An order-N electronic structure theory with generalized eigenvalue   equations and its application to a ten-million-atom system
  A linear-algebraic theory called 'multiple Arnoldi method' is presented and realizes large-scale (order-N) electronic structure calculation with generalized eigen-value 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 (http://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 upto 1,024 CPU cores. 
T. Hoshi, S. Yamamoto, T. Fujiwara, T. Sogabe, S.-L. Zhang
cond-mat.mtrl-sci physics.comp-ph