Efficient low-order scaling method for large-scale electronic structure calculations with localized basis functions

An efficient low-order scaling method is presented for large-scale electronic structure calculations based on the density-functional theory using localized basis functions, which directly computes selected elements of the density matrix by a contour integration of the Green's function evaluated with a nested dissection approach for resultant sparse matrices. The computational effort of the method scales as $\text{O}[N{({\text{log}}_{2}\text{ }N)}^{2}]$, $\text{O}({N}^{2})$, and $\text{O}({N}^{7/3})$ for one-, two-, and three-dimensional systems, respectively, where $N$ is the number of basis functions. Unlike $\text{O}(N)$ methods developed so far the approach is a numerically exact alternative to conventional $\text{O}({N}^{3})$ diagonalization schemes in spite of the low-order scaling, and can be applicable to not only insulating but also metallic systems in a single framework. It is also demonstrated that the well separated data structure is suitable for the massively parallel computation, which enables us to extend the applicability of density-functional calculations for large-scale systems together with the low-order scaling.