O(N) methods in electronic structure calculations

TL;DR

This paper reviews linear scaling (O(N)) methods in electronic structure calculations, emphasizing their theoretical basis, recent developments, applications, and future challenges for simulating large systems efficiently.

Contribution

It provides a comprehensive survey of the theory, recent advances, and practical implementations of O(N) methods in electronic structure calculations.

Findings

Linear scaling methods enable simulations of larger systems.

Recent developments improve efficiency and applicability.

Challenges remain in implementation and accuracy.

Abstract

Linear scaling methods, or O(N) methods, have computational and memory requirements which scale linearly with the number of atoms in the system, N, in contrast to standard approaches which scale with the cube of the number of atoms. These methods, which rely on the short-ranged nature of electronic structure, will allow accurate, ab initio simulations of systems of unprecedented size. The theory behind the locality of electronic structure is described and related to physical properties of systems to be modelled, along with a survey of recent developments in real-space methods which are important for efficient use of high performance computers. The linear scaling methods proposed to date can be divided into seven different areas, and the applicability, efficiency and advantages of the methods proposed in these areas is then discussed. The applications of linear scaling methods, as well as the implementations available as computer programs, are considered. Finally, the prospects for and the challenges facing linear scaling methods are discussed.