# Machine Learning Potential for Electrochemical Interfaces with Hybrid
Representation of Dielectric Response

Jia-Xin Zhu<EMAIL_ADDRESS>[  Jun Cheng<EMAIL_ADDRESS>[

###### Abstract

Understanding electrochemical interfaces at a microscopic level is essential
for elucidating important electrochemical processes in electrocatalysis,
batteries and corrosion. While ab initio simulations have provided valuable
insights into model systems, the high computational cost limits their use in
tackling complex systems of relevance to practical applications. Machine
learning potentials offer a solution, but their application in
electrochemistry remains challenging due to the difficulty in treating the
dielectric response of electronic conductors and insulators simultaneously. In
this work, we propose a hybrid framework of machine learning potentials that
is capable of simulating metal/electrolyte interfaces by unifying the
interfacial dielectric response accounting for local electronic polarisation
in electrolytes and non-local charge transfer in metal electrodes. We validate
our method by reproducing the bell-shaped differential Helmholtz capacitance
at the Pt(111)/electrolyte interface. Furthermore, we apply the machine
learning potential to calculate the dielectric profile at the interface,
providing new insights into electronic polarisation effects. Our work lays the
foundation for atomistic modelling of complex, realistic electrochemical
interfaces using machine learning potential at ab initio accuracy.

Xiamen University] State Key Laboratory of Physical Chemistry of Solid
Surfaces, iChEM, College of Chemistry and Chemical Engineering, Xiamen
University, Xiamen 361005, China Xiamen University] State Key Laboratory of
Physical Chemistry of Solid Surfaces, iChEM, College of Chemistry and Chemical
Engineering, Xiamen University, Xiamen 361005, China [IKKEM] Laboratory of AI
for Electrochemistry (AI4EC), IKKEM, Xiamen 361005, China [Xiamen University]
Institute of Artificial Intelligence, Xiamen University, Xiamen 361005, China

##  1 Introduction

In modern society, electrochemistry plays an increasingly important role in
many areas including material synthesis 1, 2, energy conversion 3, 4, and
energy storage 5. However, our understanding of the structures and activity of
electrochemical interfaces, where the electrochemical reactions happen, is far
short of expectation 6. Even for platinum, one of the most important catalysts
in electrochemistry 7, 8, 9, the understanding of how the interfacial
structures vary with electrode potentials and electrolyte compositions is
still lacking, although it has been widely observed and currently under active
investigations that the interfacial structures can be tuned by protons, ions
and additives, thus affecting the activity. 10, 11, 12, 13, 14. Indeed, the
electrochemical characterisations, such as the cyclic voltammograms 15, 16, 17
and the electrochemical impedance spectroscopy 18, 19, provide initial but
valuable insights into interfacial electrochemistry. Development and
application of the in-situ spectroscopic 20, 21, 22 and scanning probe
microscopic 23, 24 techniques make it possible to investigate the microscopic
structures of interfaces. On the other hand, computation and simulation
complement experimental measurements in recent decades 3, 10, 25, helping
interpret the spectra and elucidate the interfacial processes at the atomic
level.

Although atomistic modelling can offer detailed microscopic information on
electrochemical interfaces, it often faces a dilemma between accuracy and
efficiency. For example, the hydrogen bonding network has been proposed to
play an important role in the thermodynamics and the kinetics of the
interfacial processes 26, 27, 28, 29. While the atomic-level picture of the
hydrogen bonding network is difficult to probe in experiment, it can be
obtained from molecular dynamics (MD) simulations 30, 27. Notably, electronic
structures should be included in simulation to describe the potential-
dependent hydrogen bonding network accurately in many important systems. For
instance, it has been shown that water chemisorption on Pt can induce electron
redistribution at the interface and thus have important impacts on interfacial
water structures and the differential capacitances of the electric double
layer (EDL) 10. Ab initio molecular dynamics (AIMD), accounting for both
electronic structures and molecular dynamics, can in principle provide
accurate descriptions of electrochemical interfaces. However, its high
computational cost limits its applications to systems of hundreds of atoms at
the time scale of tens of picoseconds, which is insufficient for complex
electrochemical interfaces at realistic conditions.

Fortunately, emerging machine learning (ML) techniques provide us with
opportunities to achieve a better compromise between accuracy and efficiency
31, 32, 33, 34. While maintaining the ab initio accuracy, the calculations
with machine learning potentials (MLPs) are not only $10^{3}-10^{5}$ times
faster than those with the density functional theory (DFT) but show nearly
linear scaling between computational cost and system size 35, 36. As a result,
slow processes in systems with complex structures become accessible by
simulations with MLPs 37, 38, 39.

