Abstract
Electrosorption of solvated species at metal electrodes is a most fundamental class of processes in interfacial electrochemistry. Here, we use its sensitive dependence on the electric double layer to assess the performance of ab initio thermodynamics approaches increasingly used for the firstprinciples description of electrocatalysis. We show analytically that computational hydrogen electrode calculations at zero netcharge can be understood as a firstorder approximation to a fully grand canonical approach. Notably, higherorder terms in the applied potential caused by the charging of the double layer include contributions from adsorbateinduced changes in the work function and in the interfacial capacitance. These contributions are essential to yield prominent electrochemical phenomena such as nonNernstian shifts of electrosorption peaks and noninteger electrosorption valencies. We illustrate this by calculating peak shifts for H on Pt electrodes and electrosorption valencies of halide ions on Ag electrodes, obtaining qualitative agreement with experimental data already when considering only second order terms. The results demonstrate the agreement between classical electrochemistry concepts and a firstprinciples fully grand canonical description of electrified interfaces and shed new light on the widespread computational hydrogen electrode approach.
Introduction
In recent years, calculations based on ab initio thermodynamics have increasingly contributed to unraveling key processes in interfacial electrochemistry; e.g., in batteries, fuel cells, and other electrocatalytic systems. In such studies, the electrochemical environment and operation conditions are suitably represented in the form of thermodynamic reservoirs^{1,2,3,4,5}. Within a grandcanonical setup these reservoirs are then coupled to predictivequality (typically densityfunctional theory (DFT)) total energy calculations for the electrode, to deduce electrochemical stabilities and activities; see e.g., refs. ^{2,6,7,8,9,10,11}. In the application to interfacial electrochemistry a key challenge to this general ab initio thermodynamics concept is the necessity to exchange electrons with a reservoir representing the electrode potential. In principle, this requires to perform DFT calculations in various charge states. This clashes with the common representation of the electrode as finite slab using periodicboundary conditions, for which straightforward calculations can only be performed at zero total charge of the cell. For this reason, in most practical application, the early computational hydrogen electrode (CHE) approach^{2,6,12} relies solely on the energetics of chargeneutral electrode calculations, in the absence of the electrochemical double layer. The dependence on the electrode potential is then included in the analysis as an a posteriori shift of the electrochemical potential of the electrons taken from the reservoir, whose number is a priori fixed according to the chargeneutrality condition.
Conceptually, this restriction can be readily lifted through the introduction of meaningful countercharge models in the vacuum region between the periodic slabs^{1,13}. This then allows for a surface polarization and charging of the electrode slab while still maintaining the overall charge neutrality of the supercell. In the recent fully grandcanonical (FGC) approaches this is realized in practice by the use of an implicit solvation model in the DFT calculations^{5,9,11,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34}, which captures the effects of both electrolyte polarization and solvent screening, and renders the calculations largely insensitive to details of the countercharge model employed^{35} at conventional thicknesses of the vacuum/dielectric region^{32}. Within this approach, variablecharge calculations can nowadays be performed at high computational efficiency in a number of major DFT codes, e.g., ENVIRON^{18,36}, VASPsol^{21}, FHIaims^{25}, Jaguar^{29}, BigDFT^{24}, GPAW^{31,37}, or JDFTx^{38}.
The capabilities of FGC approaches have been highlighted in several publications and include a potentialdependence of chemical reaction steps^{8,29,30,31,39,40}, potentialinduced surface reconstructions or shifts of stable adsorption sites^{5,11,13,41}. However, there is presently still limited understanding of the methodological differences between the still prevalent zerocharge CHE and the variablecharge FGC approaches. In this work we therefore present a consistent thermodynamic framework to compare both types of approaches and highlight the simplifications made in practical calculations. In particular, we show by analysis of a generic secondorder Taylor expansion of the interface energies with respect to surface charge^{7,11,20,34,39,42,43,44,45} that the CHE approach can be understood as a firstorder approximation to the FGC energetics, while to second order, doublelayer charging is represented by changes in work function and interfacial capacitance.
We illustrate these differences between CHE, secondorder expansion and FGC in the application to two classic probes of interfacial electrochemistry at metal electrodes, namely nonNernstian shifts of proton electrosorption peaks at lowindex Pt crystals^{5,46,47,48,49,50} and the noninteger electrosorption valency of halide ions at Ag(111)^{51,52,53,54}. While neither of these observations can be reproduced at the CHE level, already the secondorder expansion introduces them qualitatively, if not even in quantitative agreement with existing experimental data. We find that with current simple implicit solvation models higherorder terms captured in the FGC approach do not generally lead to further significant improvements. Apart from the intrinsic shortcomings of the continuum solvation itself^{55}, we also discuss alternative sources for the remaining discrepancies with experiments. As an important corollary, the present work also demonstrates analytically that the FGC energetics does not tend toward CHE results in the limit of single adsorbates in infinitely large surface unitcells. Instead, the discussed applications suggest that secondorder calculations are a computationally efficient approximation to FGC calculations.
Results
Ab initio thermodynamics and interfacial free energy
The central quantity in our ab initio thermodynamics approach to surface electrochemistry is the Gibbs excess energy of interface configuration α^{4,5,13},
The specific interface configuration α is hereby characterized by its surface geometry (e.g., the position of adsorbates) and chemical composition. In this work we will focus on adsorbates at metal electrodes and we will specify this composition in terms of the number of neutral substrate atoms \({N}_{{\rm{s}}}^{\alpha }\), the number of (possibly charged) adsorbate species \({N}_{{\rm{a}}}^{\alpha }\), and the number of excess electrons \({N}_{e}^{{\rm{abs}},\alpha }\) (conveniently measured as absolute surface charge, cf. below). The additional dependence on temperature T and pressure p in the grandcanonical setup is henceforth dropped for ease of notation. Appropriately normalized to the surface area A, the intensive interface excess free energy is
and denotes the cost of creating the electrode surface α in the electrochemical environment at the applied potential Φ_{E}, as it compares the Gibbs free energy of the electrode \({G}_{{\rm{surf}}}^{\alpha }\) with the sum of the free energies of its constituents in their corresponding (bulk) reservoirs. These free energies are defined by the bulk chemical potential μ_{s} of the (neutral) substrate atoms and the electrochemical potentials (highlighted with a tilde) of the (charged) adsorbate species \({\tilde{\mu }}_{{\rm{a}}}\) and of the excess electrons \({\tilde{\mu }}_{e}\). The latter relates to the applied electrode potential via \({\tilde{\mu }}_{e}=e{\Phi }_{E}\). Without loss of generality, we choose the extensive Gibbs excess energies for the entire electrode as central base quantities in this work, and will therefore explicitly include a surface area dependence in the equations wherever required.
