Linear algebraic calculation of the Green’s function for large-scale electronic structure theory

Abstract

A linear algebraic method named the shifted conjugate-orthogonal-conjugate-gradient method is introduced for large-scale electronic structure calculation. The method gives an iterative solver algorithm of the Green's function and the density matrix without calculating eigenstates. The problem is reduced to independent linear equations at many energy points and the calculation is actually carried out only for a single energy point. The method is robust against the round-off error and the calculation can reach the machine accuracy. With the observation of residual vectors, the accuracy can be controlled, microscopically, independently for each element of the Green's function, and dynamically, at each step in dynamical simulations. The method is applied to both semiconductor and metal. 71.15.Nc, 71.15.Pd

Efficient and accurate linear algebraic methods for large-scale electronic structure calculations with nonorthogonal atomic orbitals

“…However, the orthogonality is broken in the numerical calculation procedure. This problem causes several issues such as the existence of constant background of error in the spectrum, 14 appearance of "ghost" structure in spectrum due to erroneous mixing of states, and a broken normalization of the partial density of states. Figure 7 shows the examples of this broken orthonormality in a system of 256 atoms of fcc Au by using the NRL Hamiltonian.…”

“…14,15,21 It was shown that the convergence behavior can be monitored by observing the behavior of the "residual norm." The shifted COCG method is generalized for Eq.…”

“…(14), that of GsCOCG with Eq. (37), and that of the exact diagonalization method for the NRL Hamiltonian of fcc Au system of 256 and 864 atoms.…”

“…3,4 Development of efficient linear algebraic methods has been, so far, mainly based on the orthogonal basis sets. 13,14,[17][18][19] However, localized basis wave functions are generally nonorthogonal and it is much more desirable to generalize the methods to the case of a nonorthogonal basis set. The most popular strategy of the generalized eigenvalue problem (represented by the nonorthogonal basis set) would be the transformation to the standard eigenvalue problem.…”

“…We developed the subspace diagonalization method and the shifted conjugate orthogonal conjugate gradient (COCG) method. [13][14][15][16] Then, the methods were applied to the fracture propagation and surface formation in Si crystals with the tight-binding Hamiltonian based on an orthogonal basis set. 1,2 On the other hand, since its Hamiltonian is described by the tight-binding Hamiltonian based on a nonorthogonal basis set, the problem of the formation of Au multishell helical nanowires was solved by the exact diagonalization method.…”