Perfect Scaling of the Electronic Structure Problem on a SIMD Architecture

We report on benchmark tests of computations of the total electronic density of states of a micro-crystallite of rutile TiO2 on MasPar MP-1 and MasPar MP-2 autonomous SIMD computers. The 3D spatial arrangement of atoms corresponds to the two dimensional computational grid of processing elements (PE) plus memory (2D + 1D) while the interactions between the constituent atoms correspond to the communication between the PEs. The largest sample we study consists of 491,520 atoms and its size is 41.5 × 41.5 × 1.5nm. Mathematically, the problem is equvalent to solving an n × n eigenvalue problem, where n ~ 2,500,000. The program is scalable in the number of atoms, so that the time required to run it is nearly independent of the size of the system in x and y directions (2D PE mesh) and is step-wise linear in z direction (memory axis). The total CPU time for the largest sample on a MasPar MP-2 computer with 16,384 processing elements is ~ 2.1 hour.