Linear Algebra and Charge Self-consistent Tight-binding Method for Large-scale Electronic Structure Calculations

We review our recently developed electronic structure calculation methods used for the dynamics of large-scale solids or liquids with an efficient algorithm for large scale simultaneous linear equations. The electronic structure calculation method is the `atomic superposition and electron delocalization molecular orbitals theory' (ASED), using the Mulliken charge density. Very crucial algorithm is the generalized shifted COCG (conjugate orthogonal conjugate gradient) method based on the Krylov subspace extended to non-orthogonal basis set. The most important techniques for applications are the shifted equation and the seed switching method, which make the computational cost be reduced much. We, then, present some applications to electronic structure calculations with MD simulation. The applications are given to the fracture propagation in nano-scale Si crystals and the proton transfer in water.