Efficient hybrid-functional-based force and stress calculations for periodic systems with thousands of atoms

We present an efficient linear-scaling algorithm for evaluating the analytical force and stress contributions derived from the exact-exchange energy, a key component in hybrid functional calculations. The algorithm, working equally well for molecular and periodic systems, is formulated within the framework of numerical atomic orbital (NAO) basis sets and takes advantage of the localized resolution-of-identity (LRI) technique for treating the two-electron Coulomb repulsion integrals. The linear-scaling behavior is realized by fully exploiting the sparsity of the expansion coefficients resulting from the strict locality of the NAOs and the LRI ansatz. Our implementation is massively parallel, and enables efficient structural relaxation based on hybrid density functionals for bulk materials containing thousands of atoms. In this work, we will present a detailed description of our algorithm and benchmark the performance of our implementation using illustrating examples. By optimizing the structures of the pristine and doped halide perovskite material CsSnI$_3$ with different functionals, we find that in the presence of lattice strain, hybrid functionals provide a more accurate description of the stereochemical expression of the lone pair.