Note that the sign convention followed in Eq. (1) considers negative Gibbs free energies and (electro)chemical potentials, with more negative values corresponding to more stable or more dilute configurations. The electrode potential Φ_{E} is measured according to electrochemistry conventions, with increasing values away from the zeroreference vacuum level, such that e.g., the experimental standard hydrogen electrode (SHE) lies at +4.44 V on this absolute scale^{56}. In multicomponent systems different neutral and charged species may exist and appropriate sums over the indices s and a with appropriate stoichiometries need to be introduced in Eq. (1). We stress that this need extends also generally to all interfacial species in the double layer, such as water or solvated ions^{57,58,59}. However, in this work we restrict ourselves to implicit solvation models in which only the specifically bound ionic adsorbates a are treated explicitly in Eq. (1).
It is important to point out that the sodefined excess energies have often no direct thermodynamic relevance, as they still depend on the specific interface configuration α. The true thermodynamic excess energy G_{exc} corresponds to the minimum \({G}_{{\rm{exc}}}^{\alpha }\) in the space of all possible interface configurations α^{5}, and this then also defines the true groundstate electrode configuration in terms of N_{s}, N_{a}, and \({N}_{e}^{{\rm{abs}}}\). We note in passing that the two mathematical steps of formulating the excess energies \({G}_{{\rm{exc}}}^{\alpha }\) as in Eq. (1) and then minimizing with respect to configurations α correspond to a Legendre transform, which is why G_{exc} approximates in fact the grand canonical free energy of the system. In a comprehensive allexplicit simulation the optimum interface configuration that defines G_{exc} would eventually emerge directly from an appropriate, ideally exhaustive sampling of all corresponding degrees of freedom. In practice, G_{exc} is instead often approximately determined by computing \({G}_{{\rm{exc}}}^{\alpha }\) for a set of candidate configurations α with fixed \({N}_{{\rm{s}}}^{\alpha },{N}_{{\rm{a}}}^{\alpha },{N}_{e}^{{\rm{abs}},\alpha }\), and the configuration with minimal \({G}_{{\rm{exc}}}^{\alpha }\) is declared to be the most stable one^{3,4,5,60,61}. It is also possible to compare candidate configurations and excess energies that already minimize \({G}_{{\rm{exc}}}^{\alpha }\) in a subspace of all interface configurations α. In the electrochemical context, this refers notably to chargeequilibrated excess energies \({{\mathcal{G}}}_{{\rm{exc}}}^{\alpha }\) that minimize \({G}_{{\rm{exc}}}^{\alpha }\) with respect to the number of electronic charges \({N}_{e}^{{\rm{abs}},\alpha }\) for a configuration α with defined \({N}_{{\rm{s}}}^{\alpha }\) and \({N}_{{\rm{a}}}^{\alpha }\)
\(\mathcal{G}_{{\rm{exc}}}^{\alpha }\) then defines the cost of creating an interface with composition \({N}_{{\rm{s}}}^{\alpha }\) and \({N}_{{\rm{a}}}^{\alpha }\) at a given applied potential Φ_{E}, as would be obtained equivalently in a constant potential calculation^{27,30,62}.
As common in ab initio thermodynamics approaches to surface systems^{4}, the difference in the solidstate terms in Eq. (1), i.e., \({G}_{{\rm{surf}}}^{\alpha }\) and any bulklike reservoir μ_{s}, is generally approximated as
Here, \({E}_{{\rm{surf}}}^{{\rm{DFT}},\alpha }\) is the total energy of candidate structure α and \({E}_{{\rm{bulk}},{\rm{s}}}^{{\rm{DFT}}}\) is the total energy per atom of the bulklike reservoir of neutral species s, both total energies being typically calculated with DFT. ΔF^{corr,α} is a correction term due to the changes in the vibrational and configurational degrees of freedom of the adsorbates in candidate structure α as compared to their corresponding reservoir. This free energy term is commonly assumed to be independent of the charge state \({N}_{e}^{{\rm{abs}},\alpha }\) of the candidate structure. One should note that even though \({E}_{{\rm{surf}}}^{{\rm{DFT}},\alpha }\) is denoted here as a total energy, it does contain free energy contributions due to solvent screening and electrolyte response when an implicit solvation model is employed in the DFT calculation^{5,32}.
FGC calculations
In Eq. (1) the absolute surface charge \({N}_{e}^{{\rm{abs}},\alpha }\) of candidate configuration α measures the number of electrons in excess or deficient with respect to the number of electrons of the corresponding chargeneutral electrode surface at the applied potential (other references are possible without affecting the interpretation^{5}). \({N}_{e}^{{\rm{abs}},\alpha }\) is thus the number of electrons that need to be exchanged with the external electron reservoir. In the present application to adsorbates at metal electrodes
i.e., \(e{N}_{e}^{{\rm{abs}},\alpha }\) is the sum of charges to compensate for the ion charges \({q}_{{\rm{a}}}{N}_{{\rm{a}}}^{\alpha }\) of all adsorbates of charge q_{a} and the net electronic surface excess charge \(e{N}_{e}^{{\rm{net}},\alpha }\) that is itself compensated exactly by the electrolyte counter charges in the diffuse layer. Without meaningful countercharge model, any net charge of a metallic surface poses a significant challenge in calculations with periodic boundary conditions^{63}. In its practical realization, without explicitly represented electrolyte ions (c.f. ref. ^{64}) the CHE^{2,6,61} approach therefore considers only chargeneutral candidate structures with \({N}_{e}^{{\rm{net}},\alpha }=0\). In order to do so, it thus assumes that each ion of charge q_{a} also drags q_{a}/e electrons onto the surface upon adsorption. Any electrochemical reaction involving a proton transfer would then, for instance, necessarily become a protoncoupled electron transfer (PCET). For such chargeneutral structures, solvation effects, at least on the level of implicit solvation models, are often small. This is why corresponding CHE work is often also based on DFT calculations performed without any solvation treatment at all.
