Machine-learning interatomic potentials achieving CCSD(T) accuracy for systems with extended covalent networks and van der Waals interactions

1. Problem Statement

Machine-learning interatomic potentials (MLIPs) have revolutionized atomistic simulations by offering near-ab initio accuracy at a fraction of the computational cost. However, most existing MLIPs are trained on Density Functional Theory (DFT) data. DFT suffers from intrinsic errors in exchange-correlation functionals and often fails to capture long-range van der Waals (vdW) interactions accurately without semi-empirical corrections.

While Coupled Cluster with Single, Double, and perturbative Triple excitations (CCSD(T)) is the "gold standard" for chemical accuracy (1 kcal/mol) and inherently includes vdW interactions, its computational cost scales steeply ($O(N^7)$), restricting its application to small molecules.

• The Core Challenge: There is a lack of MLIPs trained on CCSD(T) data for systems with extended covalent networks (e.g., Covalent Organic Frameworks - COFs, polymers, MOFs).
• The Bottleneck: Generating CCSD(T) reference data for periodic systems is computationally prohibitive. Standard fragmentation strategies (cutting a periodic system into molecules) fail for extended networks because they introduce unpaired valence electrons, fundamentally altering the electronic structure.

2. Methodology

The authors propose a novel $\Delta$-learning strategy combined with a tight-binding baseline to train MLIPs with CCSD(T) accuracy for extended systems without requiring periodic CCSD(T) calculations.

A. Quantum-Chemical Reference Data

• Method: The reference energies are computed using PNO-LCCSD(T)-F12 (Pair Natural Orbital Local Coupled Cluster with Single, Double, and perturbative Triple excitations, augmented with F12 explicit correlation).
• Basis Set: Heavy-aug-cc-pVTZ (augmented with diffuse functions for non-hydrogen atoms).
• Key Features:
  • F12 Correction: Dramatically reduces basis-set incompleteness error, allowing the use of triple-$\zeta$ basis sets to approach the Complete Basis Set (CBS) limit.
  • All-Electron Treatment: Core electrons are correlated to ensure high accuracy for atomization energies and vibrational frequencies.
  • BSSE Handling: The combination of PNO and F12 methods significantly suppresses Basis Set Superposition Error (BSSE), rendering the Counterpoise (CP) correction unnecessary for the training data generation.

B. The $\Delta$-Learning Strategy

Instead of training an MLIP directly on total CCSD(T) energies, the authors decompose the energy:
$$E_{\text{CCSD(T)}} = E_{\text{GFN2-xTB}} + \Delta E$$

• Baseline: GFN2-xTB (a semi-empirical tight-binding method). It captures the bulk of the covalent bonding and includes a D4 dispersion correction for vdW interactions.
• Target: The MLIP (specifically a Moment Tensor Potential, MTP) is trained only on the energy difference ($\Delta E$) between the high-level CCSD(T) and the GFN2-xTB baseline.
• Advantage: Since GFN2-xTB already describes the local covalent network and long-range dispersion reasonably well, the $\Delta E$ is small and highly local. This allows the MLIP to be trained solely on molecular fragments (monomers, dimers, trimers) derived from the target periodic system, while remaining transferable to the full periodic structure.

C. Training Dataset Construction

• Target System: A prototypical quasi-2D Covalent Organic Framework (COF) composed of Carbon and Hydrogen ($C_{48}H_{30}$).
• Fragments: The training set includes:
  • Monomers (benzene, biphenyl, terphenyl, etc.) with hydrogen termination to avoid unpaired electrons.
  • Multimers (dimers, trimers, tetramers) to capture inter-molecular vdW interactions.
  • Dihydrogen ($H_2$) to model hydrogen absorption.
• Configuration Generation: Molecular dynamics (MD) simulations were performed using GFN2-xTB at 300 K to generate diverse configurations. Snapshots were selected for high-level single-point energy calculations.
• Dataset Sizes: Four training sets were created with increasing complexity (up to 5 benzene rings), with the largest set (#5) containing ~1,872 training configurations.

D. Model Architecture

• Potential: Moment Tensor Potential (MTP).
• Cutoff Radius: 7 Å.
• Optimization: Parameters were optimized using the BFGS algorithm, with a pre-optimization step (linear fitting at MTP level 2) to accelerate convergence.

