Machine learned force-fields for an Ab-initio quality description of metal-organic frameworks

The study addresses the need for accurate yet computationally efficient modeling of metal-organic frameworks (MOFs), whose large unit cells and dynamical properties challenge ab-initio methods like DFT. Classical transferable force fields (e.g., Dreiding, UFF/UFF4MOF) are fast but often inaccurate for dynamical and vibrational properties. System-specific force fields (e.g., MOF-FF, QuickFF) can be accurate but are cumbersome to parameterize and may lack flexibility. Recent machine-learning potentials (MLPs), especially neural-network potentials, have shown promise but often require large DFT datasets and can be less efficient than classical force fields. The research questions are: how to most efficiently generate MLPs for MOFs; how to parameterize them with minimal DFT effort; how they perform for elastic and phonon-related properties; and what computational cost they entail relative to traditional force fields. The proposed solution combines on-the-fly active learning in VASP (kernel-based GAP-type potentials) to generate high-value reference data with training of both VASP MLPs and moment tensor potentials (MTPs), aiming for near-DFT accuracy at high efficiency.

- Transferable force fields (Dreiding, UFF, UFF4MOF) are convenient and broadly applicable but yield sizable errors for MOF dynamical properties; examples include poor elastic properties for MIL-53 and thermal conductivity for MOF-5 (UFF4MOF overestimates by factor ~2.6). - More specialized potentials (GAFF, COMPASS for organics; BTW-FF for MOFs) improve some aspects but struggle with accurate vibrational properties. - System- or fragment-specific force fields (MOF-FF, QuickFF) can achieve high precision but require careful selection of potential terms, reference data, and fitting procedures; ReaxFF adds flexibility but can suffer from energy conservation and stability issues. - Machine-learned potentials are increasingly successful across materials. In MOFs, neural network potentials have modeled thermal conductivity and diffusion but often need large DFT datasets; GPU-enabled implementations (GPUMD, DeePMD-kit) offer speed. - GAPs and MTPs are well-studied in conventional materials, but systematic evaluation for MOFs has been lacking. Active learning and efficient phase-space sampling can significantly reduce reference data needs. - The work builds on VASP’s kernel-based on-the-fly active learning approach and MLIP’s MTPs, leveraging their complementary strengths.

- Systems: MOF-5 (Zn4O-bdc, Fm-3m, 106 atoms), UiO-66 (Zr, Fm-3m, 114 atoms), MOF-74 (Zn, R-3, 54 atoms), MIL-53(Al) large-pore (Imma) and narrow-pore (Cc) phases (76 atoms). Closed-shell metals chosen to avoid spin complications. - Reference data generation (active learning): VASP on-the-fly MLP during MD in NPT (Parrinello-Rahman barostat, Langevin thermostat), time step 0.5 fs. Temperature ramp 50→900 K over 50,000 steps (25 ps). Bayesian force error threshold triggers DFT single-point (energies, forces, stresses) and addition of local reference configurations; threshold adapts to maintain efficiency across temperatures. Optional extended dataset: additional NPT MD at 300 K for 100,000 steps with fixed threshold 0.02 eV Å^-1. - Dataset sizes (initial/final reference structures): MOF-5 974/1110; UiO-66 739/1373; MOF-74 998/2549; MIL-53(lp) 783/2832; MIL-53(np) 827/3073. - VASP MLP training: After active learning, reselection of local reference configurations (VASP ≥6.4.1, ML_MODE=select), increased cutoff for radial descriptors to 8 Å (5 Å during AL), CUR sparsification (10^-11 for refits), cap of 1500 local references per atom type (3000 during AL). Production refit with SVD/Tikhonov regularization and error prediction disabled (ML_MODE=refit) for speed. Atom-type schemes: both element-only and chemically distinct environments tested; default used full separation. - MTP training: Same DFT reference data used to avoid reference bias. MLIP package with BFGS minimization of a weighted cost function over energies, forces, and stresses (weights: forces 0.01, energies 1.0, stresses 1.0; stress weight increased from default). Baseline MTP level 22, radial basis size 10, cutoff 5 Å; levels 10–24 explored for speed/accuracy tradeoffs. Both element-only and environment-separated atom types examined. Multiple random initializations; selection based on best performance on an independent validation set and thermal stability. - Validation set: Independent VASP active-learning MD at 300 K with 100 DFT-labeled configurations per system. - DFT settings: VASP, PBE functional with Grimme D3(BJ) dispersion, 900 eV plane-wave cutoff, SCF convergence 1e-6 eV, ionic forces ≤1e-3 eV Å^-1; system-specific k-meshes. - Benchmarks computed: 0 K unit cells vs DFT; 300 K NPT cell volumes vs experiment; energies/forces/stresses vs DFT on validation set; phonon dispersions and DOS (phonopy finite differences, carefully converged supercells); elastic stiffness tensors (finite strain, stress–strain relations); thermal expansion (NTP MD 100–700 K); thermal conductivity for MOF-5 (NEMD and AEMD) with finite-size extrapolation. - Efficiency tests: Large supercells (MOF-5: 27,136 atoms; MOF-74: 31,104 atoms) in NPT at 300 K; hardware: 64-core AMD EPYC 7713 node; compared MTPs at multiple levels, VASP MLP variants, MOF-FF, UFF4MOF, Dreiding.

