Abstract
The versatile BellEvansPolanyi (BEP) relation stipulates the kinetics of a reaction in terms of thermodynamics. Herein, we establish the BEP relation for the hydrogen evolution reaction (HER) from fundamental electrochemical principles leveraging the ButlerVolmer relation for a onestep, oneelectron process and the transition state theory. Based on firstprinciples investigations of HER mechanisms on fourteen metal electrodes, we firmly justify the BEP relation solely using an easyto compute hydrogen adsorption free energy and universal electrochemical constants.
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Introduction
Descriptorbased approaches carry enormous predictive power in materials design as these can be rapidly deployed to identify materials with desired properties^{1,2,3,4,5,6,7,8}. The empirical Bell (Brønsted)EvansPolanyi (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)EvansPolanyi 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 easytocompute or easilyaccessible thermodynamic data^{15,16,17,18,19,20}. In the past two decades, BEP relationships have been widely observed for different reactions such as the dissociation of diatomic and triatomic molecules^{21,22,23,24}.
Hydrogen evolution reaction (HER) is the critical step in electrolysis to produce green hydrogen, which is considered a nextgeneration clean energy carrier^{25,26}. Leveraging the BEP relation as the key ingredient for computational screening of HER catalysts, Nørskov et al. proposed the hydrogen adsorption free energy ΔG_{H} as the descriptor for the volcano trend – the logarithm of the exchange current is linearly increasing or decreasing with ΔG_{H}^{27}. 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 firstprinciples computational studies have shown that the BEP relation holds for HER on metal surfaces^{28,29}. Experimentally, similar trends were also shown to be valid by studying the cyclic voltammograms on metal surfaces under varying electrochemical conditions^{30,31}; 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 ButlerVolmer relation – the norm of the electrochemical theory that describes currentpotential 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 easytocompute hydrogen adsorption free energy ΔG_{H} and universal electrochemical constants. Further, we verify the BEP relation using firstprinciples calculations by finding a direct correlation between the activation free energy of the ratedetermining 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 BellEvansPolanyi 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,
Similar steps exist in alkaline mediums but will not be discussed further^{32}. 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 H_{2} (g). The Tafel reaction is the formation of H_{2}(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 r_{f/b} is the rate for the forward/backward redox reaction. \({C}_{{{\rm{H}}}^{+}}\) and C_{H} 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 H_{2} molecule for the Volmer or Heyrovsky reaction, respectively. This is confirmed by firstprinciples modeling of minimum reaction pathways, as will be discussed later on. Using the ButlerVolmer relation for a onestep, one electron transfer process, we write the reaction rates as,
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,
At the overall HER equilibrium potential E_{eq} where the forward current density is the same as the backward current density, the exchange current density is defined as,
Here, k_{0} is the standard rate constant defined under the condition that the forward r_{f} or the backward r_{b} rates are the same at E_{eq}, i.e., k_{0} = k_{f} = k_{b}^{33}. Under the assumption that all the elementary steps are at equilibrium if the overall reaction is at equilibrium, as it has been proven theoretically^{33}, we obtain the following (see SI for the detailed derivation):
Here, ΔG_{H} 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 datadriven approach based on experimental data^{21}. This form is in agreement with the Eyring’s interpretation where the prefactor 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 state^{34,35,36}. 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 calculations^{38}. Further, this absolute form of k_{0} is verified by applying Eq. (3) to experimental exchange currents for metal surfaces^{37,39}. Using the absolute form of k_{0} and applying Eq. (4) into Eq. (3), we can reexpress the first term of Eq. (3) as,
Equation (5) has the Arrhenius form for an activation process,
provided that we define an “acting” or effective activation energy \(\Delta {G}_{{tot}}^{+}\) as,
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 ΔG_{H} that can be readily accessed from DFT^{27}. 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 firstprinciples 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 studied^{37,39,40}. We employ the Helmholtz doublelayer model constructed using a water monolayer layer, with 1/8 proton concentration, placed ~3 Å above the metal surface^{41}. For the metals with \({\rm{Fm}}\bar{3}{\rm{m}}\) and P63/mmc symmetry^{42,43}, we use (111) and (001) termination, respectively; for Bi and In, we use (111) termination^{44}. In the water monolayer, each H_{3}O^{+} molecule is bonded with three H_{2}O molecules in the Eigen form^{45}. Similar to what was done for Pt (111)^{46}, the water molecules are arranged in an icelike configuration, involving half of H_{2}O molecules with their OH bonds perpendicular to the surface, while the other half are parallel to the surface, referenced to their molecular plane.
BEP validation
Using ΔG_{H} 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 selfconsistency 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 r^{2} 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 surfaces^{37,39,40}. 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 firstprinciples modeling.
Discussion
Figure 1a shows the linear trend where \(\Delta {G}_{{\rm{DFT}}}^{+}\) increases with increasing or decreasing ΔG_{H} for the cases of the Volmer (ΔG_{H} > 0) or the Heyrovsky (ΔG_{H} < 0) reaction. Previous findings of the linear trends between ΔG_{H} 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 model^{47} to study the activation free energy defined at 0.5 vs. NHE for hollow or atop sites^{28}, which is different from our computed \(\Delta {G}_{{\rm{DFT}}}^{+}\) that is determined at 0 vs. NHE. Also, using NEB with the potential correction scheme^{48}, Tang et al. referenced the activation free energy to the protons in bulk solution^{29}, 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 welldefined ΔG_{H}, \(\Delta {G}_{0}^{+}\) and the electrochemical quantity α.
The BEP relation of Eq. (7) can be applied to quantify the exchange current j_{0} for catalysts solely from computing ΔG_{H} 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 C_{tot} 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 H_{2} 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,
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 currents^{27,37,39}.
Figure 2 compares the experimental j_{0} and the j_{0} calculated using Eq. (8) for fourteen metal surfaces. As seen from the figure, the differences between the experimental and calculated j_{0} 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 nearsurface alloys^{49}, (2) pristine, Ti and Irdoped βMo_{2}C^{50}, (3) MoS_{2} nanoparticles supported on Au (111) with different loadings^{51}, (4) two dimensional metal free catalysts such as nitrogen doped graphene (NG), gC_{3}N_{4}, and C_{3}N_{4}/NG hybrid^{52}. These studies are selected based on: (1) the reported ΔG_{H} 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 N_{site} is the number of active sites and A_{surf} is the surface area. For 3dimentional electrodes such as metals and βMo_{2}C, we approximate N_{site} using the most stable termination with the active sites that are determined using ab initio thermodynamics^{53}. For NG, 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 atoms^{52}. For C_{3}N_{4}@NG and GC_{3}N_{4}, N_{site} is assumed to be the number of the bridge sites of N atoms^{52}. The N_{site} for MoS_{2} is calculated based on the reported densities of active Moedge sites^{51}. As shown in the figure, the compute j_{0} 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 j_{0} with high fidelity using the easytocompute ΔG_{H} without conducting the highcost investigations of reaction kinetic under electrochemical conditions.
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 ButlerVolmer (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 DFTbased hydrogen adsorption free energy ΔG_{H}. This is done by introducing the absolute form of the standard rate constant into the BV equation and describing the electrochemical potential using ΔG_{H}. To further confirm the BEP relation, we employ firstprinciples 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 ΔG_{H} 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 doublelayer models, we compute the hydrogen adsorption free energy as,
where \(\varDelta {E}_{{\rm{ZPE}}}\) is the zeropoint energy difference between hydrogen in adsorbed state H and gas state \({{\rm{H}}}_{2}({\rm{g}})\), and is approximately 0.04 eV^{27}. At T= 298 K, we use experimental results to estimate the entropic contribution TΔS ≈ −0.2 eV, as done before^{54}. 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 E_{slab} 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 H_{2} molecule in the gas phase. The solvation energy \(\varDelta {E}_{{{\rm{H}}}_{2}{\rm{O}}}\) due to surfacewater interactions are found to be less than 0.1 eV. The reaction that is rate limiting can be apriori screened by examining ΔG_{H}: the rds is the Volmer or the Heyrovsky reaction for ΔG_{H}> 0 or ΔG_{H}< 0, respectively. Such justification was first confirmed by a firstprinciples study of the reaction pathways on metal surfaces^{29}. Also, this is consistent with our previous studies on the exchange current model based on the theoretical ButlerVolmer relation^{37,39}.
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 H_{tran}, 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,

