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Virtual materials design requires not only the simulation of a huge number of systems, but also of systems with ever larger sizes and through increasingly accurate models of the electronic structure. These can be provided by density functional theory (DFT) using not only simple local approximations to the unknown exchange and correlation functional, but also more complex approaches such as hybrid functionals, which include some part of Hartree–Fock exact exchange. While hybrid functionals allow many properties such as lattice constants, bond lengths, magnetic moments and band gaps, to be calculated with improved accuracy, they require the calculation of a nonlocal potential, resulting in high computational costs, that scale rapidly with the system size. This limits their wide application. Here, we present a new highly-scalable implementation of the nonlocal Hartree-Fock-type potential into FLEUR—an all-electron electronic structure code that implements the full-potential linearized augmented plane-wave (FLAPW) method. This implementation enables the use of hybrid functionals for systems with several hundred atoms. By porting this algorithm to GPU accelerators, we can leverage future exascale supercomputers which we demonstrate by reporting scaling results for up to 64 GPUs and up to 12,000 CPU cores for a single k-point. As proof of principle, we apply the algorithm to large and complex iron garnet materials (YIG, GdIG, TmIG) that are used in several spintronic applications.

We present a general method for solving the modified Helmholtz equation without shape approximation for an arbitrary periodic charge distribution, whose solution is known as the Yukawa potential or the screened Coulomb potential. The method is an extension of Weinert’s pseudo-charge method [Weinert M, J Math Phys, 1981, 22:2433–2439] for solving the Poisson equation for the same class of charge density distributions. The inherent differences between the Poisson and the modified Helmholtz equation are in their respective radial solutions. These are polynomial functions, for the Poisson equation, and modified spherical Bessel functions, for the modified Helmholtz equation. This leads to a definition of a modified pseudo-charge density and modified multipole moments. We have shown that Weinert’s convergence analysis of an absolutely and uniformly convergent Fourier series of the pseudo-charge density is transferred to the modified pseudo-charge density. We conclude by illustrating the algorithmic changes necessary to turn an available implementation of the Poisson solver into a solver for the modified Helmholtz equation.