SlaKoNet: ML-based Slater-Koster Model

• SlaKoNet is a machine-learned tight-binding model that replaces conventional analytic functions with neural network predictors within the Slater–Koster framework.
• It employs smooth cutoff functions and a minimal non-orthogonal spd basis to ensure continuity and computational efficiency for molecular dynamics and defect studies.
• Trained on a diverse set of materials, SlaKoNet achieves a bandgap MAE of roughly 0.8 eV, enabling high-throughput screening for semiconductor and insulator applications.

SlaKoNet is a tight-binding (TB) formalism built on the two-center Slater–Koster (SK) framework, in which the conventional analytic functions for radial hopping integrals and on-site energies are replaced by machine-learned predictors. Each SK Hamiltonian matrix element between orbital $u$ on atom $i$ and orbital $v$ on atom $j$ at interatomic vector $\mathbf{R}_{ij}$ is computed as

$$H_{i u,\,j v}(R) = \sum_{\alpha,\beta} V_{\alpha\beta}(R) S_{i u,\,j v}^{\,\alpha\beta}(\hat{\mathbf{R}})$$

where $S^{\alpha\beta}(\hat{\mathbf{R}})$ are the SK angular functions (e.g. $ss\sigma$, $sp\sigma$, $pp\pi$, etc.) and $i$0 are the radial hopping integrals, predicted by a feed-forward neural network as a function of interatomic separation and, in principle, local coordination. SlaKoNet targets efficient, transferable predictions of bandgaps for large-scale electronic structure calculations, especially in the context of semiconductor and insulator materials (Park et al., 25 Nov 2025).

1. Functional Formulation and Parametrization

SlaKoNet adopts the standard two-center Slater–Koster form for constructing the TB Hamiltonian. The central innovation is the prediction of each $i$1 radial integral by a dedicated neural network, implemented in PyTorch, incorporating interatomic distance $i$2 (and optionally coordination) as inputs. Likewise, on-site energy parameters $i$3 are machine-learned analogously.

Each $i$4 is multiplied by a smooth, differentiable cutoff function $i$5: $i$6 This guarantees continuity and differentiability of matrix elements with respect to $i$7, which is essential for molecular dynamics and defect studies. Precise values of $i$8 and $i$9 are material-dependent and specified in primary SlaKoNet sources.

2. Basis Set and Overlap Treatment

The basis set comprises non-orthogonal atomic $v$0 orbitals, using a single radial function per angular momentum channel. The radial form and “confinement” (cutoff) radii are derived from DFTB-style Slater-type orbitals but are re-optimized as part of the network training. Overlap matrices $v$1 are computed via the SK angular projections in this basis, and then scaled by a learned on-site shift. No orthogonalization schemes (such as Löwdin transformations) are applied beyond this scaling in the implementation described in (Park et al., 25 Nov 2025).

3. Training Protocols and Optimization Strategies

SlaKoNet is trained on a set of 50+ materials curated from the JARVIS-DFT dataset, spanning elemental metals (e.g., Al, Cu, Au, Ti, Ni), canonical semiconductors (e.g., Si, Ge, GaAs, SiC, ZnO), and wide-gap insulators (e.g., MgO, Al$v$2O$v$3, BN). The primary training target is the bandgap computed from meta-GGA DFT (TB-mBJ values) from JARVIS-DFT, ensuring that the parametrization is not merely fitted to semi-local functionals (e.g., PBE, OptB88vdW) but reflects higher-level electronic structure.

The objective (loss) function is the mean-absolute error (MAE) of the predicted bandgap across the training set: $v$4 with the possibility of enhanced weighting for bands near the Fermi level. Optimization is conducted using the Adam algorithm (gradient-based) in PyTorch, supporting both GPU and CPU training.

4. Quantitative Performance and Comparative Benchmarks

SlaKoNet’s performance is evaluated relative to experiment, DFT (OptB88vdW), and meta-GGA bandgaps (TB-mBJ). The following metrics were reported for 48–51 materials (Park et al., 25 Nov 2025):

Reference	MAE (eV)	RMSE (eV)	Pearson $v$5	$v$6
Experiment	0.81	≃1.18	0.86	0.74
OptB88vdW (OPT-DFT)	1.46	–	–	–
TB-mBJ (meta-GGA DFT)	0.76	–	–	–

Notably, >60% of materials have bandgaps within ±1.0 eV of experiment. By comparison, alternative TB models such as TB3PY and PTBP exhibit higher MAE values (1.11 eV and 1.33 eV, respectively). No bulk modulus data are available for SlaKoNet in (Park et al., 25 Nov 2025).

SlaKoNet underperforms primarily for ultra-wide-gap oxides (e.g., MgO, Al$v$7O$v$8, SiO$v$9), with errors often exceeding 1.5 eV, likely due to the non-trivial dependence of conduction bands on coordination in these cases.

5. Transferability, Limitations, and Scope

SlaKoNet’s accuracy is maintained at ∼0.8 eV MAE across diverse semiconductors and moderate-gap insulators. For ultra-wide-gap materials (bandgaps >7 eV) or those with sensitive conduction band alignment, outliers persist.

There are no direct benchmarks for metallic systems within (Park et al., 25 Nov 2025). A plausible implication is that SlaKoNet, unless specifically retrained, may not predict Fermi-level crossings or metallic bandwidths reliably. Similarly, mechanical properties (bulk modulus, elastic constants) are not in the training set, and the transferability of SlaKoNet to systems involving stress, strain, or lattice deformations is unverified in this benchmark.

6. Computational Efficiency and Practical Implications

SlaKoNet retains the computational efficiency of TB models, scaling as $j$0 with respect to system size. The core cost per matrix element is boosted only by the feed-forward neural network evaluation (per unique atom pair and separation), allowing GPU acceleration. This enables %%%%28$i$129%%%%–10$j$3x speedup relative to plane-wave DFT for systems with $j$4–$j$5 atoms.

The achieved accuracy (∼0.8 eV MAE) is suitable for high-throughput band-alignment and device screening, while insufficient precision for applications such as deep-UV optoelectronics (where a ±1 eV error on a 6 eV gap is significant) is a limitation.

7. Prospects for Extension and Application Domains

SlaKoNet’s architectural paradigm—machine-learning Slater–Koster parameters using a neural network—provides a pathway for further retraining to include defects, surfaces, strained structures, or additional properties. This suggests the model could be tailored to new device-relevant domains where large supercells and configurational diversity preclude DFT. The smooth cutoff and spd minimal basis facilitate integration into molecular dynamics or device transport frameworks where $j$6 scaling and differentiability are essential.

SlaKoNet’s deployment within the benchmarking framework of the CHIPS-TB project situates it among the leading machine-learned TB variants for semiconductors and selected insulators, establishing a baseline for future improvements and extensions (Park et al., 25 Nov 2025).