Introduction

Descriptor-based approaches carry enormous predictive power in materials design as these can be rapidly deployed to identify materials with desired properties1,2,3,4,5,6,7,8. The empirical Bell (Brønsted)-Evans-Polanyi (BEP) relationship (Brøonsted and Pederson recognized the BBEP relation from studying the decomposition of nitroamide catalyzed by series of anionic bases in aqueous, and was later derived individually by Bell, and by Evan and Polanyi. Generally referred as Bell (Brønsted)-Evans-Polanyi relation)9,10,11,12,13,14, – a linear trend between the activation energy and the reaction energy of a given process, has been widely employed to stipulate reaction kinetics based on easy-to-compute or easily-accessible thermodynamic data15,16,17,18,19,20. In the past two decades, BEP relationships have been widely observed for different reactions such as the dissociation of diatomic and triatomic molecules21,22,23,24.

Hydrogen evolution reaction (HER) is the critical step in electrolysis to produce green hydrogen, which is considered a next-generation clean energy carrier25,26. Leveraging the BEP relation as the key ingredient for computational screening of HER catalysts, Nørskov et al. proposed the hydrogen adsorption free energy ΔGH as the descriptor for the volcano trend – the logarithm of the exchange current is linearly increasing or decreasing with ΔGH27. Such behavior is in line with the Sabatier’s principle, which stipulates that the optimum reaction rates that correspond to moderate adsorption strength between catalysts and reactants. Recent first-principles computational studies have shown that the BEP relation holds for HER on metal surfaces28,29. Experimentally, similar trends were also shown to be valid by studying the cyclic voltammograms on metal surfaces under varying electrochemical conditions30,31; namely, the logarithm of the exchange current was found to linearly correlate with the hydrogen binding energy which was determined from the desorption peak potential or with the HER onset potential. However, the BEP relation has been demonstrated using various kinetic and thermodynamic descriptors. Such different measures underscore its empirical nature, and calls for deeper understanding of its roots and connections, if any, with fundamental electrochemistry at the atomic level.

This study presents a derivation of the BEP relation for HER based on the theoretical Butler-Volmer relation – the norm of the electrochemical theory that describes current-potential characteristics for electrochemical reactions. Utilizing the absolute form of the standard rate constant that follows the transition state theory, we not only correlate between HER thermodynamics and kinetics, but further quantify the BEP relationship solely with the easy-to-compute hydrogen adsorption free energy ΔGH and universal electrochemical constants. Further, we verify the BEP relation using first-principles calculations by finding a direct correlation between the activation free energy of the rate-determining step (rds) and the hydrogen adsorption energy on fourteen metal electrode surfaces. Last, we show that the BEP relation can be utilized to reproduce experimental exchange currents for a wide variety of catalysts with HER activities that vary by 10 orders of magnitude.

Results

Derivation of the Bell-Evans-Polanyi relation

Hydrogen evolution reaction \({{\rm{H}}}^{+}+{e}^{-}\to \frac{1}{2}{{\rm{H}}}_{2}\) is the cathodic reaction in water splitting. In acidic environments, HER takes place through the Volmer–Heyrovsky or Volmer–Tafel pathways,

$$\begin{array}{cc}{{\rm{H}}}_{3}{{\rm{O}}}^{+}+{e}^{-}\leftrightarrow {{\rm{H}}}^{* }+{{\rm{H}}}_{2}{\rm{O}} & ({\rm{Volmer}})\end{array}$$
$$\begin{array}{cc}{{\rm{H}}}_{3}{{\rm{O}}}^{+}+{e}^{-}+{{\rm{H}}}^{* }\leftrightarrow {{\rm{H}}}_{2}{\rm{O}}+{{\rm{H}}}_{2} & ({\rm{Heyrovsky}})\end{array}$$
$$\begin{array}{cc}{{\rm{H}}}^{* }+{{\rm{H}}}^{* }\leftrightarrow {{\rm{H}}}_{2} & ({\rm{Tafel}})\end{array}$$

Similar steps exist in alkaline mediums but will not be discussed further32. The Volmer reaction describes the reduction of hydrogen ions H+ on an electrode surface, whereas the Heyrovsky process describes the reduction of H+ and its interaction with an adsorbed hydrogen atom H* to form hydrogen gas H2 (g). The Tafel reaction is the formation of H2(g) through combining adsorbed hydrogen atoms.