FGC simulations in implicit solvation models instead bypass the overall supercell charge neutrality restriction by balancing any \({N}_{e}^{{\rm{net}},\alpha }\;\ne\;0\) through added electrolyte counter charges in the dielectric region between the periodically repeating slabs. At sufficiently high electrolyte concentrations, the details of the electrolyte model are thereby largely irrelevant for the overall energetics^{5,19,32,35,39}. Practical calculations then perform the minimization of \({G}_{{\rm{exc}}}^{\alpha }\) in Eq. (1) with respect to \({N}_{e}^{{\rm{abs}},\alpha }\) for every candidate structure by explicitly calculating a number of finite charge states. Within the approximation in Eq. (4) and assuming a symmetric slab setup the minimization of Eq. (3) yields the condition
where the underbraced equality arises from Janak’s theorem and the employed reference for the electrostatic potential (see Methods). Equation (6) thus states that the minimization is equivalent to choosing a system such that its work function eΦ^{α} is equal to the externally applied (target) potential Φ_{E} times the elementary charge e. Thus, the chargeequilibrated \({{\mathcal{G}}}_{{\rm{exc}}}^{\alpha }\) can be directly obtained by interpolating the dependence of the energy and the charge on the work function and evaluating \({G}_{{\rm{exc}}}^{\alpha }\) for charge values where the work function corresponds to Φ_{E}^{5}, similar to what is done in the generalized CHE and related constantpotential approaches^{27,30,62}.
Obviously, this procedure increases the computational burden, as each candidate structure has to be calculated in different charge states. On the other hand, certain electrochemical observables can be simulated that would otherwise be inaccessible within the CHE approximation. Noteworthy, these include nonlinear variations of the interface energy with applied potential (cf. Lippmann equation^{5,47,65,66,67}) potentialinduced surface reconstructions^{5,13,41} that maintain the overall stoichiometry in terms of N_{a} and N_{s}, and pHshifts of electrosorption peaks on the RHE scale as well as noninteger electrosorption valencies^{51,52,54}, as will be explicitly illustrated here.
Grand canonical energetics within a quadratic approximation
Further insight into the differences between FGC and CHE results can be gained by analyzing a secondorder Taylor expansion of \({E}_{{\rm{surf}}}^{{\rm{DFT}},\alpha }\) with respect to the net electronic surface excess charge \({N}_{e}^{{\rm{net}},\alpha }\)^{7,11,34,39,42,43,44,45}. Within a symmetric slab setup—with a total surface area of 2A and a unique electrostatic reference level in the implicit region—\({E}_{{\rm{surf}}}^{{\rm{DFT}},\alpha }\) can be expanded around the chargeneutral point \({N}_{e}^{{\rm{net}},\alpha }=0\) as
where \({E}_{{\rm{surf}},0}^{{\rm{DFT}},\alpha }\) is the total energy of the uncharged system, \({\Phi }_{0}^{\alpha }\) the work function and \({c}_{0}^{\alpha }\) the extensive interfacial capacitance at the potential of zero charge (PZC). \({c}_{0}^{\alpha }\) relates to the more common, areanormalized capacitance \({C}_{0}^{\alpha }\) measured in μF/cm^{2} via \({c}_{0}^{\alpha }=2A{C}_{0}^{\alpha }\). Here and in the following, a subscript _{0} will refer to quantities determined at the PZC. The explicit dependence of the total and excess energies on \({N}_{{\rm{s}}}^{\alpha }\) and \({N}_{{\rm{a}}}^{\alpha }\) is dropped for ease of notation.
Within this secondorder expansion, the minimization of the excess free energy of candidate structure α with respect to the number of electrons can be carried out analytically following Eq. (6) to obtain
Inserting Eqs. (7) and (8) into Eq. (1) then yields the chargeequilibrated excess energy \({{\mathcal{G}}}_{{\rm{exc}}}^{\alpha }\) up to secondorder as
The first bracket in Eq. (10) is identical to the excess free energy \({G}_{{\rm{exc,CHE}}}^{\alpha }({\Phi }_{E})\) as it would result within the CHE approach. The second term \({G}_{{\rm{exc,DL}}}^{\alpha }\) corresponds to the energy cost of charging the double layer.
This derivation highlights that the CHE approximation is equivalent to a firstorder approximation in Φ_{E} of the FGC energetics. For each candidate structure α the CHE energy expression is therefore accurate around the PZC of α (\({\Phi }_{0}^{\alpha }={\Phi }_{E}\leftrightarrow {N}_{e}^{{\rm{net}}}=0\)), where the term \({G}_{{\rm{exc,DL}}}^{\alpha }\) becomes small. On the other hand, for potentials much different than the PZC (\({\Phi }_{0}^{\alpha }\;\ne\;{\Phi }_{E}\leftrightarrow {N}_{e}^{{\rm{net}}}\;\ne\;0\)) the additional energy contributions \({G}_{{\rm{exc,DL}}}^{\alpha }\) can become relevant. Within the CHE model, the only potentialdependence of \({{\mathcal{G}}}_{{\rm{exc}}}^{\alpha }({\Phi }_{E})\) arises from the term \({N}_{{\rm{a}}}^{\alpha }({q}_{{\rm{a}}}{\Phi }_{E}{\tilde{\mu }}_{{\rm{a}}})\). This dependence is thus always linear in Φ_{E}, at variance with the typical parabolic potential dependence of electrochemical interface energies^{5,47,68}.
From the previous derivation we furthermore see that the relative stability of two electrode configurations α, β differing only in the number of adsorbates a and at applied potential Φ_{E} is
Within the CHE approximation the relative stability of systems with the same number of adsorbates \({N}_{{\rm{a}}}^{\alpha }={N}_{{\rm{a}}}^{\beta }\) is thus potentialindependent, excluding the possibility to simulate potentialdependent diffusion barriers, potentialinduced shifts of stable adsorption sites, or potentialinduced surface reconstructions that conserve the number of adsorbates (and generally also substrate atoms) at the electrode, as e.g., observed for the quasi hexagonal reconstruction of Au(100)^{5}. Intriguingly, all these shortcomings are lifted when including the next higher order term in Φ_{E}, namely the capacitive charging contribution of the double layer \({G}_{{\rm{exc,DL}}}^{\alpha \beta }\)–an energy contribution that is widespread in the classical electrochemical literature^{51,52,53,68,69,70,71,72,73,74,75,76,77,78}. We will explicitly demonstrate below how this contribution leads to a shift of the electrosorption peaks with pH, and how it relates to the socalled electrosorption valency, another classic concept in phenomenological electrochemistry^{51,52,53}.
