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

Abstract

The need for large-scale electronic structure calculations arises recently in the field of material physics, and efficient and accurate algebraic methods for large simultaneous linear equations become greatly important. We investigate the generalized shifted conjugate orthogonal conjugate gradient method, the generalized Lanczos method, and the generalized Arnoldi method. They are the solver methods of large simultaneous linear equations of the one-electron Schrödinger equation and map the whole Hilbert space to a small subspace called the Krylov subspace. These methods are applied to systems of fcc Au with the NRL tight-binding Hamiltonian [F. Kirchhoff et al., Phys. Rev. B 63, 195101 (2001)]. We compare results by these methods and the exact calculation and show them to be equally accurate. The system size dependence of the CPU time is also discussed. The generalized Lanczos method and the generalized Arnoldi method are the most suitable for the large-scale molecular dynamics simulations from the viewpoint of CPU time and memory size.

Solution of generalized shifted linear systems with complex symmetric matrices

“…The basic idea has been described shortly in our recent paper [35]. Our contribution in this subsection is a simple derivation of the algorithm from the shifted BiCG method.…”

\mathcal{O}(N) methods in electronic structure calculations

“…Once the vectors |x j have been generated for one energy, further energies are almost trivial, and certainly scale as O(1). An overview and summary of applications using tight binding have been given [319,320], and the method has been extended to non-orthogonal orbitals [321]. The recursion methods were the first set of linear scaling methods proposed, and Lanczos approaches are widely used in many areas of physics and mathematics.…”

Efficient methods for computing integrals in electronic structure calculations

“…which appears in an electronic structure calculation of a nanoscale amourphous-like conjugated polymer [4] The values of λ 1 , λ 2 , · · · , λ 4686 (λ 1 −1.16, λ 4686 5.58) are given on the website [5] as the data set "APF4686". (The actual computation of G(z) is done by using the expression (2), that is, by solving the large system of linear equations (zI − H)x = b, which leads to relatively large numerical errors. Since we here concentrate on examining numerical errors caused by the numerical integration, we give G(z) as the rational expression as above, the computation of which produces small numerical errors.…”