Assuming that \({{\rm{H}}}^{+}+{e}^{-}\to \frac{1}{2}{{\rm{H}}}_{2}\) is primarily determined by the rds, involving proton reduction \({{\rm{H}}}^{+}+{e}^{-}\to {\rm{H}}\)17, we can write the current density as \(j=e[{{C}_{{{\rm{H}}}^{+}}r}_{f}-{{C}_{{\rm{H}}}r}_{b}]\) where rf/b is the rate for the forward/backward redox reaction. \({C}_{{{\rm{H}}}^{+}}\) and CH are the concentration of H+ and H near the electrode surface, respectively. Such description for the overall HER current is justified under the condition that the proton reduction follows a consecutive step with a negligible activation barrier to be either stabilized on the surface or to form H2 molecule for the Volmer or Heyrovsky reaction, respectively. This is confirmed by first-principles modeling of minimum reaction pathways, as will be discussed later on. Using the Butler-Volmer relation for a one-step, one electron transfer process, we write the reaction rates as,

$$\left\{\begin{array}{ll}{r}_{f}={k}_{f}\exp \left[-\alpha f\left(E-{E}^{{0}^{{\prime} }}\right)\right]\\{r}_{b}={k}_{b}\exp \left[\left(1-\alpha \right)f\left(E-{E}^{{0}^{{\prime} }}\right)\right]\end{array},\right.$$
(1)

where α and \({E}^{{0}^{{\prime} }}\) are respectively the transfer coefficient and the formal potential for the redox process of the rds, and E is the applied potential; \({k}_{f/b}\) is the forward/backward reaction rate constant and f = F/RT where F is the Faraday’s constant, R is the gas constant and T is the temperature. Thus, we can express the current density as,

$$j=e\left\{{C}_{{{\rm{H}}}^{+}}{k}_{f}\exp [-\alpha f\left(E-{E}^{{0}^{{\prime} }}\right)]-{C}_{{\rm{H}}}{k}_{b}\exp [\left(1-\alpha \right)f\left(E-{E}^{{0}^{{\prime} }}\right)]\right\}.$$
(2)

At the overall HER equilibrium potential Eeq where the forward current density is the same as the backward current density, the exchange current density is defined as,

$${j}_{0}=e{C}_{{{\rm{H}}}^{+}}{k}_{0}\exp [-\alpha f({E}_{{eq}}-{E}^{{0}^{{\prime} }})]=e{C}_{{\rm{H}}}{k}_{0}\exp [(1-\alpha)f({E}_{{eq}}-{E}^{{0}^{{\prime} }})].$$
(3)

Here, k0 is the standard rate constant defined under the condition that the forward rf or the backward rb rates are the same at Eeq, i.e., k0 = kf = kb33. Under the assumption that all the elementary steps are at equilibrium if the overall reaction is at equilibrium, as it has been proven theoretically33, we obtain the following (see SI for the detailed derivation):

$${E}_{{eq}}-{E}^{{0}^{{\prime} }}=\left\{\begin{array}{cc}{\space\space\space}\Delta {G}_{{\rm{H}}}/{|e|} & ({\rm{Volmer}})\\ -\Delta {G}_{{\rm{H}}}/{|e|} & ({\rm{Heyrovsky}})\end{array}\right.$$
(4)

Here, ΔGH is defined as the free energy difference between the hydrogen at the adsorbed and gas states. It can be quantified from density functional theory (DFT) calculations, as shown in Eq. (9).