The doublelayer correction \(\Delta {G}_{{\rm{exc,DL}}}^{\alpha \beta }({\Phi }_{E})\) can be further simplified by a change of variables according to \(\Delta {c}_{0}={c}_{0}^{\alpha }{c}_{0}^{\beta }\), \({\bar{\Phi }}_{0}=\frac{1}{2}({\Phi }_{0}^{\alpha }+{\Phi }_{0}^{\beta })\), and \(\Delta {\Phi }_{0}={\Phi }_{0}^{\alpha }{\Phi }_{0}^{\beta }\), which yields
In this notation, \(\Delta {G}_{{\rm{exc,DL}}}^{\alpha \beta }\) becomes very suggestive. The first term corresponds to the (average) applied potential drop across the electric double layer \(({\Phi }_{E}{\bar{\Phi }}_{0})\) times the charge difference \(\Delta {q}^{{\rm{net}}}={c}_{0}^{\beta }\Delta {\Phi }_{0}\) necessary to bring both systems α and β to an equal work function. We denote it with the superscript ^{net}, as the term corresponds to the difference in net surface charges (8) of both systems. As shall be shown below it can also be directly related to the adsorbate dipole^{79} and the electrosorption valency. The second term results from a change in the capacitive energy due to different interfacial capacitances Δc_{0} of the two electrode configurations. Note that identical interface energy contributions were already proposed by Frumkin^{68} to understand electrosorption of chargeneutral species.
pH shifts of proton electrosorption peaks at platinum
Electrosorption peaks in cyclic voltammetry (CV) measurements are known to be sensitive probes of the electric double layer^{5,46,47}. They therefore form ideal observables to assess critically the capabilities and quantitative performance of the CHE and the FGC approaches. Here, we concentrate in particular on proton electrosorption at two lowindex single crystal surfaces of platinum, Pt(111) and Pt(100). For both systems, small but significant shifts of the electrosorption peaks with pH even on the RHE scale have been observed experimentally^{48,49,50}. Such shifts go beyond what would simply be predicted by the Nernst equation and are certainly an intriguing object of study.
Within ab initio thermodynamics we approximate the electrosorption peak position^{5,46} as the electrode potential \({U}_{{\rm{RHE}}}^{{\rm{sorp}}}\) on the reversible hydrogen electrode (RHE) scale for which the chargeequilibrated excess energy \({{\mathcal{G}}}_{{\rm{exc}}}^{{\rm{H}}}\) of the hydrogencovered electrode surface, with \({N}_{{\rm a}}^{{\rm{H}}}={N}_{{\rm{H}}}\) specifically adsorbed H atoms, becomes equally stable as the chargeequilibrated excess energy \({{\mathcal{G}}}_{{\rm{exc}}}^{{\rm{clean}}}\) of the pristine, clean surface
Electrode potentials on the absolute and on the RHE scale are hereby related as \({\Phi }_{E}={U}_{{\rm{RHE}}}+4.44\ {\rm{V}}\frac{{k}_{{\rm{B}}}T}{e}\mathrm{ln}\,(10){\rm{pH}}\)^{5,9,80}. Inserting the secondorder expansion of \(\Delta {{\mathcal{G}}}_{{\rm{exc}}}^{{\rm{Hclean}}}\) (Eq. (11)) we thus arrive at the condition
where the firstorder CHE term is
and the doublelayer correction is
Here, we have specified the general variables in Eqs. (11) and (14) for proton electrosorption as q_{a} = +e and \({\tilde{\mu }}_{{\rm{a}}}={\tilde{\mu }}_{{\rm{H}}}\), and have introduced the charge difference between the clean and Hcovered surface \(\Delta {q}^{{\rm{net}}}={c}_{0}^{{\rm{clean}}}\Delta {\Phi }_{0}\), together with the extensive interfacial capacitance \({c}_{0}^{{\rm{clean}}}=2A{C}_{0}^{{\rm{clean}}}\) of the clean surface, the difference in extensive interfacial capacitances \(\Delta {c}_{0}={c}_{0}^{{\rm{H}}}{c}_{0}^{{\rm{clean}}}\), and the difference \(\Delta {\Phi }_{0}={\Phi }_{0}^{{\rm{H}}}{\Phi }_{0}^{{\rm{clean}}}\) and average \({\bar{\Phi }}_{0}=1/2({\Phi }_{0}^{{\rm{H}}}+{\Phi }_{0}^{{\rm{clean}}})\) of the work functions of the Hcovered and clean surfaces, respectively.
Within the CHE approximation (\(\Delta {G}_{{\rm{exc,DL}}}^{{\rm{H}}{\rm{clean}}}=0\)), the condition is readily resolved for \({U}_{{\rm{RHE}}}^{{\rm{sorp}},{\rm{CHE}}}\) by exploiting the electrochemical potential of a proton, which is given by the SHE conditions as^{5}
where μ(H_{2}(g)) is the chemical potential of hydrogen gas at normal conditions (298 K, 1 bar). This then yields
where in the second step we have followed the general approximation of Eq. (4) and introduced the average H adsorption energy \({\bar{E}}_{{\rm{ads}}}^{{\rm{H}}}\) plus the (negligibly small) free energy correction ΔF^{corr,H} due to the change of vibrational and configurational degrees of freedom of the adsorbed hydrogen. The CHE approximation thus essentially predicts the proton electrosorption peak on the RHE scale simply at the potential corresponding to minus the average hydrogen adsorption energy. In fact, this holds in general, i.e., the CHE approximation would always equate electrosorption and adsorption for any kind of adsorbate. This is why in the CHE literature electrosorption and adsorption are often used interchangeably.