Despite the promising progress that has been made, the application of the MLPs
in electrochemistry is non-trivial. Currently, the most widely used MLPs are
short-ranged due to the finite cutoff distance (i.e., typically smaller than
$8\text{\,}\mathrm{\SIUnitSymbolAngstrom}$) used to construct the descriptors
40, 41, 33. Such short-range MLPs, as shown in previous work 42, 43, are
capable of describing the structures at the neutral metal/water interfaces,
probably due to the large dielectric constant of the water at the zero
electric field limit 44, 45. The lack of explicit long-range interactions in
these MLPs, however, has been shown to lead to breaking of electroneutrality
at the bulk electrolyte region when ions are present in the system 46.
Therefore, the short-range MLPs would fail in modelling the electrochemical
interfaces under the potential of zero charge (pzc) conditions, let alone the
EDL under a potential bias. Strategies to include long-range interactions have
been proposed in some ML models in literature 47, 48, 49, 50. One of the
commonly used recipes is to separate the total energy into the short- and
long-range interactions. While the long-range interaction is approximated with
the electrostatic interaction and can be calculated analytically, the
remaining short-range interaction can be predicted via the short-range MLPs
47, 50, 49. Instead of separating the short-range and the long-range
interactions, it is also possible to include the long-range interaction by
embedding the long-range information into the descriptors 48.

Although the long-range interactions have been included in some existing ML
models, there is another challenge in simulating the electrochemical interface
with MLPs, i.e., properly describing the dielectric response of the system.
This means that the MLPs should be able to describe the interfacial
polarisation at given atomic configurations and (electrostatic) boundary
conditions. Recently, many efforts have been made to develop the MLPs with
dielectric response 49, 50, 51, 52, 53, 54, 55. For example, some MLPs
accounting for electronic polarisation based on local chemical environments49,
51, 53, 54, 55, are suitable for describing the dielectric response of
insulators, but fail in treating the non-local charge transfer in conductors
due to the short-sightedness of the descriptors. In order to overcome the
shortcoming of short-sightedness, other forms of MLPs have been proposed on
the basis of the charge equilibrium (QEq) scheme 56, 50, 57, 50, allowing for
the non-local charge transfer. However, these MLPs inherit the intrinsic
limitations of the QEq method originating from the semi-local approximations
of kinetic energies. As a consequence, not only would the QEq-based model
overestimate the polarisability of insulators, but also incorrectly predict
fractional rather than integer charged fragments when a molecule dissociates,
which limits its applicability to isolated molecules or clusters 58, 59, 60.

In this work, we propose a hybrid scheme of MLPs to treat electrochemical
interfaces (ec-MLP), which integrates the dielectric responses of
electronically insulating electrolytes and conducting metal electrodes. In
particular, the dielectric response of electrolytes can be obtained by the
shift of the positions of the Wannier centroids, which is defined as the
average positions of the maximally localised Wannier centres (MLWCs) and
predicted via a Deep Wannier (DW) deep neural network model based on the local
chemical environment49. Meanwhile, the dielectric response of the metal
electrode can be taken into account via the Siepmann–Sprik model61. With this
method, the atomic charges of the electrode can be solved by minimising the
electrostatic energy under the given atomic configurations and the boundary
condition. Knowing the charge distributions of both electrolytes and
electrodes, the long-range electrostatic energy can be calculated analytically
and extracted from the total energy. The residual energy term, which is
regarded to be dependent on the local chemical environment, is described via a
short-range ML model. The feasibility of the ec-MLP proposed in this work is
illustrated in an important and challenging model system, i.e., the
Pt(111)/KF(aq) interfaces. For this model system, the ec-MLP can describe not
only the water chemisorption phenomenon but also the potential-dependent
interfacial structures and differential capacitances. Based on the well-
validated ec-MLP, we further calculate the dielectric profile at the Pt/water
interface, which reveals the importance of electronic effects and offers new
insights into the molecular origins of interfacial dielectric properties.
Overall, the ec-MLP proposed in this work provides access to going beyond the
timescale and the length scale of ab initio simulation, showing great
potential in future electrochemical studies.

##  2 Results

###  2.1 Machine learning model with hybrid representation of dielectric
response at metal electrode/electrolyte interfaces

At electrochemical interfaces, there can exist an extremely large electric
field62, 63, the influence of which should be considered when simulating the
electrochemical interface. The first challenge to be overcome is to describe
how the interfacial polarisation varies with the electric field, i.e., the
dielectric response of the ionically conducting electrolyte and electronically
conducting electrode at the interface. In the theory of dielectrics, the
species in the dielectrics responds to the electric field in the following
ways, i.e., the translation of ions, the reorientation of solvent molecules,
the vibration of bonds and the polarisation of electrons 63. Concerning the
electrolyte, which is an ionic conductor and electronic insulator, the first
three terms predominate in the dielectric response. The metal electrode, on
the contrary, is the electronic conductor and responds to the electric field
by electronic polarisation. The distinct dielectric response of electrolyte
and electrode leads to different treatments when describing the response in
the ec-MLP, i.e., the Wannier centroid method for electrolytes and the
polarsiable electrode method for electrodes.