The critical step in deriving the BEP relation is to utilize the absolute form of the standard rate constant \({k}_{0}=\frac{{k}_{B}T}{h}\exp (-\varDelta {G}_{0}^{+}/{k}_{B}T)\), which has long been considered empirical using a data-driven approach based on experimental data21. This form is in agreement with the Eyring’s interpretation where the pre-factor of a chemical reaction is mainly \(\frac{{k}_{B}T}{h}\), and the reaction rate is dominated by the exponential of the energy difference between the transition and initial state34,35,36. In our previous studies, we find that \(\varDelta {G}_{0}^{+}=0.7\) eV is universal, and is the overall HER activation barrier for the optimum catalysts with \(\Delta {G}_{{\rm{H}}} = 0\)37,38. Similar values are found for the activation energy barriers of the Volmer, Heyrovsky and Tafel reactions on Pt (111) surface using DFT calculations38. Further, this absolute form of k0 is verified by applying Eq. (3) to experimental exchange currents for metal surfaces37,39. Using the absolute form of k0 and applying Eq. (4) into Eq. (3), we can re-express the first term of Eq. (3) as,

$${j}_{0}=\left\{\begin{array}{cc}e{C}_{{{\rm{H}}}^{+}}\frac{{k}_{B}T}{h}\exp \left(-\frac{\varDelta {G}_{0}^{+}+\alpha \Delta {G}_{{\rm{H}}}}{{k}_{B}T}\right) & ({\rm{Volmer}})\\ e{C}_{{{\rm{H}}}^{+}}\frac{{k}_{B}T}{h}\exp \left(-\frac{\varDelta {G}_{0}^{+}-\alpha \Delta {G}_{{\rm{H}}}}{{k}_{B}T}\right) & ({\rm{Heyrovsky}})\end{array}\right.$$
(5)

Equation (5) has the Arrhenius form for an activation process,

$${j}_{0}=e{C}_{{{\rm{H}}}^{+}}\frac{{k}_{B}T}{h}\exp (-\Delta {G}_{{tot}}^{+}/{k}_{B}T),$$
(6)

provided that we define an “acting” or effective activation energy \(\Delta {G}_{{tot}}^{+}\) as,

$$\Delta {G}_{{tot}}^{+}=\left\{\begin{array}{cc}\Delta {G}_{0}^{+}+\alpha \Delta {G}_{{\rm{H}}} & ({\rm{Volmer}})\\ \Delta {G}_{0}^{+}-\alpha \Delta {G}_{{\rm{H}}} & ({\rm{Heyrovsky}})\end{array}\right.$$
(7)

Based on fundamental electrochemistry principles and the transition state theory, the derived Eq. (7) provides a universal and direct link between the thermodynamics and kinetics of HER. Importantly, \(\Delta {G}_{{tot}}^{+}\) can be quantified solely using ΔGH that can be readily accessed from DFT27. Hence, we posit that Eq. (7) explains the BEP relationship that will be further verified and elucidated using DFT, as detailed below.

While the physical interpretation of \(\Delta {G}_{{tot}}^{+}\) is not clear from the derivation, we posit that \(\Delta {G}_{{tot}}^{+}\) is the activation free energy of the proton reduction of the rds. To show this is the case, we investigate the HER kinetics and thermodynamics using first-principles calculations on fourteen different metal surfaces including metals with moderate (Pt, Ir, Pd and Rh), strong (Re, Ru, Co and Ni) and light (Cu, Au, Ag, In, Cd and Bi) interactions with hydrogen. These metals are of high interest for materials design, and their electrochemical properties for catalyzing HER have been thoroughly studied37,39,40. We employ the Helmholtz double-layer model constructed using a water monolayer layer, with 1/8 proton concentration, placed ~3 Å above the metal surface41. For the metals with \({\rm{Fm}}\bar{3}{\rm{m}}\) and P63/mmc symmetry42,43, we use (111) and (001) termination, respectively; for Bi and In, we use (111) termination44. In the water monolayer, each H3O+ molecule is bonded with three H2O molecules in the Eigen form45. Similar to what was done for Pt (111)46, the water molecules are arranged in an ice-like configuration, involving half of H2O molecules with their O-H bonds perpendicular to the surface, while the other half are parallel to the surface, referenced to their molecular plane.

