**Arash A. Mostofi, Chris-Kriton Skylaris, Peter D. Haynes and Mike C. Payne**

**Theory of Condensed Matter, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK**

We present a novel real space formalism for *ab initio* electronic structure calculations. We use localised non-orthogonal functions that are expressed in terms of a basis set that is equivalent to a plane-wave basis. As a result, advantages of the plane-wave approach also apply to our method: its applicability to any lattice symmetry, and systematic basis set improvement via the kinetic energy cut-off parameter. The localisation of our functions enables the use of fast Fourier transforms over small regions of the simulation cell to calculate the total energy with efficiency and accuracy. With just one further variational approximation, namely the truncation of the density matrix, the calculation may be performed with a cost that scales linearly with system size for insulating systems.

PACS numbers: 71.15.-m, 31.15.-p, 31.15.Ar

Keywords: Electronic structure; Density functional theory; Linear-scaling; FFT; Systematic basis set

PACS numbers: 71.15.-m, 31.15.-p, 31.15.Ar

Keywords: Electronic structure; Density functional theory; Linear-scaling; FFT; Systematic basis set

- Introduction
- Basis set
- Total energy
- Total energy using the FFT box technique
- The FFT box
- Projection operators
- Kinetic energy
- Non-local pseudopotential energy
- Charge density
- Hartree, local pseudopotential, and exchange-correlation energy

- Total energy optimisation
- Results and discussion
- Conclusions
- Acknowledgements
- Basis Set
- Definition of basis functions
- Localisation and Orthogonality
- Analytic Integrals
- Basis for the fine grid

- Bibliography
- About this document ...