In the electronically insulating electrolyte, the dielectric response is
mainly undertaken by the motion of atoms and ions. We therefore choose the WC
method to describe this dielectric response. In this method, the positions of
WCs relative to the associated nuclei are predicted based on the local
chemical environment 49. In contrast to the electrolyte, the motion of the
atoms is usually neglectable and the dielectric response from the electronic
polarisation (i.e., the electronic dielectric response) plays an important
role in the metal electrode. Previously, there have been some methods
developed for describing the electronic dielectric response of metal (i.e.,
polarisable electrode method) 64, 61, 65. For example, in the Siepmann–Sprik
model61, the charge distribution in the electrode is approximated with the
spherical Gaussian charges located at the atom sites. This leads to a
quadratic form of the electrostatic energy with respect to the magnitude of
the Gaussian charges. The charges obtained by minimising the electrostatic
energy vary with the atomic configuration and the boundary condition,
accounting for the electronic dielectric response of the electrode.
Remarkably, the Siepmann–Sprik model has been used in conjunction with DFT in
a hybrid quantum mechanics/molecular mechanics (QM/MM) calculation 66. When
this QM/MM scheme is used to investigate the molecular adsorption on the metal
slab, it can not only capture the correct long-range interactions but
reproduce the adsorption energies in the corresponding full DFT calculation.
The promising result indicates that the Siepmann–Sprik model is a good choice
for describing the electronic dielectric response of the metal electrode.

The WC method and the polarisable electrode method can be integrated for a
hybrid representation of the dielectric response of interfaces, which is used
for the further derivation of total energies in the ec-MLP (see Fig. 1). For a
given atomic configuration, the positions of WCs are predicted from a ML model
for atomic tensorial properties67, 68, and the charge distribution in
electrolytes is approximated as the point charges located at the positions of
nuclei and WCs. Based on this charge distribution, the electrostatic potential
generated by the electrolyte at the sites of electrode atoms can be
calculated, with which the atomic charges of the electrode atoms can be solved
at a given electrostatic boundary condition. The charge distribution at the
interface (i.e., for both electrolytes and electrodes) allows the calculation
of the long-range electrostatic contribution $E_{\text{elec}}(r,w,q)$ in the
total energy $E(r,w,q)$, and the residual short-range energy
$E_{\text{SR}}(r)$ is predicted via a MLP model based on the local chemical
environment47, 50, 49:

$\displaystyle E(r,w,q)=E_{\text{SR}}(r)+E_{\text{elec}}(r,w,q).$ (1)

$r$ and $w$ are the positions of the nuclei and the WCs, respectively. $q$ is
the magnitude of the charges located at the sites of the nuclei or WCs. In
this work, $E_{\text{elec}}$ in Eq. 1 is defined as the reciprocal term in the
Ewald summation algorithm. The Deep Wannier (DW) deep neural network model 68
and the Siepmann–Sprik model 61 are chosen to describe the dielectric response
in electrolytes and metal electrodes, respectively, in the interest of the
practical implementation. Nevertheless, other similar methods can be used
instead. For example, the atomic charges projected from the MLWCs can be
predicted via a ML model to capture the dielectric response of electrolyte,
while the charge equilibrium method65 can be regarded as a generalised form of
the Siepmann-Sprik model to describe the dielectric response of electrode.

Figure 1: Schematic illustration of the machine learning model for
electrochemistry (ec-MLP) The architecture of the ec-MLP is illustrated. The
charge distributions in the electrode (i.e., atomic charges $q$) can be
predicted with the polarisable electrode method, while those in the
electrolyte (i.e., the positions of Wannier centroids $w$ and the positions of
nuclei $r$) can be predicted with the ML-based Wannier centroids method. The
charge distribution can be used to calculate the long-range electrostatic
contribution $E_{\text{elec}}$ in the total energy and the residual short-
range energy $E_{\text{SR}}$ is predicted by the MLP based on the local
chemical environment.

###  2.2 Validations of machine learning potentials for electrochemistry

In the following, the Pt(111)/KF(aq) interface is chosen as the model system
to demonstrate the validity of the ec-MLP proposed above. This choice is
motivated by the widespread utilisation of the Pt(111)/electrolyte interface
as the model system in both experimental and computational electrochemistry.
Most importantly, it is well known in the literature that Pt shows a bell-
shaped Helmholtz differential capacitance curve 69 caused by dynamic
chemisorption of water induced by the applied potential 10. This phenomenon on
Pt highlights the simultaneous importance of electronic structure and
molecular dynamics effects, which cannot be accounted for by static DFT or
classical molecular dynamics calculation. Therefore, it serves as an excellent
model system for validating ec-MLP in comparison with AIMD.

The accuracy of the ec-MLP is first evaluated by comparing the predicted
energies, atomic forces and positions of WCs from the MLP with the
corresponding values obtained from DFT calculations. A concurrent learning
workflow is utilised to explore the configuration space and collect
representative configurations for training the ec-MLP70. The details of the
training of the ec-MLP and the DFT calculations are provided in the Methods
section. As listed in Table 1, the root mean square errors (RMSEs) in the
Wannier centroids are $1.9\text{\,}\mathrm{m}\mathrm{\SIUnitSymbolAngstrom}$
for the training dataset and
$2.0\text{\,}\mathrm{m}\mathrm{\SIUnitSymbolAngstrom}$ for the testing
dataset, which are comparable to the uncertainty reported in previous work 49.
Additionally, the potential energy surface of the system can be predicted
accurately by the MLP. When applied to the training dataset, the ec-MLP yields
a RMSE of
$1.305\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$
for total energies and
$75.00\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\SIUnitSymbolAngstrom}$
for atomic forces. The testing dataset demonstrates even smaller RMSEs for
both energies and atomic forces, highlighting the generalisability of the MLP.

