  An efficient O($N$) divide-conquer (DC) method based on localized natural
orbitals (LNOs) is presented for large-scale density functional theories (DFT)
calculations of gapped and metallic systems. The LNOs are non-iteratively
calculated by a low-rank approximation via a local eigendecomposition of a
projection operator for the occupied space. Introducing LNOs to represent the
long range region of a truncated cluster reduces the computational cost of the
DC method while keeping computational accuracy. A series of benchmark
calculations and high parallel efficiency in a multilevel parallelization
clearly demonstrate that the O($N$) method enables us to perform large-scale
simulations for a wide variety of materials including metals with sufficient
accuracy in accordance with development of massively parallel computers.

Computational Physics;Materials Science