Implementation of the hybrid exchange-correlation functionals in the siesta code

We present an efficient and accurate implementation of hybrid exchange-correlation (XC) functionals in the SIESTA code, enabling large-scale simulations based on Hartree-Fock-type exact exchange combined with strictly localized numerical atomic orbitals (NAOs). Our approach exploits a fitted representation of the NAOs in terms of Gaussian-type orbitals (GTOs), which allows for the analytical evaluation of four-center electron repulsion integrals (ERIs) via the LIBINT library. This framework is seamlessly integrated with SIESTA's real-space grid and sparse-matrix infrastructure, and is combined with multiple screening techniques to control the computational complexity. We also introduce a fully analytical formulation of hybrid-functional forces and a dynamic parallel distribution scheme that ensures excellent scalability. We validate our implementation through benchmark calculations on a broad set of systems (including semiconductors, insulators, and two-dimensional materials) and demonstrate that the HSE06 functional significantly improves the prediction of band gaps compared to PBE, in close agreement with G0W0 and experimental data. We analyze in detail the trade-offs between accuracy and computational efficiency as a function of the number of Gaussians, basis set range, and integral screening thresholds. Our results confirm that hybrid functional calculations in SIESTA are now feasible for large extended systems, making accurate first-principles predictions of electronic and structural properties accessible at scale.