Submatrix and GPU-accelerated implementation of density matrix tight-binding

Effective single-particle theories, such as Hartree-Fock, density functional theory, and tight-binding, are limited by the computational cost of the self-consistent field (SCF) procedure, which typically scales cubically with the system size. This makes large-scale applications impractical without specialized algorithms and hardware. Here, we present the submatrix and graphical processing unit (GPU)-accelerated software implementation of the PTB tight-binding potential, realized in the open-source ptb codebase [M. Mueller, A. Katbashev, and S. Ehlert (2025). "grimme-lab/ptb: v3.8.1," Zenodo. https://zenodo.org/records/17015872]. We first benchmark a traditional diagonalization-based SCF solver against density-matrix-based purification approaches, systematically varying both system size and computer hardware. Our findings show that the usage of GPUs permits shifting the boundaries to much larger systems than previously thought feasible, achieving an overall 10-15-fold performance speedup. Second, we introduce the implementation of a decomposition-type submatrix method, specifically designed for efficient operation on mid- to large-sized systems, to address the computational overhead associated with full-system diagonalization. We demonstrate that, from a certain dimension (≈104 basis functions) on, our submatrix method reduces the overall computational cost while maintaining acceptable numerical accuracy. Our study demonstrates the significance of the interplay between modern hardware, algorithmic considerations, and novel tight-binding methods, paving the way for further development in this direction.