<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math> ) <i>ab initio</i> calculation scheme for large-scale moiré structures

We present a two-step method specifically tailored for band structure calculation of the small-angle moir\'{e}-pattern materials which contain tens thousands atoms in unit cell. In first step, self-consistent field ground state is performed with $O(N)$ Krylov subspace implemented OpenMX. Secondly, crystal momentum dependent Bloch Hamiltonian and overlap matrix are constructed from results obtained step only small number eigenvalues near Fermi energy solved shift-invert Lanczos techniques. By systematically tuning two key parameters, cutoff radius electron hopping interaction dimension subspace, we structures both rigid corrugated twisted bilayer graphene at magic angle ($\theta=1.08^\circ$) other three larger ones satisfied accuracy on affordable costs. The good agreement those tight binding models, continuum plane-wave pseudo-potential based $ab~initio$ calculations, experimental observations. This efficient to play crucial role two-dimensional materials, where much more complex than effective model hard be constructed.