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.Comment: 85 pages, 15 figures, 488 references. Resubmitted to Rep. Prog. Phys
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