Capturing Nuclear Quantum Effects in Hydrogen Diffusion inside MoS$_2$ via Machine-Learning-Enhanced Path-Integral Simulations

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AbstractHydrogen transport through layered two-dimensional (2D) materials is central to technologies such as hydrogen storage, fuel cells, and isotope separation. Among thesematerials, MoS2 exhibits tunable interlayer diffusion properties, whose accurate theoretical description requires accounting for nuclear quantum effects (NQEs), including zero-point motion and tunneling. Here, we present a machine-learning-enhanced atomistic study of hydrogen and deuterium diffusion in layered MoS2 based on interatomic potentials trained on r2SCAN+rVV10 density-functional-theory data. Combining well-tempered metadynamics with path-integral molecular dynamics, we investigate diffusion across multiple MoS2 polytypes and twisted bilayer structures while explicitly incorporating NQEs. Our simulations show that NQEs substantially lower free-energy barriers for hydrogen diffusion at 300 K, significantly increasing the hydrogen self-diffusion coefficient compared to classical nuclei simulations. We further identify a pronounced kinetic isotope effect, with a 31 meV difference between hydrogen and deuterium quantum free-energy barriers. In twisted bilayer MoS2, hydrogen transport exhibits strong spatial variations governed by the local stacking environments within the moiré superlattices. These results highlight the critical role of NQEs in hydrogen transport through layered materials and provide atomistic insight into isotope-selective diffusion in structurally complex 2D system OverviewThis repository contains the dataset, computational data, scripts, and results supporting our research as indicated in main results and supporting information All data is compressed into .tar.gz archives, identifiable with specific material type, treatment of the nuclei (or special condition). Please note that, classical nuclei reslts includes full trajectory, while PIMD results includes only the centroid at 1000 steps to keep data size minimum. Training dataset made with CP2K and MACE training files: Training.tar.gz Phonon calculations using MLIP within ASE: phonon_calculations_with_ase.tar.gz Phonon calculations usind DFT and Phonopy in CP2K: Phonon_calculations_with_CP2K_and_Phonopy.tar.gz Deuterium Calculations: Hhh_MoS2_Classical_Deuterium.tar.gz and Hhh_MoS2_PIMD_deuterium.tar.gz Simulations of high-symmetry stackings: H_h^h Hhh_MoS2_Classical.tar.gz Hhh_MoS2_PIMD.tar.gz H_h^X HhX_MoS2_Classical.tar.gz HhX_MoS2_PIMD.tar.gz R_h^M RhM_MoS2_Classical.tar.gz RhM_MoS2_PIMD.tar.gz 3R 3R_MoS2_Classical.tar.gz 3R_MoS2_PIMD.tar.gz Geometry optimization of all high-symmetry stackings using MLIP: geometry_relaxation.tar.gz Simulations of high-angle twisted bilayers; 21.79 Moire_21-79_MoS2_Classical.tar.gz Moire_21-79_MoS2_PIMD.tar.gz 38.21 Moire_38-21_MoS2_Classical.tar.gz Moire_38-21_MoS2_PIMD.tar.gz Simulations of low-angle twisted bilayers; 3.89: TBL_3-89_all_tests.tar.gz Staggered Eclipsed Soliton 56.11: TBL_MoS2_56-11-all_tests.tar.gz Staggered 1 Staggered 2 Eclipse Soliton TBL_MoS2_3-89_PIMD.tar.gz TBL_MoS2_56-11_PIMD.tar.gz Simulations for the convergence of beads (please refer to 1 bead and 16 beads previously): 4 beads: Hhh_MoS2_4_beads.tar.gz 8 beads: Hhh_MoS2_8_beads.tar.gz Supporting simulations if the twisted bilayers fixed: Moire_3-89_fixed.tar.gz Moire_56-11_fixed.tar.gz

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