Linear Algebraic Calculation of 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.

10 pages, 9 figures. To appear in Phys. Rev. B. A PDF file with better graphics is available at http://fujimac.t.u-tokyo.ac.jp/lses/

References in corpus (5)

• Krylov Subspace Method for Molecular Dynamics Simulation based on Large-Scale Electronic Structure Theory
• Nanoscale structures formed in silicon cleavage studied with large-scale electronic structure alculations; surface reconstruction, step and bending
• Dynamical brittle fractures of nanocrystalline silicon using large-scale electronic structure calculations
• Theory of composite-band Wannier states and order-N electronic-structure calculations
• Million-atom molecular dynamics simulation by order-N electronic structure theory and parallel computation

Cited by in corpus (27)

• O(N) methods in electronic structure calculations
• Linear scaling Krylov subspace method for large scale {\it ab initio} electronic structure calculations of metals
• Accurate ionic forces and geometry optimisation in linear scaling density-functional theory with local orbitals
• Vortex pinning and dynamics in the neutron star crust
• Electrostatic Interactions in Finite Systems treated with Periodic Boundary Conditions: Application to Linear-Scaling Density Functional Theory
• An order-N electronic structure theory with generalized eigenvalue equations and its application to a ten-million-atom system
• Stochastic Estimation of Nuclear Level Density in the Nuclear Shell Model: An Application to Parity-Dependent Level Density in ${}^{58}$Ni
• Coordinate-Space Solver for Superfluid Many-Fermion Systems with Shifted Conjugate Orthogonal Conjugate Gradient Method
• Charge and spin stripe in La${}_{2-x}$Sr${}_x$NiO${}_4$ (x=1/3,1/2)
• Restarted Hessenberg method for solving shifted nonsymmetric linear systems
• Filter diagonalization of shell-model calculations
• Large-scale electronic structure theory for simulating nanostructure process
• Domain boundary formation in helical multishell gold nanowire
• Improved recursive Green's function formalism for quasi one-dimensional systems with realistic defects
• Efficient and Accurate Linear Algebraic Methods for Large-scale Electronic Structure Calculations with Non-orthogonal Atomic Orbitals
• Reduced-Shifted Conjugate-Gradient Method for a Green's Function: Efficient Numerical Approach in a Nano-structured Superconductor
• Development of simulation package 'ELSES' for extra-large-scale electronic-structure calculation
• An Approximate Framework for Quantum Transport Calculation with Model Order Reduction
• Efficient calculation of the Green's function in scattering region for electron-transport simulations
• Accuracy control in ultra-large-scale electronic structure calculation
• A hierarchical research by large-scale and ab initio electronic structure theories -- Si and Ge cleavage and stepped (111)-2x1 surfaces --
• Novel linear algebraic theory and one-hundred-million-atom quantum material simulations on the K computer
• GPU-accelerated large-scale quantum molecular dynamics simulation of 3-dimensional C60 polymers
• Large scale simulation of quantum-mechanical molecular dynamics for nano-polycrystalline diamond
• Efficient methods for computing integrals in electronic structure calculations
• Shifted Lanczos method for quadratic forms with Hermitian matrix resolvents
• A SQMRCGstab Algorithm for Families of Shifted Linear Systems