BEP validation

Using ΔGH obtained from Eq. (9), we compute \(\Delta {G}_{{tot}}^{+}\) from Eq. (7) with the universal parameters \(\Delta {G}_{0}^{+}=\) 0.7 eV and α = 0.5. As shown in Fig. 1a, \(\Delta {G}_{{tot}}^{+}\,\approx\, \Delta {G}_{{\rm{DFT}}}^{+}\) within ±0.2 eV uncertainties, strongly supporting that the acting \(\Delta {G}_{{tot}}^{+}\) can be interpreted as the activation free energy of the rds for the overall HER reaction. We further show that \(\Delta {G}_{{tot}}^{+}\,\approx\, \Delta {G}_{\exp }^{+}\) where \(\Delta {G}_{\exp }^{+}\) is computed from Eq. (7) with the values of α= 0.52 ± 0.15 and \(\Delta {G}_{0}^{+}=\)0.69 ± 0.10 eV obtained from analyzing the experimental cyclic voltammograms of the fourteen metals using Eq. (2)40. The finding \(\Delta {G}_{{tot}}^{+}\,\approx\, \Delta {G}_{{\rm{DFT}}}^{+}\,\approx\, \Delta {G}_{\exp }^{+}\) suggests the self-consistency of the BEP relationship of Eq. (7), and can further be confirmed by establishing statistical significance. In Fig. 1b, we individually fit the two trends from Fig. 1a with the BEP relation of Eq. (7) assuming that α and \(\Delta {G}_{0}^{+}\) as fitting parameters. The fitted α values are found 0.55 and 0.53, and the fitted \(\Delta {G}_{0}^{+}\) is 0.66 for both trends. The high coherency of the coefficient of determination r2 values (0.88 and 0.87) confirms our previous findings of the absolute reaction rate as well as showing that α≈ 0.5 can be treated as the universal value for all surfaces37,39,40. Importantly, we have confirmed our assumption that \(\Delta {G}_{{tot}}^{+}\,\approx\, \Delta {G}_{{\rm{DFT}}}^{+}\) corresponds to the activation free energy of the proton transfer of the rds that can be obtained from first-principles modeling.

Fig. 1: Correlation between the activation energy and the hydrogen adsorption free energy.
figure 1

a The trend of \(\Delta {G}_{{tot}}^{+}\) with the computed ΔGH. \(\Delta {G}_{{\rm{DFT}}}^{+}\) is computed from first principles modeling (red scatters) and \(\Delta {G}_{\exp }^{+}\) is calculated from Eq. (7) with the α and \(\Delta {G}_{0}^{+}\) values obtained from experimental cyclic voltammograms (gray). The black line is computed using Eq. (7) with the universal values of \(\Delta {G}_{0}^{+}=\) 0.7 eV and α = 0.5, and the black dashed lines are the boundaries of ±0.1 eV difference of \(\Delta {G}_{{tot}}^{+}\). b The fittings of \(\Delta {G}_{{\rm{DFT}}}^{+}\) and \(\Delta {G}_{\exp }^{+}\) with the BEP relation of Eq. (7). For the DFT and the experimental sets, the r2 values are 0.88 and 0.87, respectively.

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Discussion

Figure 1a shows the linear trend where \(\Delta {G}_{{\rm{DFT}}}^{+}\) increases with increasing or decreasing ΔGH for the cases of the Volmer (ΔGH > 0) or the Heyrovsky (ΔGH < 0) reaction. Previous findings of the linear trends between ΔGH and the activation free energies are qualitatively in line with our findings while the activation free energies are defined differently. For example, Cheng et al. used an implicit solvation model47 to study the activation free energy defined at 0.5 vs. NHE for hollow or atop sites28, which is different from our computed \(\Delta {G}_{{\rm{DFT}}}^{+}\) that is determined at 0 vs. NHE. Also, using NEB with the potential correction scheme48, Tang et al. referenced the activation free energy to the protons in bulk solution29, which is also different from \(\Delta {G}_{{\rm{DFT}}}^{+}\) that is referenced to the protons near electrode surface. Different from these previous studies where the linear trends are built with fitting parameters with no connection to theory, our theoretical BEP relation of Eq. (7), that follows naturally from fundamental electrochemical principles, is established with the well-defined ΔGH, \(\Delta {G}_{0}^{+}\) and the electrochemical quantity α.