Most importantly, Eq. (20) does not depend anymore on pH, i.e., within the CHE approximation the proton electrosorption peak does not exhibit any shifts on the RHE scale. If we were to transform \({U}_{{\rm{RHE}}}^{{\rm{sorp}},{\rm{CHE}}}\) to an absolute scale, the peak would thus shift simply as predicted by the Nernst equation, at variance with the experimental observations (cf. further below). To this end, we acknowledge that lateral interactions among the adsorbed H atoms and configurational entropy terms might introduce a variation of the average adsorption energy and free energy correction term with surface coverage^{9}. However, this will rather lead to a broadened electrosorption peak than natively introduce a pHdependence in \({U}_{{\rm{RHE}}}^{{\rm{sorp}},{\rm{CHE}}}\). Similarly, different interface models in the underlying DFT calculations, e.g., the surface in vacuum, in various implicit solvation models or with differing adsorbed icelike water layers, will primarily affect only the absolute value of the calculated adsorption energy and therewith the predicted pHindependent electrosorption peak position. The only way nonNernstian shifts can be rationalized within the CHE model is by a pHinduced change in the adsorption energy, assuming e.g., a pH dependence of the interfacial water structure, or secondary effects such as the adsorption of coions, as further discussed below.
In contrast, already the secondorder doublelayer correction \(\Delta {G}_{{\rm{exc,DL}}}^{{\rm{H}}{\rm{clean}}}({U}_{{\rm{RHE}}}^{{\rm{sorp}}})\) introduces intrinsic pHdependencies, and leads to a quadratic equation defining \({U}_{{\rm{RHE}}}^{{\rm{sorp}},2{\rm{nd}}}\) (cf. Eqs. (16)–(18)). This quadratic equation and even more so the higherorder terms contained in the FGC approach yield therefore complex nonNernstian shifts of electrosorption peaks on the RHE scale. As an example, we point out that already the linear term in \(\Delta {G}_{{\rm{exc,DL}}}^{{\rm{H}}{\rm{clean}}}({U}_{{\rm{RHE}}}^{{\rm{sorp}}})\) (∝Δq^{net}) effectively renormalizes the CHE peak position, while introducing an additional, linear dependence on pH as
where we have introduced the peradsorbate quantity \(\Delta {Q}^{{\rm{net}}}=\frac{\Delta {q}^{{\rm{net}}}}{{N}_{{\rm{H}}}}\). ΔQ^{net} is largely independent of the surface coverage, given that the extensive quantity Δq^{net} is expected to scale approximately linearly with N_{H} (\(\propto\Delta {\Phi }_{0}^{\alpha }({N}_{{\rm{H}}})\), see e.g., Supplementary Fig. 10). Thus, whenever ΔQ^{net}/e is not vanishingly small, the corrections and linear pH shift of Eq. (21) become significant.
Figure 1 illustrates the performance of the various approaches, by collecting the calculated proton electrosorption peak positions on Pt(111) and Pt(100). Concentrating first on the data obtained with an electrode surface model containing 1/4 ML hydrogen coverage in the energetically most stable fcc (Pt(111)) and bridge sites (Pt(100)), respectively, we see qualitatively the same trends among the theoretical approaches on both surfaces. In contrast to the constant \({U}_{{\rm{RHE}}}^{{\rm{sorp}},{\rm{CHE}}}\) predicted by the CHE approximation, already the secondorder approximation yields nonNernstian shifts of \({U}_{{\rm{RHE}}}^{{\rm{sorp}},2{\rm{nd}}}\) at both surfaces. We note that a significant contribution to these shifts arises from the second, quadratic term in Eq. (18), i.e., from the large change in the interfacial capacitance ΔC_{0} upon proton electrosorption. In fact, for Pt(100), this contribution is large enough that it actually changes the shift direction from an upwards shift to higher \({U}_{{\rm{RHE}}}^{{\rm{sorp}}}\) from the linear Δq^{net} term only (cf. Eqs. (18) and (21) and the signs of the work function change for H on Pt in Table 1) to a small shift downwards. However, this holds, of course, only, if an implicit solvation model is used in the underlying DFT calculations as done in this work. Using a surface slab in vacuum would yield highly underestimated differential interfacial capacitances and would correspondingly then also yield only minor shifts of the electrosorption potential even within the secondorder approximation.
At both surfaces, the FGC approach yields shifts of the electrosorption potential \({U}_{{\rm{RHE}}}^{{\rm{sorp}},{\rm{FGC}}}\) that almost coincide with the secondorder approximation at low pH. Towards higher pH, the two approaches then exhibit increasingly different results, with the FGC approach leading to smaller absolute shifts. These findings are consistently obtained also for other H adsorption sites and at other surface coverages (see Supplementary Figs. 5–9). Figure 1 illustrates this explicitly by also showing the results calculated for full ML coverage. The primary effect of changing coverage is an offset of the various \({U}_{{\rm{RHE}}}^{{\rm{sorp}}}\), as expected from the different average H adsorption energies. \({U}_{{\rm{RHE}}}^{{\rm{sorp}},2{\rm{nd}}}\) and \({U}_{{\rm{RHE}}}^{{\rm{sorp}},{\rm{FGC}}}\) exhibit only small changes of their slope and curvature. This is in agreement with the often rigid shift of experimental CV peaks at overall unaltered shape^{5}. From the above derivations, we see that this will occur whenever the change of work function and interfacial capacitance scales approximately linearly with the surface coverage. Then the doublelayer correction \({G}_{{\rm{exc,DL}}}^{{\rm{H}}{\rm{clean}}}({U}_{{\rm{RHE}}}^{{\rm{sorp}}})\) will scale similarly with N_{H} as the CHE term \({G}_{{\rm{exc,CHE}}}^{{\rm{H}}{\rm{clean}}}({U}_{{\rm{RHE}}}^{{\rm{sorp}}})\), and the pH terms in the overall condition for \({U}_{{\rm{RHE}}}^{{\rm{sorp}},2{\rm{nd}}}\), Eq. (16), become coverage independent. Such a behavior can be expected for most adsorbates, whenever adsorbateinduced work function changes derive from local dipoles and capacitance variations from local alterations of the capacitance.
