Theory of large-scale matrix computation and applications in electronic structurecalculation

Abstract

We review the methods that we recently developed for making large-scale electronicstructure calculations, both using one-electron theory and using many-electron theory. Themethods are based on the density matrix representation, together with the Wannier staterepresentation and the Krylov subspace method, using one-electron theory ofsystems on a scale of a few tens of nanometers. The hybrid method of quantummechanical molecular dynamical simulation is explained. The Krylov subspacemethod, the CG (conjugate gradient) method and the shifted COCG (conjugateorthogonal conjugate gradient) method can be applied in the investigation ofthe ground state and the excitation spectra using many-electron theory. Themathematical foundation of the Krylov subspace method for large-scale matrixcomputation is focused on, and the key techniques of the shifted COCG method,i.e. using the collinear residual and ‘seed-switching’, are explained. A wide variety ofapplications of the extended novel algorithm are also explained. These include studies offracture formation and propagation, liquid carbon, and formation processes ofgold nanowires, together with the application to the extended Hubbard model.