The BEP relation of Eq. (7) can be applied to quantify the exchange current j0 for catalysts solely from computing ΔGH in conjunction with Eq. (6). For the Volmer reaction, we approximate \({C}_{{{\rm{H}}}^{+}}={C}_{{tot}}\left(1-\theta \right)\) as H+ is closely in contact with the empty stable sites for adsorption process, where Ctot is the areal concentration of total active sites and θ is the hydrogen adsorption fraction. For the Heyrovsky reaction, we approximate \({C}_{{{\rm{H}}}^{+}}={C}_{{tot}}\theta\) as H2 formation must associate with an adsorbed hydrogen atom. This is based on the assumption that there is no mass transfer effects, as discussed in SI. Thus, we can rewrite Eq. (6) as,

$${j}_{0}=\left\{\begin{array}{ll}e{C}_{{tot}}(1-\theta )\frac{{k}_{B}T}{h}\exp (-\Delta {G}_{{tot}}^{+}/{k}_{B}T) & ({\rm{Volmer}})\\ e{C}_{{tot}}\theta \frac{{k}_{B}T}{h}\exp (-\Delta {G}_{{tot}}^{+}/{k}_{B}T) & ({\rm{Heyrovsky}})\end{array}\right.$$
(8)

where \(\Delta {G}_{{tot}}^{+}\) follows Eq. (7) with the universal parameters \(\Delta {G}_{0}^{+} =\) 0.7 eV and α = 0.5. Further, we use the Langmuir isotherm \(\theta =K/(1+K)\) with the equilibrium constant \(K=\exp (-\Delta {G}_{{\rm{H}}}/{k}_{B}T)\), which has been confirmed to be a suitable approximation on metal surfaces based on ab initio thermodynamics (see SI). Note that the heterogeneous nature of the adsorption sites on realistic electrodes violates the Langmuir isotherm that describes the homogeneous adsorption events. However, nevertheless, this model has been shown to provide a reasonable approximation for experimental exchange currents27,37,39.

Figure 2 compares the experimental j0 and the j0 calculated using Eq. (8) for fourteen metal surfaces. As seen from the figure, the differences between the experimental and calculated j0 are within two orders of magnitude. Further, we posit that the BEP relation is general and is not only limited to pure metal surfaces. This is supported by examining a variety of catalysts with different structures and activities. Namely, (1) Pt/Cu surface alloys and near-surface alloys49, (2) pristine, Ti- and Ir-doped β-Mo2C50, (3) MoS2 nanoparticles supported on Au (111) with different loadings51, (4) two dimensional metal free catalysts such as nitrogen doped graphene (NG), g-C3N4, and C3N4/NG hybrid52. These studies are selected based on: (1) the reported ΔGH corresponds to the active sites that dominantly contribute to the overall HER, and (2) the concentration of the active sites \({C}_{{tot}}\approx {N}_{{\rm{site}}}/{A}_{{\rm{surf}}}\) can be estimated by the reported information, where Nsite is the number of active sites and Asurf is the surface area. For 3-dimentional electrodes such as metals and β-Mo2C, we approximate Nsite using the most stable termination with the active sites that are determined using ab initio thermodynamics53. For N-G, we consider the reported doping percentage of 6.2% for determining the number of active sites shown to be the atop sites near the doped N atoms52. For C3N4@NG and G-C3N4, Nsite is assumed to be the number of the bridge sites of N atoms52. The Nsite for MoS2 is calculated based on the reported densities of active Mo-edge sites51. As shown in the figure, the compute j0 values are in line with the experimental values for all the investigated electrodes despite the differences of 10 orders of magnitude in the exchange currents. This suggests that one can easily screen the experimental j0 with high fidelity using the easy-to-compute ΔGH without conducting the high-cost investigations of reaction kinetic under electrochemical conditions.

