Linear-scaling density-functional theory (DFT) calculations for insulators can be performed in principle by exploiting the short spatial range of the single-particle density-matrix of the Kohn-Sham system. The density-matrix is usually expressed in a separable form in terms of a matrix of generalised occupation numbers and a set of localised non-orthogonal functions or generalised Wannier functions.
The focus of this paper will be the representation of the localised functions with a view to how they may be optimised most efficiently during a calculation. Two schemes will be considered: first, the use of a localised spherical-wave basis set [1,2]; and second, the use of restricted regions of a real-space grid [3]. The first scheme highlights the problems which arise when non-orthogonal basis sets are used to optimise localised functions [4]. In the second, fast Fourier transforms may be used to obtain a linear-scaling method, as long as certain conditions are met to ensure the Hermiticity and consistency of the various components of the Hamiltonian.
[1] P. D. Haynes and M. C. Payne, Comput. Phys. Commun. 102, 17-27 (1997).
[2] C. K. Gan, P. D. Haynes and M. C. Payne, Phys. Rev. B 63, 205109 (2001).
[3] C.-K. Skylaris, A. A. Mostofi, P. D. Haynes, C. J. Pickard and M. C. Payne, Comput. Phys. Commun. 140, 315-322 (2001).
[4] C. K. Gan, P. D. Haynes and M. C. Payne, Comput. Phys. Commun. 134, 33-40 (2001).
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