12. Methods for Hartree-Fock and Density Functional Theory Electronic Structure Calculations with Linearly Scaling Processor Time and Memory Usage

We discuss algorithms that can be used to calculate electron densities using computer resources – memory and processor time – that increase only linearly with system size. We focus on the Hartree-Fock and density functional theories and calculations using Gaussian basis sets. However, many of the approaches discussed here are applicable also for other methods and for any local basis. Particular attention is directed towards error control and how to avoid the use of the ad-hoc selected parameters and threshold values often associated with computational approximations employed to reach linear scaling. The discussed aspects include multipole methods, linear scaling computation of the Hartree-Fock exchange and density functional theory exchange-correlation matrices, hierarchic representation of sparse matrices, and density matrix purification. The article also describes how these different parts are put together to achieve linear scaling for the entire Hartree-Fock or density functional theory calculation, controlling errors in the self-consistent field procedure by considering rotations of the occupied subspace.