Fig. 2: Correlation between the calculated and the experimental exchange current density.
figure 2

Comparison of the experimental j0 and the calculated j0 using Eq. (8) with the universal values \({\Delta G}_{0}^{+}=\) 0.7 eV and α = 0.5. The solid line shows parity and the dashed lines are the boundaries of the difference within two orders of magnitude.

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In summary, we derive the BEP relation of the hydrogen evolution reaction – the linear trend between the activation energy and the reaction energy. From the Butler-Volmer (BV) relation - the empirical equation that explains the relation between electrochemical current and potential using the fundamentals of electrochemistry, we identify that the activation energy of the electrochemical current \(\Delta {G}_{{tot}}^{+}\) is linearly proportional to, and solely a function of the DFT-based hydrogen adsorption free energy ΔGH. This is done by introducing the absolute form of the standard rate constant into the BV equation and describing the electrochemical potential using ΔGH. To further confirm the BEP relation, we employ first-principles modeling to compute the kinetic barriers of the minimum reaction pathways in conjunction with a charge transfer correction scheme to quantify \(\Delta {G}_{{tot}}^{+}\). We show that the \(\Delta {G}_{{tot}}^{+}\) values computed using first principles are in agreement with the values obtained from experimental cyclic voltammograms. Last, we show that the BEP relation can be utilized to compute the exchange current with high fidelity solely using ΔGH for a variety of electrodes. Our framework of building the theoretical BEP relation is general and is applicable for other reactions.

Methods

Computational hydrogen adsorption free energy

Employing the Helmholtz double-layer models, we compute the hydrogen adsorption free energy as,

$$\varDelta {G}_{{\rm{H}}}=\varDelta {E}_{{\rm{ads}}}+\varDelta {E}_{{\rm{ZPE}}}-T\varDelta S,$$
(9)

where \(\varDelta {E}_{{\rm{ZPE}}}\) is the zero-point energy difference between hydrogen in adsorbed state H and gas state \({{\rm{H}}}_{2}({\rm{g}})\), and is approximately 0.04 eV27. At T= 298 K, we use experimental results to estimate the entropic contribution TΔS ≈ −0.2 eV, as done before54. The adsorption energy is defined as \(\Delta {E}_{{\rm{ads}}}=\frac{1}{{\rm{n}}}({E}_{n{\rm{H}}/{\rm{slab}}}-{E}_{{\rm{slab}}}-0.5n{E}_{{{\rm{H}}}_{2}({\rm{g}})})+\varDelta {E}_{{{\rm{H}}}_{2}{\rm{O}}}\) where \({E}_{n{\rm{H}}/{\rm{slab}}}\) and Eslab are respectively the DFT energies of the surface with n adsorbed hydrogen and of the empty slab, and \({E}_{{{\rm{H}}}_{2}({\rm{g}})}\) is the energy of H2 molecule in the gas phase. The solvation energy \(\varDelta {E}_{{{\rm{H}}}_{2}{\rm{O}}}\) due to surface-water interactions are found to be less than 0.1 eV. The reaction that is rate limiting can be apriori screened by examining ΔGH: the rds is the Volmer or the Heyrovsky reaction for ΔGH> 0 or ΔGH< 0, respectively. Such justification was first confirmed by a first-principles study of the reaction pathways on metal surfaces29. Also, this is consistent with our previous studies on the exchange current model based on the theoretical Butler-Volmer relation37,39.

Computational activation energy

We employ the nudged elastic band theory (NEB)55 to search for the minimum reaction pathways of the rds. From a careful security, we investigate three different routes for each of the rds, as shown schematically in Fig. 3. We note that these reaction paths were not fully investigated in previous studies, see e.g., refs. 38,56 As shown in Fig. 3, all the reaction routes commence by transferring a hydrogen atom, referred as Htran, from water layer to surface, but they differ by the nature of the active site(s) associated with charge transfer. For the Volmer reaction, the three potential routes correspond to,

  1. I.

    Htran directly adsorbs at the most stable site (V-direct).