While the comparison of the various theoretical approaches sketches a rather consistent picture for both Pt(111) and Pt(100), clear differences arise in Fig. 1 in the comparison to the existing experimental data^{48,49,50}. For Pt(111), the nonNernstian shift of the proton electrosorption peak is reproduced very nicely by the FGC approach. Even though this holds also for the absolute positions, we note that this comparison is less welldefined. First, we infer only approximately the absolute experimental positions from the better resolved half peakcurrents of the CVs in refs. ^{48,49}. (see also Supplementary Fig. 11). Second, absolute theoretical positions are sensitively affected by the employed approximate DFT exchangecorrelation functional, through the change in the H adsorption energy. The good agreement reached with the PBE functional could thus be partially fortuitous as discussed further below. More important is the fact that the FGC approach yields nonNernstian shifts of the correct sign and order of magnitude without having to invoke a pHdependent change of the adsorption energy. To this end, it is important to point out that a previous theoretical CHE study by Karlberg et al.^{46} found no impact on the adsorption energy upon application of an electrostatic field and inclusion of explicit interfacial water molecules. In contrast, most recent results from ab initio molecular dynamics suggest a small influence of explicit water on average adsorption energies^{81}. Nevertheless, even such refined dynamical simulations on neutral slabs, i.e., remaining at the CHElevel in terms of electrode charge equilibration, could not explain the nonNernstian pH shifts of \({U}_{{\rm{RHE}}}^{{\rm{sorp}}}\) at this surface. We therefore believe that the rationalization of these shifts within the FGC approach is indeed correct for the right reason, i.e., that the appropriate account of charge equilibration in the FGC scheme is key to describe proton electrosorption here.
On Pt(100) the performance of the FGC approach as measured by the existing experimental data is not as good. The peak position is significantly misaligned and the nonNernstian shift with pH is predicted in the wrong direction and with wrong magnitude. The FGC approach predicts an almost negligible shift, in contrast to experiments where the magnitude of the shift is of comparable size as at Pt(111), just in the opposite direction. As already pointed out, the misalignment is most likely related to the exchangecorrelation functional. A reevaluation of \({U}_{{\rm{RHE}}}^{{\rm{sorp}},{\rm{CHE}}}\) with the revPBE functional^{82,83} yields, for instance, a peak position lowered by 140 meV to 0.33 V, which agrees very well with the average experimental position. At the same time though, this functional would also lower the predicted Pt(111) peak position by 130 meV and thus worsen the apparent agreement with experiment for this surface. Accepting a typical ±200 meV uncertainty of GGAlevel absolute adsorption energies, the truly critical point assessing the description of the electrochemistry are therefore not the absolute positions, but the nonNernstian shifts of the electrosorption peaks. To this end, the inability of our calculations to describe the experimental pH shifts at Pt(100) are most likely related to the absence of the explicit water structure in the present calculations. As shown convincingly in a recent study by Cheng et al.^{84}, an increasing distance between metallic surface and interfacial water for higher pH values lead to a decrease in the adsorption energy and thereby to a peak shift of correct magnitude on the RHE scale. In general, any such influence of surfacespecific interfacial water^{58,59,85,86,87,88,89,90,91,92} can not be captured in calculations based only on implicit solvation. This highlights that presentday FGC calculations still need to be seen as a numerically efficient approximation—that as a next step might need to be refined by appropriately extending them to mixed explicit/implicit solvation models^{39,58}. We also note in passing that Pt(100) has no proper double layer region (coadsorbed H, OH)^{49}; coion effects might thus also contribute to the overall pHshift on the RHE scale^{93}. As such, proton electrosorption will remain an intriguing test system for future more explicit calculations to address the complexity of the interface.
Electrosorption valencies of halide ions at Ag(111)
Electrosorption valencies are another fundamental and, as we shall see, closely related concept in the context of electrosorption in interfacial electrochemistry^{51,52}. The electrosorption valency l_{a} (or formal charge number^{77}) denotes the number of electrons that flows onto the electrode upon electrosorption of one adsorbate a at applied potential Φ_{E}. While some debate exists of how to actually measure l_{a} or relate it to other concepts like partial charge transfer^{51,52,53,54,70,71,72,73,74,75,76,77,78}, l_{a} is a welldefined thermodynamic quantity. According to the Lippmann equation^{47,65}, the electronic surface charges are given by the derivative of the interface energy with respect to the applied potential. Within the present ab initio thermodynamics framework, this definition translates to obtaining the absolute number of electrons \({N}_{e}^{\alpha ,{\rm{abs}}}({\Phi }_{E})\) of an electrode configuration α from the derivative of the chargeequilibrated excess energy \({{\mathcal{G}}}_{{\rm{exc}}}^{\alpha }({\Phi }_{E})\) with respect to Φ_{E},
Without loss of generality, we will focus here again on a symmetric slab setup, where on each of the two identical surfaces one adsorbate a electrosorbs to an empty site of an electrode configuration α characterized by N_{a} adsorbates of the same type and at the same site type. This results in a new electrode configuration with 2(N_{a} + 1) adsorbates (and, as before, without changing the number of substrate atoms N_{s}). The electrosorption valency of this process is then
where the factor 1/2 renormalizes the symmetric slab setup. If we insert the general secondorder expansion (Eqs. (11) and (14)), the electrosorption valency can again be separated into a CHE and a DL part
At the CHE level we recover the assumption dictated by chargeneutrality that each adsorbed ion of charge q_{a} drags q_{a}/e electrons onto the surface. The electrosorption valency is thus purely given by the formal ionic charge: it is a constant integer number, independent of the applied potential and surface structure or composition. This simplification is lifted by the twoterm DL correction. The first potentialindependent term renormalizes the formal ionic charge by the charge difference \(\Delta {q}^{{\rm{net}}}={c}_{0}^{2{N}_{{\rm{a}}}}\Delta {\Phi }_{0}\) and therewith allows for noninteger electrosorption valencies. The second term introduces an explicit potential dependence. To this end, it is important to point out that in this derivation both DL correction terms formally contain extensive differential capacitances. However, the intensive electrosorption valency is still well defined, as we consider the idealized situation where only one adsorbate electrosorbs on each side of the symmetric electrode slab.
