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We review our recently developed methods for large-scale electronic structurecalculations, both in one-electron theory and many-electron theory. The methodare based on the density matrix representation, together with the Wannier staterepresentation and the Krylov subspace method, in one-electron theory ofa-few-tens nm scale systems. The hybrid method of quantum mechanical moleculardynamical simulation is explained.The Krylov subspace method, the CG (conjugategradient) method and the shifted-COCG (conjugate orthogonal conjugate gradient)method, can be applied to the investigation of the ground state and theexcitation spectra in many-electron theory. The mathematical foundation of theKrylov subspace method for large-scale matrix computation is focused and thekey technique of the shifted-COCG method, e.g. the collinear residual and seedswitching, is explained. A wide variety of applications of these extended novelalgorithm is also explained. These are the fracture formation and propagation,liquid carbon and formation process of gold nanowires, together with theapplication to the extend Hubbard model.

Theory of large-scale matrix computation and applications in electronic structure calculation