Conclusions

We have presented a novel formalism for performing DFT calculations on a real space grid. Features of our method include a systematic basis set, non-orthogonal functions localised in spherical regions, and the use of fast Fourier transforms over small regions of the simulation cell (the FFT box) for efficient and accurate calculation of matrix elements of the Hamiltonian. The total energy may be obtained in $O(N)$ operations with only a single further variational approximation, namely the truncation of the density kernel.