This is further clarified when switching to a meanfield picture that characterizes the adsorbate layer only in terms of the surface coverage θ = N_{a}*A_{site}/A, where A_{site} is the crystallographically determined surface area per adsorption site. This allows to describe the finite change in adsorbate numbers in terms of changing coverage, e.g.,
This expression then contains only intensive quantities: A_{site}, the differential capacitance \({C}_{0}^{\theta }\), and the derivative of the work function with respect to θ. In this formulation, the DLcorrection to the electrosorption valency becomes
i.e., in essence a charge per adsorbate necessary to keep the work function constant upon electrosorption. ΔQ^{net} is identical to the term introduced before in the discussion of linear pH shifts for proton electrosorption in Eq. (21), where the derivative was approximated by the average change per adsorbate. Considering that \(\frac{{\rm{d}}{\Phi }_{0}^{\theta }}{{\rm{d}}\theta }=\frac{1}{{\epsilon }_{0}{A}_{{\rm{site}}}}{d}_{z}\) can be related to the normal component of the adsorbate dipole d_{z}^{79} (with electrochemical sign convention), we recognize that the terms in Eq. (27) are closely related to the dipolar and capacitive contributions to the electrosorption valency that are recurrently discussed in classical electrochemistry works^{47,51,52,94,95}. In fact, the relation between d_{z} and l_{a} is identically reported by Schmickler^{95}.
To illustrate the effects of the DL corrections, we consider the electrosorption of halide ions (Cl^{−}, Br^{−}, I^{−}) on Ag(111) in the lowcoverage regime. The experimentally wellestablished increase of their electrosorption valencies with increasing Pauling electronegativity^{53,54}, i.e., \({l}_{{\rm{I}}}^{\exp }<{l}_{{\rm{Br}}}^{\exp }<{l}_{{\rm{Cl}}}^{\exp }\), up to values that are only a fraction of the formal charge of −1, poses a challenging test case. While the electrosorption valency is generally a function of applied potential and coverage (cf. Eq. (27) for the DL correction), we focus here on the l_{a} values in the coverage range up to 1/3 ML, at a given fixed potential. Specifically, we choose a potential value for which Foresti et al. observe ordered halide overlayers corresponding to 1/3 ML^{54}. After correcting for the underestimation of the PZC of Ag(111) with the current solvent parameters (−0.42 eV), we have: Φ_{E}(I) = 4.0 V, Φ_{E}(Br) = 4.1 V and Φ_{E}(Cl) = 4.1 V. Interpolation of \({\Phi }_{0}^{\theta }\) and \({C}_{0}^{\theta }\) computed at coverages between 1/12 an 1/3 ML (see Supplementary Figs. 12 and 13) allows to determine the derivatives in Eq. (27) and therefore the doublelayer correction to the electrosorption valencies. Figure 2 provides the corresponding valencies, computed with the CHE approximation, with the full secondorder doublelayer correction and only considering the secondorder ΔQ^{net} term. By construction, the CHE approximation yields the formal charge −1 for the entire halide ion series. Instead, the secondorder terms introduce varying noninteger values which vary with coverage. To this end, it is important to note that the extrapolations of the corresponding lines to the limit θ → 0 do not coincide with the CHE results, supporting the fundamental differences between FGC and CHE calculations also for individual adsorbates in the limit of infinite cell size. In this respect, it is reassuring to see that already the consideration of the secondorder ΔQ^{net} term reproduces the expected trend of the computed electrosorption valencies with the Pauling electronegativity over the entire lowcoverage range. This also extends to the full DL correction, but the actual values obtained depend then on the chosen applied potential (cf. Eq. (27)). Deferring a detailed comparison to experimental data to a forthcoming publication, we include in Fig. 2 measured “integral” electrosorption valencies for low coverages^{54}. Also in light of the large uncertainties of different measurement methods^{53,54}, the agreement obtained with the DL corrected calculations is very encouraging, and confirms the importance of FGC approaches toward a reliable firstprinciples description of electrified interfaces.
Discussion
Ab initio thermodynamics offers a computationally efficient and ideally predictive access to questions in interfacial electrochemistry. In the CHE approximation it enjoys high popularity, in particular in applications to electrocatalysis. In this work, we used electrosorption as a fundamental elementary step in electrocatalysis to compare this approach to FGC ab initio thermodynamics. The present analysis underscores the importance of doublelayer charging, missing in the zero netcharge CHE approach. In the context of electrosorption this manifests itself in the ability to describe nonNernstian pHshifts of electrosorption peaks and noninteger electrosorption valencies; both effects can be captured at the variablecharge FGC level. As evaluated against existing experimental data for proton electrosorption at Pt electrodes, and halide ion electrosorption at Ag electrodes, the present firstprinciples description yields in general results closer to experiment than pure CHE calculations and is partly already semiquantitative. Analysis of a secondorder approximation to the FGC energetics allows to rationalize the relevance of the potential of zero charge, the interfacial capacitance, and work function shifts induced by adsorbates e.g., through their dipoles. All the expressions derived here for the electrochemical stability of adsorbates and their electrosorption valencies are in agreement with classical electrochemistry works^{47,51,52,53,68,69,70,71,72,73,74,75,76,77,78,94,95}, closing the gap between phenomenological and firstprinciples treatments of electrified interfaces.
On a more practical level, the analysis with the secondorder model provides clear insight into the limitations of zerocharge calculations and guidelines for their usage:

Increased deviations of the CHE energetics from FGC results are expected whenever the electrode potentials Φ_{E} considered are significantly different from the potential of zero charge Φ_{0}, as the correction terms scale with (Φ_{E} − Φ_{0}).

Energetic differences between CHE and FGC calculations are proportional to the magnitude of the interfacial capacitance C_{0}; CHE energetics thus becomes problematic for high electrolyte concentrations with large C_{0} values. Furthermore, a nonnegligible influence of the electrolyte chemistry can be expected, due to ionspecific interfacial capacitances^{96,97}. DFT slab calculations in vacuum underestimate C_{0} significantly and do therefore not allow to assess the effects of an applied potential.

Major differences between CHE and FGC calculations will occur for adsorbates with significant dipole moments, whenever the work function change per adsorbate is large. This is in line with other studies^{30,40} and was noted early^{7,98}.
