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. 3.8, two O($N$) methods are available: a divide-conquer (DC) method [39] and a Krylov subspace method [32]. In the following subsections each O($N$) method is illustrated by examples.