Table 1: Root mean square errors (RMSEs) in the positions of WCs (m$\mathrm{\SIUnitSymbolAngstrom}$), the potential energies ($\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$) and the atomic forces ($\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\SIUnitSymbolAngstrom}$) by comparing the results from the MLP and DFT calculation, in either the training set or the testing set. The details of the dataset are referred to Supplementary Sec.1. | WC positions | energies | forces
---|---|---|---
training set | 1.9 | 1.305 | 75.00
testing set | 2.0 | 0.9638 | 57.19

Good performance of the ec-MLP in predicting energies, atomic forces and
positions of WCs lays a solid foundation for reproducing the potential-
dependent interfacial water structures. Here, we perform MD simulations on the
atomic models as shown in Fig. 2a. The Pt electrodes are described with a
(6$\times$6) 6-atomic-layer slab while the
$\sim$$2.2\text{\,}\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{/}\mathrm{L}$ KF
aqueous solution is filled between the space of
$30\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ between two Pt surfaces. To
electrify the interfaces, we adopt a similar method, i.e., counterions in
symmetric supercells, as in the previous work 10, 71, 72. In this way, the
electrodes carry the total charges in the opposite sign but the same magnitude
as the electrolytes, which can be regarded as a reasonable approximation of a
compact EDL at the high concentration limit. In the constant charge method,
the total charges of electrodes should be defined by users in the ec-MLP,
while those in AIMD are induced automatically by the electroneutrality
constraint. The electrostatic potential profile is calculated based on the
charge distribution by solving the Poisson equation. The timescales for all MD
simulations are set as $100\text{\,}\mathrm{p}\mathrm{s}$, which are longer
than those used in the previous AIMD simulations (i.e.,
10$\sim$$20\text{\,}\mathrm{p}\mathrm{s}$) and lead to better statistical
accuracy. The details of the setup for MD simulations are referred to the
Methods section.

In the simulations under the pzc, the chemisorbed water can be observed in the
water density distribution profile in the direction perpendicular to the
surface, corresponding to a peak at
$\sim$$2.3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ away from the Pt surface
(see the insert with red frame in Fig. 3a). Furthermore, the potential-
dependent behaviours of the water chemisorption can also be captured. When the
potential varies from negative to positive values, the coverage of the
chemisorbed water increases, which agrees with the previous AIMD simulations
(see the blue curve and the red triangles in Fig. 3b). Meanwhile, the
molecular orientation of the chemisorbed water remains almost unchanged. In
addition to water chemisorption, water reorientation at the interface can also
be described accurately. The water molecules located at
2.7-$4.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ away from the Pt surface,
identified as the physisorbed water, reorientate when changing the electrode
potential, leading to a total dipole moment aligning with the electric field.
Further details about the potential-dependent orientation of the interfacial
water can be found in Supplementary Sec.2. All the results mentioned above,
generated from the simulations with the ec-MLP, agree with those from the
previous AIMD simulations not only in the qualitative trend but also in the
quantitative values 73, 74, 10.

Figure 2: Representative snapshots of electrochemical interfaces with
different electrifying methods. a The electrochemical interfaces can be
electrified by adding counterions adjacent to electrodes in a symmetric
supercell. The electrodes carry the total charges in the opposite sign but the
same magnitude as the electrolytes.72 In this case, the electrostatic
potential distributions at two interfaces are identical at the thermodynamics
limit, as shown by the blue line. b Additionally, it is also feasible to build
up a model with both positively and negatively charged interfaces by applying
a potential drop (or a periodic electric field $\mathcal{E}$) crossing the
supercell (e.g., finite field methods)75, 76, 77, 78. In all snapshots, the
Pt, K, O, and H atoms are coloured in silver, violet, red, and white,
respectively. The Pt electrode regions are shaded in grey, while the
electrolyte region is shaded in blue.

The accurate description of the potential-dependent water chemisorption and
reorientation at the interface makes it possible to reproduce the Helmholtz
differential capacitance of the EDL. The differential capacitance is defined
as the first derivative of surface charge with respect to the potential. In
our model, the surface charge is the controlled variable in simulation, which
is equal to the number of counterions. The potential can be calculated by
referring the average electrostatic potential in the bulk electrode region to
that in the bulk electrolyte region 10, 73. As shown in Fig. 3c, the surface
charge density-potential function shows the uncertainty smaller than
$0.05\text{\,}\mathrm{V}$, guaranteeing the statistical accuracy of the
differential capacitance curve.

