Order($N$) method

The computational effort of the conventional diagonalization scheme scales as the third power of the number of basis orbitals, which means that the part could be a bottleneck when large-scale systems are calculated. On the other hand, the O($N$) methods can solve the eigenvalue problem in O($N$) operation in exchange for accuracy. Thus, O($N$) methods could be efficient for large-scale systems, while a careful consideration is always required for the accuracy. In OpenMX Ver. 3.9, three O($N$) methods are available: a divide-conquer (DC) method [50], a divide-conquer (DC) method with localized natural orbitals (LNO) [51], and a Krylov subspace method [43]. Our recommendation among the three O($N$) methods is the DC-LNO method, since the method is robust, and can be applied to a wide range of materials including metals, and the parallel efficiency is expected to be the best one among the three methods. In the following subsections each O($N$) method is illustrated by examples.