I.
H_{tran} directly adsorbs at the most stable site (Vdirect).

II.
H_{tran} adsorbs at the most stable site along a tilted path after slightly bonding to a surface metal atom (Vassist).

III.
H_{tran} 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 (Vdiffuse).
For the Heyrovsky reaction, the three routes are,

I.
H_{tran} directly interacts with H* to form H_{2} without interacting with a surface site (Hdirect).

II.
H_{tran} combines with an H* to form H_{2} after slightly bonding with a surface metal atom at an unstable site (atop site) (Hassist).

III.
H_{tran} directly combines with an H* that diffuses from the most stable site to an unstable site (atop site) without interacting with a surface site (Hdiffuse).
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 Hdirect, Hassist and Hdiffuse 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 solution^{29}. 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 scheme^{48} 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 analysis^{48}, where q_{water} 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 experimentally^{57}. Using the “corrected” activation energy, we define the DFT activation free energy \(\Delta {G}_{{\rm{DFT}}}^{+}\) as,
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 environment^{58}, as is the case in our study. Further, the preexponential factor for hydrogen recombination reaction \(\frac{{k}_{B}T}{h}\exp (\frac{T\Delta {S}^{+}}{{k}_{B}T})\) is reported to be ~10^{13} on several transition metal surfaces^{59} suggesting also a nearly zero value for ΔS^{+}.
Data availability
The structures obtained from the simulations will be shared with community upon request.
Code availability
The VASP calculation inputs and outputs will be shared with community upon request.
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Acknowledgements
We acknowledge partial financial support from the National Science Foundation (Award No. CBET2130804). Computational support was provided in part by the University of Pittsburgh Center for Research Computing through the resources provided on the H2P cluster, which is supported by NSF (Award No. OAC2117681).
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T.Y. and W.A.S. conceived the project. T.Y carried out the DFT calculations and analyzed the results. T.Y. wrote the manuscript under the direction of W.A.S.
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Yang, T.T., Saidi, W.A. The BellEvansPolanyi relation for hydrogen evolution reaction from firstprinciples. npj Comput Mater 10, 98 (2024). https://doi.org/10.1038/s41524024012443
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DOI: https://doi.org/10.1038/s41524024012443
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