Taking the first derivative of the surface charge density-potential function,
the differential capacitance can be calculated with its maximum value of
$\sim$80 $\mu$F/cm2 around the pzc. In the potential windows chosen in this
work, the differential capacitance curve converges to values of 20 and 40
$\mu$F/cm2 at the negative and positive limits, respectively. Overall, the ec-
MLP is capable of reproducing the bell shape of the differential capacitance
curve, and the capacitance value is in good agreement with the experimental
results 69 and the AIMD results 10, 71. Since the bell-shaped capacitance
curve can only be captured when taking both electronic structure and molecular
dynamics effect into account, this stringent validation strongly shows the
suitability and promise of the ec-MLP in simulating electrochemical systems.

Figure 3: Structure and capacitance of the Pt/KF interface calculated using
the ec-MLP. a Water density distribution profile in the direction
perpendicular to the Pt surface under different electrode potentials versus
pzc. The zero point in the z coordination is set at the average position of
the outermost Pt atoms. The chemisorbed water (red) and the physisorbed water
(blue) are identified with the shadows in the background, the example
configurations of which are shown in the insets. The grey dashed line refers
to 1 g/cm3 as a guide to the eye. b Potential-dependent coverage of the
chemisorbed water. The blue curve and the red triangles represent the results
from the simulations with the ec-MLP in this work and from the AIMD
simulations in the previous work 10, respectively. c Plots of surface charge
density $\sigma$ and the differential capacitance as a function of the
potential versus pzc $(U-U_{\text{pzc}})$. The blue curve is the charge
density-potential function calculated from the simulations with the ec-MLP,
with the uncertainty shown in the blue shadow. The differential capacitance
calculated from the first derivative of the blue curve is shown in the red
curve.

###  2.3 Dielectric profile of interfacial water

Water in contact with solids shows distinct dielectric behaviours from bulk
water, attracting great interest in both experiment and simulation. For
instance, water between two hydrophobic surfaces separated by a few nanometers
shows an anomalously low dielectric constant due to strong confinement
effects79. At highly heterogeneous electrochemical interfaces, the dielectric
constant of water is anticipated to be a complex function of position, which
is relevant when analysing the electrochemical conditions for interfacial
processes. However, our knowledge about how the dielectric constant varies at
the interface is still lacking at the qualitative level. Recently, MD
simulations provide access to the dielectric profile, i.e., the dielectric
constant distribution in the direction perpendicular to the surface 80, 81,
82, 83, 84, 85, 86. The dielectric profile can be further used in the
continuum modelling and lead to a better agreement between the experimental
data and the modelling results63, 87. Nevertheless, the existing studies of
dielectric profiles are limited to the cases of inert interfaces, and the
calculation of interfaces of high activity (e.g., Pt/water) is still lacking,
probably due to the following reasons. On the one hand, simulations with
classical force fields cannot describe the interfacial water structures
correctly due to the omission of the electronic structure, and the timescale
for statistically reliable results is not affordable for AIMD simulations. On
the other hand, it is not yet practical to apply an electric field between two
metal slabs in current DFT implementations. Fortunately, with the help of the
ec-MLP, we can overcome the limitations mentioned above and calculate the
dielectric profile at the Pt(111)/water interface for the first time.
Furthermore, based on the calculated profile, the impact of the electronic
polarisation on the interfacial dielectric response will be revealed.

Aiming for the dielectric profile of water at the Pt/water interface, a small
perturbing electric field is applied to the model shown in Fig. 2b. The local
electric fields induced by the applied field can be obtained by integrating
the induced charge densities. Consequently, the dielectric profile can be
calculated as follows80:

$\displaystyle\varepsilon_{\perp}^{-1}(z)=\frac{\mathcal{E}_{\perp}^{\text{ind}}(z)}{\mathcal{E}_{\perp}^{\text{ind}}(z)+\varepsilon_{0}^{-1}P_{\perp}^{\text{ind}}(z)},$
(2)

where $z$ is the distance away from the surface, $\varepsilon_{0}$ is the
vacuum permittivity, and $\varepsilon$ is the dielectric constant (i.e.,
relative permittivity). $\mathcal{E}_{\perp}^{\text{ind}}$ and
$P^{\text{ind}}$ are the induced electric field and polarisation,
respectively. The footnote “$\perp$” refers to the direction perpendicular to
the surface. Details of calculating $\mathcal{E}_{\perp}^{\text{ind}}$ and
$\varepsilon_{0}^{-1}P_{\perp}^{\text{ind}}$ are referred to Supplementary
Sec.3. Notably, a large fluctuation can be observed in the resulting
dielectric profile, which includes the effects of the intramolecular
variations in the electric field. Since it is only meaningful to consider the
response on length scales corresponding to intermolecular distances, the
dielectric profile should be averaged over the regions of interest.

For non-interacting, homogeneous dielectrics, its susceptibility (i.e.,
$\varepsilon-1$) is expected to be proportional to densities (e.g., shaded
area in Fig. 4a). However, water close to Pt(111) surfaces with higher
densities than bulk water does not show a higher dielectric constant due to
intricate metal-water interactions (see blue line in Fig. 4a). Particularly, a
negative dielectric constant is observed at the region of
2-$2.7\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, where the chemisorbed water is
located. In contrast to the normal dielectric constant $\varepsilon$ (i.e.,
$0\leq\varepsilon^{-1}\leq 1$), the negative dielectric constant indicates the
over-screening of the electric field by the chemisorbed water layer85. The
more pronounced the over-screening effect, the more negative the value of the
inverse dielectric constant $\varepsilon_{\perp}^{-1}$ in Eq. 2. The negative
dielectric constant agrees with the negative capacitive response of the water
chemisorption proposed in the previous work 10. In contrast, the physisorbed
water layer shows a small average dielectric constant of $\sim$3, which is
close to the dielectric constant used in the traditional EDL model 88.