  2. II.

    Htran adsorbs at the most stable site along a tilted path after slightly bonding to a surface metal atom (V-assist).

  3. III.

    Htran initially adsorbs at an unstable site (the top site of a surface atom, i.e., the atop site), then diffuses to the most stable adsorption site (V-diffuse).

Fig. 3: The routes for the elementary steps of the hydrogen evolution reaction.
figure 3

Schematic of potential routes for the (a) Volmer reaction and (b) Heyrovsky reaction. The oxygen and hydrogen atoms are shown in red and yellow, respectively. The surface hydrogen atoms on the stable/non-stable sites are shown in gray/yellow. The electrode surface metal atoms are shown in black. The stretched O-H bonds are represented by black dashed lines, and the red dashed lines represent the interaction of a hydrogen atom with a surface metal atom.

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For the Heyrovsky reaction, the three routes are,

  1. I.

    Htran directly interacts with H* to form H2 without interacting with a surface site (H-direct).

  2. II.

    Htran combines with an H* to form H2 after slightly bonding with a surface metal atom at an unstable site (atop site) (H-assist).

  3. III.

    Htran directly combines with an H* that diffuses from the most stable site to an unstable site (atop site) without interacting with a surface site (H-diffuse).

As shown in Supplementary Table 1, the favorable routes for the Volmer or Heyrovsky reactions differ across various metal surfaces. For example, Ru, Ir and Pt follow H-direct, H-assist and H-diffuse routes for the Heyrovsky reaction, respectively. From these investigations, we obtain the barrier of the minimum reaction pathways \(\Delta {E}_{{\rm{DFT}}}^{+}\) with an initial state where the protons are near the electrode surface. The initial state is distinguished from another study where it is referenced with respect to the protons in bulk solution29. We further correct \(\Delta {E}_{{\rm{DFT}}}^{+}\) by \(\Delta q\left[\frac{1}{2}\Delta \phi -({\phi }_{{\rm{SHE}}}-{\phi }_{I})\right]\) following an extrapolation scheme48 to account for the varying potential along the reaction pathways as imposed by the periodic supercell model. Here, \(\Delta \phi ={\phi }_{T}-{\phi }_{I}\) is the change of the work function from the initial (I) to transition state (T), and \(\Delta q={q}_{T}-{q}_{I}\) is the difference in excess surface charge between the two states, respectively. The excess surface charge \(q={q}_{{\rm{water}}}-{q}_{{\rm{surf}}}\) is determined using the Bader charge analysis48, where qwater is the total charge of an isolated water layer and \({q}_{{\rm{water}}/{\rm{surf}}}\) is the total surface charge of the water layer in contact with the surface. To extrapolate \(\Delta {E}_{{\rm{DFT}}}^{+}\) to the standard hydrogen electrode \({U}_{{\rm{SHE}}}=(\varPhi -{\varPhi }_{{\rm{SHE}}})/e\), we use \({\varPhi }_{{\rm{SHE}}}=4.44\,{\rm{eV}}\) that is measured experimentally57. Using the “corrected” activation energy, we define the DFT activation free energy \(\Delta {G}_{{\rm{DFT}}}^{+}\) as,

$$\Delta {G}_{{\rm{DFT}}}^{+}=\Delta {E}_{{\rm{DFT}}}^{+}+\Delta q\left[\frac{1}{2}\Delta \phi -({\phi }_{{\rm{SHE}}}-{\phi }_{I})\right]+\Delta {S}^{+}.$$
(10)

Rossmeisl and collaborators posit that the main source of the activation entropy ΔS+ originates from the proton transfer across the Helmholtz layer plane, and not from any of the three HER elementary steps that take place at the electrode surface. Thus, ΔS+ is negligible in a low pH environment58, as is the case in our study. Further, the pre-exponential factor for hydrogen recombination reaction \(\frac{{k}_{B}T}{h}\exp (\frac{T\Delta {S}^{+}}{{k}_{B}T})\) is reported to be ~1013 on several transition metal surfaces59 suggesting also a nearly zero value for ΔS+.