Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS2 via Machine-Learning-Enhanced Path-Integral Simulations

Ismail Eren; Ege Yigit Erbil; Maria-Judith Caisachana-Lozada; Hossein Mirhosseini; Thomas D. Kühne; Agnieszka B. Kuc

doi:10.48550/arXiv.2606.25991

Paper details

Research paper

Computational MLIPComputed PhononComputational AimdComputational Multiscale

Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS₂ via Machine-Learning-Enhanced Path-Integral Simulations

Ismail Eren, Ege Yigit Erbil, Maria-Judith Caisachana-Lozada, Hossein Mirhosseini et al.

arXiv·2026·arXiv:2606.25991

Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS2 via Machine-Learning-Enhanced Path-Integral Simulations | Matter42 Literature

Materials

MoS₂

Studied Material

studied material

Hydrogen

Reactant Species

Also known as

protiumH

reactant species

Deuterium

Reactant Species

Also known as

deuteriumD

reactant species

Characterization

Calculated Phonon

Calculated PhononCalculated

Conditions

purposevalidation of MLIP phonon band structures against DFT reference calculations

Figure References

Figure 1

Properties

Type	Label	Value	Material	Sample	Condition	Source
Free Energy Barrier	H diffusion barrier	122meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	64meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	45meV	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	18meV	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	85meV	MoS₂ H	Bulk RM_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	58meV	MoS₂ H	Bulk RM_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	87meV	MoS₂ H	Bulk 3R-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	35meV	MoS₂ H	Bulk 3R-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	60meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	D diffusion barrier	140meV	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	D diffusion barrier	95meV	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	11.410⁻³ cm2s−1	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Self Diffusion Coefficient	quantum correction factor	9.43dimensionless	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	0.19810⁻³ cm2s−1	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	1.8710⁻³ cm2s−1	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	D diffusion coefficient	0.07110⁻³ cm2s−1	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	D diffusion coefficient	0.40410⁻³ cm2s−1	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Twisted bilayer MoS₂ with twist angle 21.79 degrees.		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Twisted bilayer MoS₂ with twist angle 38.21 degrees.		—
Ir Band Position	IR band	2570cm⁻¹	MoS₂	—	—	—
Ir Band Position	IR band	1845cm⁻¹	MoS₂	—	—	—

Simulations

DFT

r2SCAN+rVV10 density-functional-theory reference calculations used to generate the training dataset and validate MLIP energetics and phonons

Settings

purposetraining_data_and_reference_validation

functionalr2SCAN+rVV₁₀

system familyMoS₂ polytypes and twisted bilayers

validation targetsrelative energetics, phonon band structures

Materials

MoS₂

Studied Material

studied material

Hydrogen

Reactant Species

Also known as

protiumH

reactant species

Deuterium

Reactant Species

Also known as

deuteriumD

reactant species

Characterization

Calculated Phonon

Calculated PhononCalculated

Conditions

purposevalidation of MLIP phonon band structures against DFT reference calculations

Figure References

Figure 1

Properties

Type	Label	Value	Material	Sample	Condition	Source
Free Energy Barrier	H diffusion barrier	122meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	64meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	45meV	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	18meV	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	85meV	MoS₂ H	Bulk RM_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	58meV	MoS₂ H	Bulk RM_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	87meV	MoS₂ H	Bulk 3R-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	35meV	MoS₂ H	Bulk 3R-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	60meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	D diffusion barrier	140meV	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	D diffusion barrier	95meV	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	11.410⁻³ cm2s−1	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Self Diffusion Coefficient	quantum correction factor	9.43dimensionless	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	0.19810⁻³ cm2s−1	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	1.8710⁻³ cm2s−1	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	D diffusion coefficient	0.07110⁻³ cm2s−1	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	D diffusion coefficient	0.40410⁻³ cm2s−1	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Twisted bilayer MoS₂ with twist angle 21.79 degrees.		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Twisted bilayer MoS₂ with twist angle 38.21 degrees.		—
Ir Band Position	IR band	2570cm⁻¹	MoS₂	—	—	—
Ir Band Position	IR band	1845cm⁻¹	MoS₂	—	—	—

Simulations

DFT

r2SCAN+rVV10 density-functional-theory reference calculations used to generate the training dataset and validate MLIP energetics and phonons