To estimate the influence of the electronic polarisation in water induced by
the interface, we calculate a dielectric profile with the water dipole moment
set to the bulk value, denoted as the “reference calculation” in the following
(see details in Supplementary Sec.4). Comparing the blue line and the red line
in Fig. 4a, over-screening of the electric field in the chemisorbed water
layer can be attributed to two reasons. On the one hand, as indicated by the
negative values of the red line, over-screening exists due to water
structuring at the interface. On the other hand, the presence of chemisorption
increases the water dipole moment and enhances the over-screening effect,
based on the fact that the blue line becomes more negative. The latter, i.e.,
how the electron polarization induced by chemisorption influences the
dielectric constant, can be validated by the probability distribution of
chemisorbed water dipole moments. As shown in Fig. 4b, the molecular dipole
moments of chemisorbed water increase in the z-direction (i.e., the direction
perpendicular to the surface), while those in the x- and y-directions are
almost identical after taking the electronic polarisation into account.
Overall, the molecular origins of the dielectric property of the chemisorbed
water layer shown by our calculation agree with the pictures proposed in
previous studies73, 10.

In contrast to the chemisorbed water layer, the dielectric property of the
physisorbed water layer is often believed to be mainly influenced by
restricted molecular reorientation at the interface rather than electronic
polarisation79, 10. Interestingly, a distinct picture can be obtained from
Fig. 4a. In the reference calculation, the physisorbed water layer exhibits an
average dielectric constant comparable to the bulk region, which is
significantly higher than the result with the effect of electronic
polarisation. This indicates that the low dielectric constant observed in the
physisorbed water layer at the interface is largely influenced by electronic
polarisation rather than the widely accepted restricted reorientation of water
molecules. A picture at the atomic level is revealed in Fig. 4c, in which the
dipole moments of physisorbed water in the z-direction become smaller due to
the influence of electronic polarisation. The reduced water dipole moment can
be further rationalised as the results of the lower number of hydrogen
bonds89, 90 in the physisorbed water layer compared to the bulk water region.
As shown by the inset in Fig. 4a, the physisorbed water with a reduced dipole
points one hydrogen atom towards electrodes and loses one hydrogen-bond
donor74.

With the help of the ec-MLP, we extend the timescale of MD simulations to
$1\text{\,}\mathrm{n}\mathrm{s}$ and hence are able to calculate the
dielectric profile at the Pt/water interface. In the chemisorbed water layer,
an over-screening of the electric field is observed due to the synergy of the
ordered molecular configurations and the electronic polarisation induced by
chemisorption. Moreover, the average dielectric constant in the physisorbed
water layer illustrates the significant influence of electronic polarisation
in the absence of chemisorption, which, to the best of our knowledge, has
rarely been discussed before. Overall, the dielectric profile calculated from
the MLP sheds light on the effect of electronic polarisation on the dielectric
response at the interface, which lays the foundation for future exploration of
tuning interfacial dielectric properties and optimising the reaction activity
in electrocatalysis.

Figure 4: The dielectric profile of water at the Pt/water interface. a The
light lines in the background represent the calculated dielectric profiles,
including the effects of the intramolecular variations in the electric field.
The dark lines in the foreground represent the regional average of the
dielectric constant. While the blue lines refer to the calculation at the
Pt/water interface, the red line refers to the dielectric profile calculated
using the same trajectories but with the bulk water dipole moment (i.e., in
the “reference calculation”). The shaded area refers to the water density
profile at the interface for the identification of the chemisorbed water layer
and the physisorbed water layer. The two insets on the top illustrate the
molecular origins of the electronic polarisation in the chemisorbed water and
the physisorbed water. Distributions of water dipole moments of the
chemisorbed water layer and the physisorbed water layer are shown in b and c,
respectively. The subscripts denote the components in the x-, y-, or
z-direction. The superscripts of “ref” refer to the results from the reference
calculation.

##  3 Discussion

In this work, we introduce the ec-MLP model, which is feasible to describe the
dielectric response at the electrode/electrolyte interfaces in a hybrid
representation. The ec-MLP shows a high accuracy in predicting not only the
water chemisorption but the potential-dependent interfacial water structures
and differential capacitances. In particular, the Pt(111)/KF(aq) interface is
chosen as the model system, which can only be simulated accurately when both
the electronic structure and the molecular dynamics effect are taken into
account. The success of ec-MLP in this challenging model system illustrates
the feasibility of the method. Furthermore, we also calculate the dielectric
profile at the Pt(111)/water interface and gain new insight into the effect of
the electronic polarisation on the dielectric properties at the interface.
While the electronic polarisation induced by chemisorption results in larger
water dipole moments and over-screening of electric fields, that induced by
fewer hydrogen bonds leads to smaller water dipole moments and lower
dielectric constant. Overall, the insight into the interfacial dielectric
properties provided by the ec-MLP is inspiring for further studies of, for
example, pH and ion effects, which are highly sensitive to the electronic
structure at the interface. Additionally, we anticipate this method will be
utilised to investigate some important processes with extended timescales at
the electrochemical interface, e.g., electrocatalytic reactions 91, 92,
formation of solid–electrolyte interphases (SEI) 93, 94, and electrode
corrosion 95, 96.

