Update ABACUS_17_04_2025.md (#245)

Adjust the formulas in the article to ensure they can be correctly displayed on the DeepModeling official website.

Author: RussellHu41

source/_posts/ABACUS_17_04_2025.md

@@ -1,5 +1,5 @@
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-title : "Can DP + ABACUS Also Do This? Exploring Nuclear Quantum Effects in Electrochemical Reactions with Machine Learning and Developing Computational Methods"
+title : "What Can DP + ABACUS Do Too? | Exploring Nuclear Quantum Effects in Electrochemical Reactions with Machine Learning and Developing Computational Methods"
 date : 2025-04-17
 categories :
 - ABACUS
@@ -13,21 +13,29 @@ Recently, the research group led by Researcher Xu Shenzhen from the School of Ma
 
 Electrocatalytic systems can efficiently and cleanly convert electrical energy into chemical energy, making them one of the widely concerned fields in new energy technologies. They cover various scenarios such as water electrolysis, CO₂ reduction, and nitrogen reduction. The thermodynamic and kinetic properties of the elementary step of proton-coupled electron transfer (PCET) on the surface of electrode or catalyst materials, which is a crucial step in energy conversion, are key factors influencing performance indicators of electrocatalytic systems, such as product formation rate, energy conversion efficiency, and selectivity. They are also fundamental scientific issues that are the focus of experimental and theoretical electrochemical research.
 
-Microscopic simulation of electrocatalytic systems is an important tool for studying the PCET mechanism. However, the electrochemical open system formed jointly by the electrode surface and the solution poses great challenges to computational simulation. How to consider the constant potential condition, sampling of the complex electrode-solution interface structure, and nuclear quantum effects (NQEs) of protons in a rigorous and efficient manner in the simulation has become a key problem to be solved urgently in the field of electrochemical theoretical calculation. For a typical elementary step of PCET on the electrode surface, such as $\text{A}^* + \text{H}^+_{\text{sol}} + \text{e}^- \rightarrow {}^* \text{AH}$ (where "*" represents the surface adsorption site), the traditional simulation method is to optimize the structures of the initial and final states of the reaction and then use the transition state search method to calculate the reaction energy barrier. The defects of this set of methods are as follows: (1) Only specific initial and final state structures are considered, and the reaction path is single, lacking statistical averaging; (2) In the simulation of the reaction path, the number of electrons in the surface system remains constant, which does not conform to the experimental condition of constant potential. Some post-treatment constant potential correction methods also cannot handle this problem strictly; (3) The nuclear quantum effects of hydrogen cannot be considered, while NQEs have been proven to have a significant impact on proton transfer kinetics in many other application scenarios (also at room temperature).
+Microscopic simulation of electrocatalytic systems is an important tool for studying the PCET mechanism. However, the electrochemical open system formed jointly by the electrode surface and the solution poses great challenges to computational simulation. How to consider the constant potential condition, sampling of the complex electrode-solution interface structure, and nuclear quantum effects (NQEs) of protons in a rigorous and efficient manner in the simulation has become a key problem to be solved urgently in the field of electrochemical theoretical calculation. For a typical elementary step of PCET on the electrode surface, such as A* + H<sup>+</sup> + sol + e<sup>-</sup> ➜ *AH (where "*" represents the surface adsorption site), the traditional simulation method is to optimize the structures of the initial and final states of the reaction and then use the transition state search method to calculate the reaction energy barrier. The defects of this set of methods are as follows: (1) Only specific initial and final state structures are considered, and the reaction path is single, lacking statistical averaging; (2) In the simulation of the reaction path, the number of electrons in the surface system remains constant, which does not conform to the experimental condition of constant potential. Some post-treatment constant potential correction methods also cannot handle this problem strictly; (3) The nuclear quantum effects of hydrogen cannot be considered, while NQEs have been proven to have a significant impact on proton transfer kinetics in many other application scenarios (also at room temperature).
 
 Based on the above problems, the research team developed a unified computational framework that can accurately handle NQEs under strict grand canonical constant potential conditions. Combined with the deep potential (DP) machine learning force field, it achieved sufficient sampling of electrochemical open systems. The DP force field can maintain an accuracy comparable to that of density functional theory (DFT) while greatly improving the computational efficiency, making it possible to conduct sufficient statistical sampling of the complex configurations of the electrode-solution interface in the reaction path at an affordable computational cost.
 
