Efficient and Accurate Linear Algebraic Methods for Large-scale Electronic Structure Calculations with Non-orthogonal 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 one-electron Schrödinger equation and maps 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 (Phys. Rev. B {\bf 63}, 195101 (2001)). We compare results by these methods and the exact calculation and show them 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 view point of CPU time and memory size.

13pages, 7figures

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Cited by in corpus (7)

• O(N) methods in electronic structure calculations
• An order-N electronic structure theory with generalized eigenvalue equations and its application to a ten-million-atom system
• Quantum algorithms for the generalized eigenvalue problem
• Ten-million-atom electronic structure calculations on the K computer with a massively parallel order-N theory
• Efficient methods for computing integrals in electronic structure calculations
• Efficient implementation of single particle Hamiltonians in exponentially reduced qubit space
• Novel linear algebraic theory and one-hundred-million-atom quantum material simulations on the K computer