The improved description at the FGC level comes at a somewhat increased computational cost. This does not so much refer to the actual DFT calculations themselves, thanks to efficient implementations of current implicit solvation models. Instead, it refers to the larger number of DFT calculations of the system in different charge states. Depending on the specific application this may typically mean ~10 times more calculations. While this will typically not be prohibitive, we note that the number of calculations required to determine the quantities for the secondorder model is generally smaller, as it only includes parameters obtained at the PZC, namely adsorption energies, \({\Phi }_{0}^{\alpha }\) and the interfacial capacitance \({C}_{0}^{\alpha }\). Considering the success of this model in describing the electrosorption phenomena studied in this work, calculations at this level might therefore be an appealing intermediate option for systems where computational cost is truly limiting. Having said this, we emphasize that the remaining inaccuracies of FGC calculations can still be significant. Notable factors are the approximate DFT exchangecorrelation functional and their errors in adsorption energies, the neglect of explicit water^{58,59,86}, or missing coions^{93}. In addition, the present analytical derivations indicate that discrepancies between allimplicit and allexplicit interfacial capacitances^{39,55,92,99} and work functions^{58} can have an impact as well. We expect many of these limitations to be overcome by FGCtype schemes with implicit/explicit hybrid descriptions of interfacial water, an approach we will pursue in the future.
Methods
DFT calculations
All DFT calculations reported below are performed with the Quantum ESPRESSO package^{100} (PWscf), the PBE exchangecorrelation functional^{101} and pseudopotentials from the SSSP library^{102} (v0.7, PBE, efficiency) with density and wave function cutoffs of 360 and 45 Ry, respectively. The two systems, H on Pt and halide ions on Ag are studied in a periodic, symmetric slab setup with substrate and adsorbate degrees of freedom described explicitly. For Pt we use (2 × 2) cells with 7 and 8 Pt layers for the (111) and (100) surfaces, respectively; for Ag(111) we use a (\(\sqrt{12}\times \sqrt{12}\)) supercell consisting of 6 Ag layers. Slabs are separated by ≈17 Å. Brillouin zone integrations are performed using Γcentered MonkhorstPack meshes (Pt: (10 × 10 × 1), Ag: (4 × 4 × 1)) and a cold smearing^{103} of 0.01 Ry (Pt) or 0.02 Ry (Ag), yielding a numerical accuracy better than 1 meV/Å^{2} in the interface energies.
As implicit solvation model we use the SCCS implementation of ENVIRON^{18,36,104,105} with optimized interfacial parameters (\({\rho }_{\min }=0.0013,{\rho }_{{\rm{max}}}=0.01025\), α = β = γ = 0) and a Helmholtzlayer representation of the electrolyte via planar counter charges. For this setup, good agreement is found between experimentally measured and predicted PZCs, interfacial capacitances, and electrosorption behavior for Pt electrodes, as reported in earlier work^{5}. In particular, the work function—when determined as the energy difference between the Fermi level and the flat electrostatic potential in the implicit region—and the electrode potential Φ_{E} on the absolute scale are identical, which is why all electrochemical potentials (ionic and electronic) in this work are expressed on the absolute scale.
As reference energies for the adsorbates we use the Gibbs free energies of the diatomic molecules (p = 1 bar, T = 298 K) in vacuum including the vibrational and rotational degrees of freedom. The free energy corrections ΔF^{corr} are determined as the Helmholtz free energy contributions due to coverageindependent vibrations of the adsorbed species (calculated via finitedifferences), neglecting small configurational entropy contributions. Specifically, in the case of H@Pt we thus discriminate only between hollow and bridge sites with the (111)fcc and (100)bridge vibrations for 0.25 monolayer (ML) coverage taken as peradsorbate values.
For the determination of the chargeequilibrated excess energies, we optimize the geometries for the interfacial candidate structures for a set of finite surface charges \({N}_{e}^{{\rm{net}}}\) (≥9 values centered around 0) and evaluate their work functions Φ^{α} and total energies \({E}_{{\rm{surf}}}^{{\rm{DFT}},\alpha }\), as exemplified in Fig. 3a for the case of H at fcc sites on Pt(111). Values are chosen to sample a potential range of ~4–5 V around the PZC (\({N}_{e}^{{\rm{net}}}=2A{C}_{0}({\Phi }_{E}{\Phi }_{0})\) with C_{0} ≈30 μF/cm^{2}) for H on Pt, and in a smaller range of ~2 V for the halide systems. As described above and more thoroughly in ref. ^{5}, chargeequilibrated excess energies \({{\mathcal{G}}}_{{\rm{exc}}}^{\alpha }\) (dots in Fig. 3c) can then be obtained directly for the set of electrode potentials Φ_{E}, which correspond to the obtained set of work functions \({\Phi }^{\alpha }({N}_{e}^{{\rm{net}},\alpha })\). This involves evaluation of Eq. (1) with the described approximations and for the DFT energies with corresponding surface charges \({E}_{{\rm{surf}}}^{{\rm{DFT}},\alpha }\left({N}_{e}^{{\rm{net}},\alpha }({\Phi }^{\alpha })\right)\) (dots in Fig. 3a). The lines in Fig. 3 correspond to polynomial interpolations using the data of the studied set of charges. The close agreement between a secondorder model and the FGC results for this system are supported by the reported residuals in Fig. 3b, and the accurate parabolic interpolation in Fig. 3c (error ~1 meV/Å^{2}) (see also Supplementary Figs. 1–4). The interfacial capacitances follow directly from the second derivative of the interpolant (cf. Eq. (9)). Table 1 reports work functions Φ_{0} and areanormalized interfacial capacitances C_{0} for the systems studied in the main text, at the lowest and highest coverage. Values for other coverages and adsorption sites are reported in the Supplementary Tables 1–4.
Data availability
All relevant computational results are provided in the reported tables.
Code availability
All results were obtained with the open source code Quantum ESPRESSO. Postprocessing scripts can be provided on request from the corresponding author.
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Acknowledgements
We acknowledge partial financial support from the Swiss National Science Foundation (SNSF) through the NCCR MARVEL, the EU through the MAX CoE for einfrastructure and the Marie SklodowskaCurie grant agreement No 754462, as well as the German Research Foundation (DFG) CoE econversion EXC 2089/1. This work was supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID s836 and the computing facilities of SCITAS, EPFL. Furthermore, the authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gausscentre.eu) by providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUWELS^{106} at Juelich Supercomputing Centre (JSC). Open access funding provided by Projekt DEAL.
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Hörmann, N.G., Marzari, N. & Reuter, K. Electrosorption at metal surfaces from first principles. npj Comput Mater 6, 136 (2020). https://doi.org/10.1038/s41524020003944
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