 ## Research Results
 
-This work is based on a rigorous statistical physics model. It uses the grand canonical hybrid Monte Carlo (GC-HMC) algorithm and the path integral Monte Carlo (PIMC) method, combined with the DP machine learning force field, to simulate the nuclear quantum effects of protons under precisely constant potential conditions. The workflow of this GC-(PI)HMC is shown in Figure 1a. In addition, based on the traditional DP machine learning potential function that takes atomic positions as inputs, the research team introduced a new input parameter degree of freedom: the total number of electrons Ne in the interface model system. Figure 1b shows the construction framework and training workflow of the DP-Ne machine learning force field used in this study. In order to calculate the work function W of the instantaneously sampled configuration in the extended space [R, Ne], an additional output term $`\frac{\partial E(R,N_{e})}{\partial N_{e}}=-W(R,N_{E})`$ was introduced. For effective grand canonical ensemble sampling, it is necessary to satisfy: $`<\frac{\partial E\left(R, N_{e}\right)}{\partial N_{e}}>_{GC}=<-W\left(R, N_{e}\right)>_{GC}=\mu_{e}`$, where $`\mu_{e}`$ represents the external electrochemical potential in equilibrium with the system and is a thermodynamic macroscopic quantity. The updated DP-Ne force field is helpful for sampling grand canonical systems with variable electron numbers.
+This work is based on a rigorous statistical physics model. It uses the grand canonical hybrid Monte Carlo (GC-HMC) algorithm and the path integral Monte Carlo (PIMC) method, combined with the DP machine learning force field, to simulate the nuclear quantum effects of protons under precisely constant potential conditions. The workflow of this GC-(PI)HMC is shown in Figure 1a. In addition, based on the traditional DP machine learning potential function that takes atomic positions as inputs, the research team introduced a new input parameter degree of freedom: the total number of electrons Ne in the interface model system. Figure 1b shows the construction framework and training workflow of the DP-Ne machine learning force field used in this study. In order to calculate the work function W of the instantaneously sampled configuration in the extended space [R, N<sub>e</sub>], an additional output term
+
+<center><img src=https://dp-public.oss-cn-beijing.aliyuncs.com/community/Blog%20Files/ABACUS_17_04_2025/pic1.png pic_center width="20%" height="20%" /></center>
+
+was introduced. For effective grand canonical ensemble sampling, it is necessary to satisfy: 
+
+<center><img src=https://dp-public.oss-cn-beijing.aliyuncs.com/community/Blog%20Files/ABACUS_17_04_2025/pic2.png pic_center width="30%" height="30%" /></center>
+
+represents the external electrochemical potential in equilibrium with the system and is a thermodynamic macroscopic quantity. The updated DP-Ne force field is helpful for sampling grand canonical systems with variable electron numbers.
 
 <center>
 <img src="https://dp-public.oss-cn-beijing.aliyuncs.com/community/Blog%20Files/ABACUS_17_04_2025/p1.webp">
 </center>
 
-*Figure 1. a GC-(PI)HMC method workflow. The trial steps of different types of degrees of freedom are selected from PIMC($`\mathbf{R}_\Delta^{(k)}`$), centroid atomic coordinates (R), and the number of electrons Ne using the random variable ξ according to a preset ratio. b The construction framework and training workflow of the DP-Ne force field used in this study. The added new degree of freedom - the total number of electrons Ne in the interface system is used as an input parameter for the fitting network.*
+*Figure 1. a GC-(PI)HMC method workflow. The trial steps of different types of degrees of freedom are selected from PIMC, centroid atomic coordinates (R), and the number of electrons Ne using the random variable ξ according to a preset ratio. b The construction framework and training workflow of the DP-Ne force field used in this study. The added new degree of freedom - the total number of electrons Ne in the interface system is used as an input parameter for the fitting network.*
 
-The research team applied the new method to the hydrogen evolution reaction (HER) on the surface of a Pt electrode - a classic electrocatalytic science problem scenario, focusing on the impact of nuclear quantum effects on the thermodynamics and kinetics of the reaction. Electrocatalytic HER involves two basic PCET processes: Volmer ($`\text{H}^+_{\text{sol}} + \text{e}^- + {}^* \rightarrow \text{H}^*`$), Heyrovsky ($`\text{H}^+_{\text{sol}} + \text{e}^- + \text{H}^* \rightarrow \text{H}_2`$), and a non-electrochemical Tafel step (H* + H* ➜ H₂). After considering NQEs in the free energy calculations of the Volmer and Heyrovsky PCET steps, the obtained activation free energy results decreased by 0.1 - 0.15 eV (see Figure 2d), indicating that traditional electrocatalytic calculation models that do not consider NQEs may underestimate the PCET reaction rate at room temperature by dozens of times. The research results also provide a clear physical picture of proton tunneling when overcoming the energy barrier in the PCET path. The characteristic is that the position of the quantized proton has obvious uncertainty, reflecting the inherent quantum nature of proton transfer (see Figure 2e, 2f).
+The research team applied the new method to the hydrogen evolution reaction (HER) on the surface of a Pt electrode - a classic electrocatalytic science problem scenario, focusing on the impact of nuclear quantum effects on the thermodynamics and kinetics of the reaction. Electrocatalytic HER involves two basic PCET processes: Volmer (H<sup>+</sup><sub>sol</sub> + e<sup>-</sup> + * ➜ H*), Heyrovsky (H<sup>+</sup><sub>sol</sub> + e<sup>-</sup> + H* ➜ H<sub>2</sub>), and a non-electrochemical Tafel step (H* + H* ➜ H₂). After considering NQEs in the free energy calculations of the Volmer and Heyrovsky PCET steps, the obtained activation free energy results decreased by 0.1 - 0.15 eV (see Figure 2d), indicating that traditional electrocatalytic calculation models that do not consider NQEs may underestimate the PCET reaction rate at room temperature by dozens of times. The research results also provide a clear physical picture of proton tunneling when overcoming the energy barrier in the PCET path. The characteristic is that the position of the quantized proton has obvious uncertainty, reflecting the inherent quantum nature of proton transfer (see Figure 2e, 2f).
 
 <center>
 <img src="https://dp-public.oss-cn-beijing.aliyuncs.com/community/Blog%20Files/ABACUS_17_04_2025/p1.webp">
@@ -47,4 +55,4 @@ This study not only provides new tools and methods for the theoretical research
 
 [2] Chen, M., Guo, G.-C. & He, L. Systematically Improvable Optimized Atomic Basis Sets for Ab Initio Calculations. J. Phys.: Condens. Matter 22, 445501 (2010).
 
-[3] Sun, M., Jin, B., Yang, X. & Xu, S. Probing Nuclear Quantum Effects in Electrocatalysis via a Machine-Learning Enhanced Grand Canonical Constant Potential Approach. Nature Communications 16, 3600 (2025)
+[3] Sun, M., Jin, B., Yang, X. & Xu, S. Probing Nuclear Quantum Effects in Electrocatalysis via a Machine-Learning Enhanced Grand Canonical Constant Potential Approach. Nature Communications 16, 3600 (2025)