3. Key Contributions

1. Methodological Breakthrough: Demonstrated that MLIPs trained on molecular fragments via $\Delta$-learning can achieve CCSD(T) accuracy for periodic systems with extended covalent networks, bypassing the need for periodic CCSD(T) calculations.
2. Validation of Transferability: Proved that a potential trained on small molecules and multimers can accurately predict properties of a complex 2D COF, including structural symmetry, binding energies, and vibrational spectra.
3. Benchmarking: Provided a rigorous comparison against DFT (PBE-D4), other MLIPs (ANI-1ccx), and high-level wavefunction methods, highlighting the limitations of semi-empirical corrections in DFT and the transferability issues of existing CCSD(T)-trained potentials (like ANI-1ccx) for non-standard systems.
4. Application to COFs: Applied the method to analyze the $C_{48}H_{30}$ COF, resolving structural ambiguities (e.g., layer twisting) and calculating hydrogen absorption energies with chemical accuracy.

4. Key Results

A. Accuracy Metrics

• Energy Errors: The best model ($\Delta$MTP#5, MTP level 20) achieved a Root Mean Square Error (RMSE) of < 0.4 meV/atom on both training and test sets (including larger molecules not seen in training).
• Electronic Total Atomization Energies (eTAEs):
  • For $H_2$ and $C_6H_6$, the TB+$\Delta$MTP model reproduced CCSD(T) values within 0.1 kcal/mol, matching experimental data.
  • In contrast, ANI-1ccx showed massive errors (e.g., -1328 kcal/mol for benzene) due to its training set lacking specific molecular environments and free-atom references.
• Bond Lengths: Predicted equilibrium bond lengths for $H_2$ and $C_6H_6$ deviated from experiment by < 0.002 Å, outperforming DFT and matching CCSD(T)-F12.
• Vibrational Frequencies: RMSEs were ~10 cm$^{-1}$ for $H_2$ and ~11 cm$^{-1}$ for benzene, comparable to CCSD(T)-F12 and significantly better than DFT or ANI-1ccx.

B. Inter-molecular Interactions

• Benzene Dimer: The model accurately reproduced the $\pi$-$\pi$ stacking interaction energy curve, matching reference CCSD(T)/CBS values with errors < 0.6 kcal/mol.
• Limitations of Alternatives: ANI-1ccx failed to capture dispersion interactions at long range (falling to zero beyond 5.6 Å) because its training set lacked multimers.

C. Application to $C_{48}H_{30}$ COF

• Structural Stability: The potential correctly identified that the fully eclipsed $P6/mmm$ structure is dynamically unstable (imaginary phonon modes) and relaxed to a twisted $C222$ structure, which is energetically favored by ~10 meV/atom.
• Geometric Parameters:
  • Inter-layer distance: 3.673 Å (vs. 3.35 Å for graphite), consistent with the sparse nature of COFs.
  • Node distance: 12.891 Å, matching experimental measurements (12.9–13.6 Å).
• Hydrogen Absorption: The model predicted an $H_2$ absorption energy of -0.9 kcal/mol at the optimal site, significantly weaker than GFN2-xTB (-1.1 kcal/mol) but consistent with the lower dispersion interaction expected in COFs compared to graphite.

5. Significance and Impact

• Chemical Accuracy for Large Systems: This work bridges the gap between high-accuracy quantum chemistry and large-scale materials simulation. It enables the study of complex, vdW-dominated materials (like COFs, MOFs, and polymers) with chemical accuracy (1 kcal/mol), a regime previously inaccessible for systems of this size.
• Efficiency: The computational cost of evaluating the TB+$\Delta$MTP potential is comparable to classical force fields and significantly lower than DFT, while retaining CCSD(T) fidelity.
• Generalizability: The workflow is transferable. By expanding the training set to include diverse chemical fragments, this approach can be applied to a wide range of materials, accelerating the discovery of new porous materials for gas storage, separation, and catalysis.
• Reliability: The use of "local extrapolation grades" allows users to quantify the uncertainty of predictions in periodic systems, ensuring the reliability of the MLIP when applied to new configurations.

In conclusion, the paper establishes a robust, practical route to performing large-scale atomistic simulations with gold-standard quantum chemical accuracy, overcoming the historical limitations of both DFT (accuracy) and CCSD(T) (cost/scalability).