Shifted Conjugate-Orthogonal–Conjugate-Gradient Method and Its Application to Double Orbital Extended Hubbard Model

Abstract

We discuss the shifted conjugate-orthogonal–conjugate-gradient (COCG) method, which can be used to solve a series of linear equations generated by a number of scalar shifts, without performing time-consuming matrix–vector operations, except at a single reference energy. This is a type of CG method and is robust and has an expression of the accuracy of the calculated result that requires no additional calculation cost. The shifted COCG is useful for calculating the Green’s function of a many-electron Hamiltonian, which has a very large dimension. We applied the shifted COCG method to the double orbital extended Hubbard model with twelve electrons on a periodic \\sqrt8×\\sqrt8 site system, with the dimension of the Hamiltonian equal to 64,128,064, and we found that the ground state is an insulator. We also discuss the main points of reducing the amount of memory required to apply COCG algorithm.