adsorption energy ${\bar{G}}_{\text{ads,}\text{0}}^{{\bar{\theta }}_{\text{a}}}$ is assumed to vary only linearly with the coverage, and where adsorbate dipoles are assumed as coverage-independent. Such systems show no shape dependence on  log(c_a), but exhibit non-Nernstian peak shifts. The according analysis is not only interesting due to its straightforward interpretability, but also as it can be treated fully analytically. We refer the reader to section 5 in the SI for details.

3.3.1. The quantity ${\underline{\delta }}_{\text{a}}$

As already stated the quantity ${\underline{\delta }}_{\text{a}}$ is a unitless measure of DL effects, and at the same time it relates trivially to the electrosorption valency via $e{l}_{\text{a}}^{\text{MFT}}={q}_{\text{a}}\left(1+{\underline{\delta }}_{\text{a}}\right)$. Its meaning is thus very accessible and becomes even more clear when we reformulate e.g. the traditional chemical reaction equation for the electrosorption of Cl⁻

$$\begin{equation}{\mathrm{C}\mathrm{l}}^{-}+{\ast}\to \mathrm{C}\mathrm{l}{\ast}+\enspace {e}^{-},\end{equation} \tag{ 44 }$$

into a more general electrosorption equation, namely

$$\begin{equation}{\mathrm{C}\mathrm{l}}^{-}+{\ast}\to {\mathrm{C}\mathrm{l}}^{{\underline{\delta }}_{\text{Cl}}}{\ast}+\enspace \left(1+{\underline{\delta }}_{\text{Cl}}\right){e}^{-},\end{equation} \tag{ 45 }$$

with ${\underline{\delta }}_{\text{Cl}}$ being identical to the according quantity used throughout this work and with ${\underline{\delta }}_{\text{Cl}}\approx -0.5$ [16]. Reaction equation (45) thus clarifies that some negative charge remains on the adsorbed Cl and thus only $\left(1+{\underline{\delta }}_{\text{Cl}}\right)\approx 0.5$ electrons are actually exchanged upon the electrosorption process. Note, however, that this is only a simplified picture, as above charge-conserving reaction equations are problematic in view of a fully grand canonical picture where charge is not a conserved quantity.

In general, ${\underline{\delta }}_{\text{a}}$ discriminates between systems with normal and anomalous work function change: ${\underline{\delta }}_{\text{a}}$ is negative for adsorbates with a normal work function change ($\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}{< }0$ for q_a > 0 and $\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}{ >}0$ for q_a < 0). On the other hand ${\underline{\delta }}_{\text{a}}{ >}0$ for adsorbates with anomalous work function change (cf equation (16)). Thus it makes a significant difference, whether adsorbates behave normally or abnormally in this respect.

3.3.2. Peak positions P

Indeed the work function changes due to the adsorption of halides on metallic surfaces can be surprisingly complex [73–75], and it is thus certainly interesting to analyse what these results mean in terms of their electrosorption behaviour.

Based on the approach outlined in section 7 of the SI, we use published theoretical data [74] for ${\bar{G}}_{\text{ads,}\text{0}}^{{\bar{\theta }}_{\text{a}}}$ and $\frac{\mathrm{d}}{\mathrm{d}{\bar{\theta }}_{\text{a}}}{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}$ in combination with our own results for halides on Ag(111), in order to estimate the values of ${\underline{\delta }}_{\text{a}}$ and ΔΦ_E at interfacial capacitances C₀ ≈ 40 μF cm⁻² for Cl, Br and I and for a wide range of substrate materials (see section 7 in the SI for more details). This allows us directly to estimate their non-Nernstian peak shifts as well as the accuracies of the CHE peak positions (equation (29)), which depend only on the sign and order of magnitude of ${\underline{\delta }}_{\text{a}}$ and the peak position relative to the PZC.