- Unit cells: Near-DFT accuracy for volumes and lattice parameters. Isotropic MOFs (MOF-5, UiO-66): <0.2% volume deviation (MTP and VASP MLP). Anisotropic MOFs (MOF-74, MIL-53): ~0.3%. UFF4MOF errors up to ~14%. - Experimental volumes at ~300 K: MTPs and VASP MLPs within <2% of averaged experimental values for most systems; MIL-53(np) underestimated by ~5% amid ±4% experimental spread. UFF4MOF performs much worse. - Energies/forces/stresses (validation at 300 K): • Energies: MTP and VASP MLP deviations typically <1% of absolute values; for MOF-5, max differences ~0.3 meV/atom (below DFT convergence of 1 meV/atom). UFF4MOF deviations up to ~50 meV/atom. • Forces (MOF-5 RMSD): MTP 0.02 eV Å^-1; VASP MLP 0.02 eV Å^-1; UFF4MOF 1.03 eV Å^-1; Dreiding 2.13 eV Å^-1; system-specific MOF-FF 0.09 eV Å^-1. • Stresses: MTP slightly better than VASP MLP (higher stress weight in fitting); UFF4MOF errors roughly two orders of magnitude larger. • Using extended reference data improved VASP MLP for MOF-74 (force RMSD 0.27→0.22 eV Å^-1; stress RMSD 0.17→0.12 kbar); minor impact on MTPs. - Vibrational properties: • Phonon band structures (low-frequency): MTPs and VASP MLPs reproduce DFT nearly perfectly; e.g., MOF-5 deviations <5 cm^-1. UFF4MOF significantly overestimates acoustic dispersions and optical frequencies. • Γ-point frequency RMSDs (cm^-1), full/≤200 cm^-1 (Table 2): – MOF-5: MTP 3.3/1.6; VASP MLP 9.0/1.5; MOF-FF 14.1/7.8; UFF4MOF 203.2/51.0. – UiO-66: MTP 3.7/1.6; VASP MLP 10.4/2.7. – MOF-74: MTP 3.1/1.4; VASP MLP 4.2/1.1; UFF4MOF 242.8/116.9. – MIL-53(lp): MTP 5.2/2.8; VASP MLP 8.8/3.2. MIL-53(np): MTP 7.6/3.9; VASP MLP 6.9/2.5. • Phonon DOS: ML potentials closely match DFT across low and full frequency ranges; UFF4MOF shows poor agreement and underestimates the large frequency gap (50–90 THz). - Elastic constants: MTPs nearly match DFT (max deviations ~1 GPa); VASP MLP slightly worse for anisotropic MOF-74 but improved with extended data. UFF4MOF performs poorly (e.g., MOF-74 C33 >5× DFT), though bulk modulus of MOF-5 appears incidentally close due to error cancellation. - Thermal conductivity (MOF-5, 300 K): MTP (level 18) yields 0.32 W m^-1 K^-1 via NEMD/AEMD, matching experiment (0.32 W m^-1 K^-1). Traditional force fields: UFF4MOF 0.847, Dreiding 1.102, MOF-FF 0.29 W m^-1 K^-1. - Efficiency: • UFF4MOF/Dreiding fastest but very inaccurate (force RMSD >1 eV Å^-1). MOF-FF improves accuracy (0.09 eV Å^-1) but ~5× slower than simple transferable FFs. • MTPs: tuning level 24→10 gives ~28× speedup with modest accuracy loss (force RMSD 0.016→0.033 eV Å^-1); level-10 MTP ~2× cost of UFF4MOF yet ~34× lower force RMSD. • VASP MLPs: similar accuracy to mid/high-level MTPs but typically 2–3× slower on tested CPU setup; speed improves drastically (∼100×) when live error estimation is disabled in production refits; accuracy benefits from SVD refit and more local references; parallel scaling is a bottleneck. • Atom-type separation: beneficial for accuracy. In MTPs, added cost is negligible; in VASP MLPs, more atom types increase cost; benefits are system dependent (larger for MOF-74). - Stability: VASP MLPs stable up to 700 K (structures intact; rare unphysical cell deformations at very high T). Some MTPs showed long-term instability in flat energy landscapes (e.g., MIL-53); adding high-temperature training data (e.g., +1009 structures at 400 K) improved stability.

