Linear Scaling Hierarchical Integration Scheme for the Exchange-Correlation Term in Molecular and Periodic Systems

An adaptive numerical integration scheme for efficient evaluation of the exchange-correlation term using localized basis functions and atom-centered grids is presented. The method treats molecules and systems with periodic boundary conditions on an equal footing. Its computational efficiency and O(N) scaling with the system size is achieved by a hierarchical spatial grouping of basis functions and grid points using an octree. This allows for an efficient screening of negligible contributions and an optimal use of hardware-optimized matrix matrix multiplication subroutines, such as BIAS. The implementation of the method within the TURBOMOLE program package demonstrates consistent accuracy and efficiency across molecular and periodic systems.