Aiming for a wider application of the method, it is worthwhile to discuss how
the MLP works under different conditions in a general way. At the
electrochemical interfaces, there usually exist (electro-)chemical reactions,
the treatment of which could be different in the MLP. For example, when the
chemisorption happens at the interfaces, the electrons of the adsorbate
polarise 97, which can be described by the displacements of the WCs in the
MLP. As illustrated in Supplementary Sec.5, the (partial) charge transfer
during the chemisorption can be captured via the electron redistribution
within the adsorbates under constant potential constraint. In contrast,
dealing with the electron transfer in the electrolyte is more complicated.
Currently, the charge of the WCs and hence the oxidation states of the
elements is fixed in the ec-MLP. Aiming for the free energy change of an
electron transfer process, two MLPs with different oxidation states can be
combined 98, 99.

At the end of the discussion, we propose some possible strategies to improve
the ec-MLP in the future. Currently, only the electrostatic interaction is
minimised when solving the atomic charges in the Siepmann-Sprik model. In
fact, we expect the chemical potential to also play a role, especially for the
substrates with multiple elements. It is therefore promising to take advantage
of the ML-based charge equilibrium model 50, 60, 57, which allows a higher
generalisability of the ec-MLP. In addition, calculation of atomic forces of
electrode atoms from the negative derivative of energy is computationally
demanding. In this work, we approximate the forces of the electrode atoms as
the sum of the electrostatic atomic forces and the short-range atomic forces
from the derivative of the short-range energy, ignoring the dependency of the
charges on the positions of the electrode atoms. The approximation can be
justified as having a minor effect on this work based on the good performance
of the ec-MLP. Nevertheless, more rigorous calculation of the atomic forces of
the electrode atoms might be necessary when investigating, for example, the
dynamic structure evolution of the electrode during the electrochemical
processes 23. In order to overcome the limitation, the automatic
differentiation framework can be integrated into the ML model in future
development100.

##  4 Methods

###  4.1 DFT calculations for the dataset used in the machine learning model
training

All the density functional theory calculations were performed using
CP2K/Quickstep package 101. In CP2K/Quickstep, a mixed Gaussian and plane
waves approach 102 is implemented for DFT calculations, in which the orbitals
are described by an atom-centered Gaussian-type basis set and an auxiliary
plane wave basis set is used to represent the electron density in the
reciprocal space. The 2s and 2p electrons of O; 3s, 3p, and 4s electrons of K;
2s and 2p electrons of F; and 5d and 6s electrons of Pt are treated as
valence, and the rest core electrons were represented by Goedecker-Teter-
Hutter pseudopotentials 103, 104. The Gaussian basis sets were double zeta
with one set of polarisation functions. The plane wave energy cutoff was set
to 800 Ry. The Perdew-Burke-Ernzerhof (PBE) functional 105 was used to
describe the exchange-correlation effects, with the Grimme D3 correction 106
accounting for the dispersion interactions. The MLWCs were calculated from the
DFT calculation by localising the occupied orbitals with a convergence
threshold 0.01. The MLWCs were then associated with certain central atoms if
the distances between the MLWCs and the atoms are smaller than
$1\text{\,}\mathrm{\SIUnitSymbolAngstrom}$.

###  4.2 Training machine learning models for Wannier centroids

In this work, the Deep Wannier (DW) model 68 was used to predict the positions
of the WCs relative to the associated nuclei based on the local chemical
environment. In this model, the nuclei carry the charges of the number of
their valance electrons, i.e., $6\text{\,}\mathrm{e}$, $1\text{\,}\mathrm{e}$,
$9\text{\,}\mathrm{e}$, $7\text{\,}\mathrm{e}$, and $0\text{\,}\mathrm{e}$ for
O, H, K, F and Pt atoms, respectively. The WCs with a charge of
$-8\text{\,}\mathrm{e}$ were associated with the O, K, and F atoms. When
training the DW model, the descriptor was chosen as the Deep Potential Smooth
Edition (DeepPot-SE) descriptor 41 with a cutoff of
$5.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. The number of neurons in each
hidden layer of the embedding net and the fitting net were [25, 50, 100] and
[240, 240, 240], respectively. The DW model was trained with $5\times 10^{5}$
batches with a batch size of 1. The learning rate was set as starting from
$1\times 10^{-3}$ and decaying to $1\times 10^{-8}$ with a step width of 5000
and rate of 0.95.