The study demonstrates that combining VASP’s on-the-fly active learning with MTP training provides a practical recipe to generate highly accurate and efficient force fields for MOFs. Using the same DFT reference data eliminates bias and enables a head-to-head comparison of kernel-based VASP MLPs and MTPs. Across structural (unit cells), energetic (energies, forces, stresses), vibrational (phonons, DOS), elastic, and transport (thermal conductivity) benchmarks, both ML potentials achieve near-DFT fidelity and vastly outperform transferable force fields. VASP MLPs are straightforward to obtain within the VASP ecosystem and exhibit strong stability; MTPs offer superior accuracy-to-cost tunability, enabling significant speedups while retaining high accuracy. The findings confirm that high-quality predictions of MOF dynamical properties—previously inaccessible with DFT due to cost or with classical FFs due to inaccuracy—are now achievable. The work also clarifies trade-offs: VASP MLP accuracy is sensitive to reference selection, refit settings, and atom-type choices; MTPs benefit from atom-type separation with minimal cost but may require additional high-temperature training for long-term stability. Discrepancies in thermal expansion highlight sensitivity to the underlying DFT reference and to anharmonic effects, guiding future refinements. Overall, the approach answers the posed questions by delivering an efficient parametrization strategy, quantifying performance on key properties, and mapping the accuracy–efficiency landscape relative to traditional force fields.

The paper provides an easy-to-implement roadmap for creating machine-learned force fields for MOFs with ab-initio quality at vastly reduced computational cost. Using VASP’s on-the-fly active learning to generate compact, informative reference sets and training both VASP MLPs and MTPs on the same data yields force fields with near-DFT accuracy for structural, vibrational, elastic, and transport properties. MTPs allow flexible tuning of speed versus accuracy and can approach the cost of simple transferable force fields while delivering orders-of-magnitude better accuracy; VASP MLPs show excellent stability and accuracy, particularly with modern refit workflows. Thermal conductivity of MOF-5 is reproduced quantitatively (0.32 W m^-1 K^-1), and phonon band structures and elastic tensors closely match DFT. Compared to UFF4MOF and Dreiding, errors are reduced by orders of magnitude. Future work should address: improving long-term stability of MTPs (e.g., enhanced sampling at high T or stability-enforcing terms), refining thermal expansion predictions (including exploring alternative DFT functionals and anharmonic treatments), expanding to more complex chemistries (e.g., open-shell metals), and incorporating reactive events and guest diffusion. With accessible software (VASP, MLIP, LAMMPS, phonopy) and provided settings, the presented strategy can be broadly adopted to elevate MOF simulations.

- Thermal expansion predictions show system-dependent discrepancies (e.g., MOF-74 pore direction overestimated vs experiment), reflecting challenges in modeling anharmonicity and sensitivity to DFT functional choices used for training data. - Some MTPs exhibited long-term instability in MD for systems with flat energy landscapes (notably MIL-53), requiring additional high-temperature training data; stability can vary across random initializations. - VASP MLP computational cost is impacted by parallel scaling and the number of atom types; live error estimation is expensive (though avoidable in production refits). - Scope limited to chemically stable, homogeneous materials with closed-shell metals; chemical reactions, guest diffusion/adsorption, and defects are not modeled. - Thermal transport benchmarking limited to MOF-5 due to availability of reliable single-crystal experimental data; NEMD/AEMD not available in VASP at the time. - Results may be influenced by the chosen DFT methodology (PBE+D3(BJ)); alternative functionals could change reference behaviors (e.g., thermal expansion sign in MOF-74).