Perfect Scaling of the Electronic Structure Problem on a SIMD Architecture

Abstract We report on benchmark tests of computations of the total electronic density of states of a micro-crystallite of rutile TiO 2 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 × 1.5 nm . 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.

[1]  M. Schmeits,et al.  Electronic structure of ideal TiO2(110), TiO2(001), and TiO2(100) surfaces , 1984 .

[2]  Morgan,et al.  Model of hydrogenated amorphous silicon and its electronic structure. , 1993, Physical review. B, Condensed matter.

[3]  F. A. Grant.  Properties of Rutile (Titanium Dioxide) , 1959 .

[4]  Munnix,et al.  Electronic structure of point defects on oxide surfaces. , 1986, Physical review. B, Condensed matter.

[5]  D. Weaire,et al.  Electrons and holes in amorphous silicon , 1993 .

[6]  B. Santo,et al.  Solid State , 2012 .

[7]  G. J. Morgan,et al.  The double-sign anomaly of the Hall coefficient in amorphous silicon : verification by computer simulations , 1992 .

[8]  D. Weaire,et al.  New applications of the equation-of-motion method: Optical properties , 1993 .

[9]  B. Hickey,et al.  Application of the equation-of-motion method to the calculation of optical properties , 1991 .

[10]  G. J. Morgan,et al.  The electronic structure and conductivity of large models of amorphous silicon , 1992 .

[11]  V. Heine,et al.  Electronic structure based on the local atomic environment for tight-binding bands. II , 1972 .

[12]  The Equation of Motion Method , 1987 .

[13]  K. Vos.  Reflectance and electroreflectance of TiO2 single crystals. II. Assignment to electronic energy levels , 1977 .

[14]  D. Weaire,et al.  A comparison of the recursion method and the equation-of-motion method for the calculation of densities of states , 1985 .

[15]  J. W. Halley,et al.  Equation of motion method for the electronic structure of disordered transition metal oxides , 1992 .

[16]  D. Weaire,et al.  The Anderson localization problem. I. A new numerical approach , 1977 .

[17]  J. W. Halley,et al.  Photoelectrochemical spectroscopy studies of titanium dioxide surfaces: theory and experiment , 1991 .

[18]  David Beeman,et al.  Vibrational properties of elemental amorphous semiconductors , 1977 .

[19]  M. T. Michalewicz,et al.  Massively parallel calculations of the electronic structure of non-periodic micro-crystallites of transition metal oxides , 1994 .

[20]  Munnix,et al.  Origin of defect states on the surface of TiO2. , 1985, Physical review. B, Condensed matter.

[21]  G. J. Morgan,et al.  Electron localization in models of hydrogenated amorphous silicon and pure amorphous silicon , 1994 .

[22]  Nevill Mott,et al.  Metal-insulator transitions , 1973 .

[23]  D. Pettifor,et al.  The Recursion Method and Its Applications , 1986 .

[24]  Michalewicz,et al.  Electronic structure of multiple vacancies in rutile TiO2 by the equation-of-motion method. , 1990, Physical Review B (Condensed Matter).

[25]  B. Hickey,et al.  The density of states and spectral function in amorphous Si obtained using the equation of motion method in k-space , 1986 .

[26]  D. Weaire,et al.  A new numerical method for the calculation of the conductivity of a disordered system , 1977 .

[27]  J. W. Halley,et al.  Electronic properties of disordered anodic TiO2 (001) surfaces: Application of the equation‐of‐motion method , 1992 .

[28]  Tom Blank,et al.  The MasPar MP-1 architecture , 1990, Digest of Papers Compcon Spring '90. Thirty-Fifth IEEE Computer Society International Conference on Intellectual Leverage.

[29]  R. Alben,et al.  Exact results for a three-dimensional alloy with site diagonal disorder: comparison with the coherent potential approximation , 1975 .