An order-Nelectronic structure theory with generalized eigenvalue equations and its application to a ten-million-atom system

Abstract

A linear algebraic theory called the 'multiple Arnoldi method' is presented and realizes large-scale (order-N) electronic structure calculations with generalized eigenvalue equations. A set of linear equations, in the form of (zS - H)x = b, are solved simultaneously with multiple Krylov subspaces. The method is implemented in a simulation package ELSES (www.elses.jp) with tight-binding-form Hamiltonians. A finite-temperature molecular dynamics simulation is carried out for metallic and insulating materials. A calculation with 10(7) atoms was realized by a workstation. The parallel efficiency is shown up to 1024 CPU cores.

Novel Linear Algebraic Theory and One-Hundred-Million-Atom Electronic Structure Calculation on The K Computer

“…[1] The mathematical foundation is the 'generalized shifted linear equation', or the set of linear equations…”

“…[1,2] A conjugated polymer system depicted in Figs. 1(b)(c), poly-(9,9 dioctyl-fluorene), was chosen as a test system.…”

“…As a reference data, the monomer and dimer were calculated by the present method and the ab initio calculation of Gaussian (TM) and the results agree reasonably among the two methods. [1] The calculated highest-occupied (HO) and lowest-unoccupied (LU) states of dimer are contributed only by the π states of benzene rings, as in benzene, and are illustrated in Fig.2(a) and (b), respectively. A molecular dynamics simulation with 2076 atoms was carried out as shown in Fig.…”

“…3(a) of Ref. [1]. The present paper presents more recent calculations for larger systems with one hundred million atoms and the calculations are called '100-nm-scale calculation', because one hundred million atoms are those in silicon single crystal with the volume of V =(126nm) 3 .…”

“…Recently, calculations with ten million atoms were realized on the K computer [1,2] by our code ELSES (http://www.elses.jp/). The computational cost is order-N or is proportional to the number of atoms N, as shown in Fig.…”