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

摘要

We review the methods that we recently developed for making large-scale electronic structure calculations, both using one-electron theory and using many-electron theory. The methods are based on the density matrix representation, together with the Wannier state representation and the Krylov subspace method, using one-electron theory of systems on a scale of a few tens of nanometers. The hybrid method of quantum mechanical molecular dynamical simulation is explained. The Krylov subspace method, the CG (conjugate gradient) method and the shifted COCG (conjugate orthogonal conjugate gradient) method can be applied in the investigation of the ground state and the excitation spectra using many-electron theory. The mathematical foundation of the Krylov subspace method for large-scale matrix computation 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 of applications of the extended novel algorithm are also explained. These include studies of fracture formation and propagation, liquid carbon, and formation processes of gold nanowires, together with the application to the extended Hubbard model.