Efficient O( N ) divide-conquer method with localized single-particle natural orbitals

An efficient O(N) divide-conquer (DC) method based on localized single-particle natural orbitals (LNOs) is presented for large-scale density functional theory (DFT) calculations of gapped and metallic systems. The LNOs are noniteratively 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.