###  4.3 Training machine learning models for potential energy surfaces

The training codes for total energies were implemented based on the Deep
Potential Long Range (DPLR) method 49 in the software DeepMD-kit 34. As
mentioned above, total energies can be decomposed into the short-range part
and the long-range electrostatic part. The electrostatic contributions in
total energies and atomic forces were calculated from the charge distributions
based on the Particle-Particle Particle-Mesh (PPPM) algorithm 107. The charge
distributions of the electrolyte were predicted using the DW model, while
those of the electrode were predicted using the Siepmann–Sprik model. The
atomic charges of the electrode in the Siepmann–Sprik model follow Gaussian
distributions centring at the atom sites. The Gaussian spread of
$1.6\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ was chosen as the atomic radius
of Pt used in the original charge equilibrium method 108. The remaining short-
range contributions can be trained via a neural network. When training the
potential, the descriptor was chosen as the Deep Potential Smooth Edition
(DeepPot-SE) descriptor 41 with a cutoff of
$5.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. The number of neurons in each
hidden layer of the embedding net and the fitting net were [25, 50, 100] and
[240, 240, 240], respectively. The machine learning potential was trained with
$5\times 10^{5}$ batches with a batch size of 1. The learning rate was set as
starting from $1\times 10^{-3}$ and decaying to $1\times 10^{-8}$ with a step
width of 5000 and rate of 0.95. The prefactor for the energy in the loss
function started from 0.001 and ends at 1, while that for the atomic forces
started from 1000 and ends at 1.

###  4.4 Molecular dynamics simulation and calculation of interfacial
dielectric properties

The MD simulations in this work were performed with the software LAMMPS 109.
The `USER_DEEPMD` 34 and `ELECTRODE` 110 packages were modified to combine the
DW model and the Siepmann–Sprik model.

The MD simulations were performed in the canonical ensemble (NVT) using a
timestep of 0.5 fs. In all simulations, the temperature was set as
$330\text{\,}\mathrm{K}$ with the damping time of
$50\text{\,}\mathrm{f}\mathrm{s}$ for the Nosé-Hoover thermostat. The Ewald
screening parameter was set as
$0.4\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{-1}$ and the grid width in each
dimension for the PPPM algorithm 107 was set as
$0.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{-1}$, which keeps consistent with
the setup used in training the ec-MLP.

The Pt(111)/electrolyte interfaces used in this work were described in a
$(16.869\times 14.609\times 41.478)$ $\mathrm{\SIUnitSymbolAngstrom}^{3}$
periodic supercell model with a (6$\times$6) 6-atomic-layer Pt slab. The
lattice constant of the Pt was $3.98\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,
which is consistent with the previous AIMD simulations 73, 74, 10. The
electrolyte was filled between the space of
$30\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ between two Pt surfaces. For all
simulations, the water density in the bulk region (i.e.,
$\leq$$9\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ from both surfaces) is
(1$\pm$0.05) g cm-3. All Pt atoms in the electrode were frozen during the
simulations, which is required by the algorithm used in the `ELECTRODE`
package.

The constant charge method was used to investigate the potential-dependent
properties at the interfaces, as shown in Fig. 3. For each potential data
point corresponding to a certain surface charge density, five simulations were
performed with independent initial configurations. In each simulation, the
system was pre-equilibrated for $20\text{\,}\mathrm{p}\mathrm{s}$ and sampled
for $100\text{\,}\mathrm{p}\mathrm{s}$.

The constant potential method was used to calculate the dielectric profile at
the Pt/water interface. The simulations with the macroscopic electric fields
of 0.0 V/$\mathrm{\SIUnitSymbolAngstrom}$ and 0.024
V/$\mathrm{\SIUnitSymbolAngstrom}$ were performed for
$1\text{\,}\mathrm{n}\mathrm{s}$. The convergence tests for the dielectric
profile calculation are shown in Supplementary Sec.6.

J.-X.Z. gratefully acknowledges Xiamen University and iChEM for a Ph.D.
studentship. J.C. gratefully acknowledges funding from the National Science
Fund for Distinguished Young Scholars (Grant No. 22225302), the National
Natural Science Foundation of China (Grant Nos. 92161113, 21991151, 21991150,
and 22021001) and the Fundamental Research Funds for the Central Universities
(Grant Nos. 20720220008, 20720220009, 20720220010), Laboratory of AI for
Electrochemistry (AI4EC), and IKKEM (Grant Nos. RD2023100101 and
RD2022070501). J.-X.Z. also thanks Dr. Jia-Bo Le, Dr.Xiao-Hui Yang, Dr.
Katharina Doblhoff-Dier and Dr. Jun Huang for helpful discussions.

##  5 Author contributions

J.C. and J.-X.Z. conceived this work and designed the project. J.-X.Z.
implemented the method and performed the simulation. J.-X.Z. and J.C. analysed
the results and wrote the manuscript.

##  6 Correspondence

Correspondence to Jia-Xin Zhu or Jun Cheng.

##  7 Competing interests

The authors declare no competing interests.

##  8 Data availability

The data that support the findings of this study are available from the
corresponding authors upon reasonable request.

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