Settings

purposetraining_data_and_reference_validation

functionalr2SCAN+rVV₁₀

system familyMoS₂ polytypes and twisted bilayers

validation targetsrelative energetics, phonon band structures

Materials

MoS₂

Studied Material

studied material

Hydrogen

Reactant Species

Also known as

protiumH

reactant species

Deuterium

Reactant Species

Also known as

deuteriumD

reactant species

Characterization

Calculated Phonon

Calculated PhononCalculated

Conditions

purposevalidation of MLIP phonon band structures against DFT reference calculations

Figure References

Figure 1

Properties

Type	Label	Value	Material	Sample	Condition	Source
Free Energy Barrier	H diffusion barrier	122meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	64meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	45meV	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	18meV	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	85meV	MoS₂ H	Bulk RM_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	58meV	MoS₂ H	Bulk RM_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	87meV	MoS₂ H	Bulk 3R-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	35meV	MoS₂ H	Bulk 3R-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	60meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	D diffusion barrier	140meV	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	D diffusion barrier	95meV	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	11.410⁻³ cm2s−1	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Self Diffusion Coefficient	quantum correction factor	9.43dimensionless	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	0.19810⁻³ cm2s−1	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	1.8710⁻³ cm2s−1	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	D diffusion coefficient	0.07110⁻³ cm2s−1	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	D diffusion coefficient	0.40410⁻³ cm2s−1	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Twisted bilayer MoS₂ with twist angle 21.79 degrees.		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Twisted bilayer MoS₂ with twist angle 38.21 degrees.		—
Ir Band Position	IR band	2570cm⁻¹	MoS₂	—	—	—
Ir Band Position	IR band	1845cm⁻¹	MoS₂	—	—	—

Simulations

DFT

r2SCAN+rVV10 density-functional-theory reference calculations used to generate the training dataset and validate MLIP energetics and phonons

Settings

purposetraining_data_and_reference_validation

functionalr2SCAN+rVV₁₀

system familyMoS₂ polytypes and twisted bilayers

validation targetsrelative energetics, phonon band structures

Materials

MoS₂

Studied Material

studied material

Hydrogen

Reactant Species

Also known as

protiumH

reactant species

Deuterium

Reactant Species

Also known as

deuteriumD

reactant species

Characterization

Calculated Phonon

Calculated PhononCalculated

Conditions

purposevalidation of MLIP phonon band structures against DFT reference calculations

Figure References

Figure 1

Properties

Type	Label	Value	Material	Sample	Condition	Source
Free Energy Barrier	H diffusion barrier	122meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	64meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	45meV	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	18meV	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	85meV	MoS₂ H	Bulk RM_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	58meV	MoS₂ H	Bulk RM_h-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	87meV	MoS₂ H	Bulk 3R-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	35meV	MoS₂ H	Bulk 3R-MoS₂ stacking used for diffusion calculations.		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	60meV	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	D diffusion barrier	140meV	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	D diffusion barrier	95meV	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	11.410⁻³ cm2s−1	MoS₂ H	Bulk HX_h-MoS₂ stacking used for diffusion calculations.		—
Self Diffusion Coefficient	quantum correction factor	9.43dimensionless	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	0.19810⁻³ cm2s−1	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	H diffusion coefficient	1.8710⁻³ cm2s−1	MoS₂ H	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	D diffusion coefficient	0.07110⁻³ cm2s−1	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Self Diffusion Coefficient	D diffusion coefficient	0.40410⁻³ cm2s−1	MoS₂ D	Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion,		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Twisted bilayer MoS₂ with twist angle 21.79 degrees.		—
Free Energy Barrier	H diffusion barrier	120meV	MoS₂ H	Twisted bilayer MoS₂ with twist angle 38.21 degrees.		—
Ir Band Position	IR band	2570cm⁻¹	MoS₂	—	—	—
Ir Band Position	IR band	1845cm⁻¹	MoS₂	—	—	—

Simulations

DFT

r2SCAN+rVV10 density-functional-theory reference calculations used to generate the training dataset and validate MLIP energetics and phonons

Settings

purposetraining_data_and_reference_validation

functionalr2SCAN+rVV₁₀

system familyMoS₂ polytypes and twisted bilayers

validation targetsrelative energetics, phonon band structures

Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS₂ via Machine-Learning-Enhanced Path-Integral Simulations

Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS₂ via Machine-Learning-Enhanced Path-Integral Simulations

Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS₂ via Machine-Learning-Enhanced Path-Integral Simulations

Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS₂ via Machine-Learning-Enhanced Path-Integral Simulations

Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS₂ via Machine-Learning-Enhanced Path-Integral Simulations

Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS₂ via Machine-Learning-Enhanced Path-Integral Simulations

Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS₂ via Machine-Learning-Enhanced Path-Integral Simulations

Capturing Nuclear Quantum Effects in Hydrogen Diffusion through MoS₂ via Machine-Learning-Enhanced Path-Integral Simulations

Samples

Bulk Hhh-MoS₂ stacking used for MLIP validation, diffusion, and isotope-effect calculations.

Bulk HXh-MoS₂ stacking used for diffusion calculations.

Bulk RMh-MoS₂ stacking used for diffusion calculations.

Bulk 3R-MoS₂ stacking used for diffusion calculations.

Twisted bilayer MoS₂ with twist angle 3.89 degrees.

Twisted bilayer MoS₂ with twist angle 21.79 degrees.

Twisted bilayer MoS₂ with twist angle 38.21 degrees.

Twisted bilayer MoS₂ with twist angle 56.11 degrees.

Materials

Characterization

Properties

Simulations

Why these are connected

Samples

Bulk Hhh-MoS₂ stacking used for MLIP validation, diffusion, and isotope-effect calculations.

Bulk HXh-MoS₂ stacking used for diffusion calculations.

Bulk RMh-MoS₂ stacking used for diffusion calculations.

Bulk 3R-MoS₂ stacking used for diffusion calculations.

Twisted bilayer MoS₂ with twist angle 3.89 degrees.

Twisted bilayer MoS₂ with twist angle 21.79 degrees.

Twisted bilayer MoS₂ with twist angle 38.21 degrees.

Twisted bilayer MoS₂ with twist angle 56.11 degrees.

Materials

Characterization

Properties

Simulations

Why these are connected

Samples

Bulk Hhh-MoS₂ stacking used for MLIP validation, diffusion, and isotope-effect calculations.

Bulk HXh-MoS₂ stacking used for diffusion calculations.

Bulk RMh-MoS₂ stacking used for diffusion calculations.

Bulk 3R-MoS₂ stacking used for diffusion calculations.

Twisted bilayer MoS₂ with twist angle 3.89 degrees.

Twisted bilayer MoS₂ with twist angle 21.79 degrees.

Twisted bilayer MoS₂ with twist angle 38.21 degrees.

Twisted bilayer MoS₂ with twist angle 56.11 degrees.

Materials

Characterization

Properties

Simulations

Why these are connected

Samples

Bulk Hhh-MoS₂ stacking used for MLIP validation, diffusion, and isotope-effect calculations.

Bulk HXh-MoS₂ stacking used for diffusion calculations.

Bulk RMh-MoS₂ stacking used for diffusion calculations.

Bulk 3R-MoS₂ stacking used for diffusion calculations.

Twisted bilayer MoS₂ with twist angle 3.89 degrees.

Twisted bilayer MoS₂ with twist angle 21.79 degrees.

Twisted bilayer MoS₂ with twist angle 38.21 degrees.

Twisted bilayer MoS₂ with twist angle 56.11 degrees.

Materials

Characterization

Properties

Simulations

Why these are connected

Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion, and isotope-effect calculations.

Bulk HX_h-MoS₂ stacking used for diffusion calculations.

Bulk RM_h-MoS₂ stacking used for diffusion calculations.

Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion, and isotope-effect calculations.

Bulk HX_h-MoS₂ stacking used for diffusion calculations.

Bulk RM_h-MoS₂ stacking used for diffusion calculations.

Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion, and isotope-effect calculations.

Bulk HX_h-MoS₂ stacking used for diffusion calculations.

Bulk RM_h-MoS₂ stacking used for diffusion calculations.

Bulk Hh_h-MoS₂ stacking used for MLIP validation, diffusion, and isotope-effect calculations.

Bulk HX_h-MoS₂ stacking used for diffusion calculations.

Bulk RM_h-MoS₂ stacking used for diffusion calculations.