Figure 4 plots the derived numerical values for ${\underline{\delta }}_{\text{a}}$ and ${l}_{\text{a}}^{\text{MFT}}=-\left(1+{\underline{\delta }}_{\text{a}}\right)$ as a function of the vacuum work function of the substrate material. Evidently, for many adsorbate-substrate combinations, $\vert {\underline{\delta }}_{\text{a}}\vert$ takes values that are significantly different from zero. In addition, negative and positive values are observed, in line with the fact that halides can induce normal and anomalous work function changes [73–75].

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As a result of equations (27) and (35), adsorbates with normal work function change exhibit larger peak shifts with  log(c_a) than expected from the ideal Nernst behaviour. The reverse is true for adsorbates with an anomalous work function shift. This is perfectly in line with the interpretations gained from the Frumkin model analysed in the SI, and the above intuitive chemical reaction equation (45): with ${\underline{\delta }}_{\text{a}}$ being directly related to the electrosorption valency ${l}_{\text{a}}^{\text{MFT}}=\frac{{q}_{\text{a}}}{e}\left(1+{\underline{\delta }}_{\text{a}}\right)$, non-zero ${\underline{\delta }}_{\text{a}}$ values are directly linked to a rescaled driving force for the adsorption process. Thus, systems with normal(anomalous) work function change exchange less(more) electrons upon adsorption, and thus necessitate larger(smaller) applied potentials to drive the adsorption process. While these results hold in general, e.g. independent on the charge of the adsorbate, we exemplify it in figure 4 for the negatively charged halides by having included also the shift with  log(c_a) on the right y-axis (equation (35)), which evidently exhibits the described dependence and demonstrates the expected magnitude of non-Nernstian behaviour for the considered systems (Nernstian: $\frac{\mathrm{d}}{\mathrm{d}\enspace \mathrm{log}\left({c}_{\text{a}}\right)}P=60\enspace \mathrm{m}\mathrm{V}/\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{d}\mathrm{e}$).

The combination of the ${\underline{\delta }}_{\text{a}}$ values with the reported [74] adsorption energies ${\bar{G}}_{\text{ads,}\text{0}}^{{\bar{\theta }}_{\text{a}}}$ allows to assess directly the expected CHE errors in the peak positions P − P^CHE (equation (29)) as well as the peak shift with DL capacitance $\frac{\mathrm{d}}{\mathrm{d}{C}_{0}}P$ (equation (38)). Note that both proxies are trivially related via the DL capacitance C₀. The DL potential drop is approximated via ${\Delta}{{\Phi}}_{\text{E}}\approx {P}^{\text{CHE}}-{{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}$ and with P^CHE from the theoretically reported values of ${\bar{G}}_{\text{ads,}\text{0}}^{{\bar{\theta }}_{\text{a}}}$ (equation (28)) and the PZC ${{\Phi}}_{0}^{{\bar{\theta }}_{\text{a}}}$ from experimentally measured values, which are available for the major part of the studied substrates [76] (see SI, section 7). The results are plotted in figure 5 for an ion concentration in solution of c_a = 1 M. Evidently, the CHE error P − P^CHE can easily vary in a range ±0.2 V ⁷ , which is far from negligible and on the same scale as inherent DFT inaccuracies. Furthermore, while the latter is often transferable (e.g. between substrates), figure 5 does not suggest this for the DL-induced shifts. However, a certain degree of correlation in the data points exists, which might enable a more accessible understanding in the future.

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Note as well, that the magnitude and sign of peak shifts $\frac{\mathrm{d}}{\mathrm{d}{C}_{0}}P$ (equation (38)) follow the same behaviour of the CHE error and thus depend on the product of ${\underline{\delta }}_{\text{a}}$ and ΔΦ_E. As a result, the magnitude and direction depends on how far the electrosorption peak is positioned left or right of the PZC, and whether the adsorbates induce a normal(anomalous) work function change (${\underline{\delta }}_{\text{a}}\left({ >}\right){{< }}0$, cf figure 5).

3.3.3. Peak shapes, H and W