Divide-and-conquer density functional theory on hierarchical real-space grids: Parallel implementation and applications

A linear-scaling algorithm based on a divide-and-conquer ! DC" scheme has been designed to perform large-scale molecular-dynamics ! MD" simulations, in which interatomic forces are computed quantum mechanically in the framework of the density functional theory ! DFT" . Electronic wave functions are represented on a real-space grid, which is augmented with a coarse multigrid to accelerate the convergence of iterative solutions and with adaptive fine grids around atoms to accurately calculate ionic pseudopotentials. Spatial decomposition is employed to implement the hierarchical-grid DC-DFT algorithm on massively parallel computers. The largest benchmark tests include 11.8! 10 6 -atom ! 1.04! 10 12 electronic degrees of freedom" calculation on 131 072 IBM BlueGene/L processors. The DC-DFT algorithm has well-defined parameters to control the data locality, with which the solutions converge rapidly. Also, the total energy is well conserved during the MD simulation. We perform first-principles MD simulations based on the DC-DFT algorithm, in which large system sizes bring in excellent agreement with x-ray scattering measurements for the pairdistribution function of liquid Rb and allow the description of low-frequency vibrational modes of graphene. The band gap of a CdSe nanorod calculated by the DC-DFT algorithm agrees well with the available conventional DFT results. With the DC-DFT algorithm, the band gap is calculated for larger system sizes until the result reaches the asymptotic value. DOI: 10.1103/PhysRevB.77.085103

[1]  P. Hohenberg,et al.  Inhomogeneous Electron Gas , 1964 .

[2]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[3]  A. Zunger,et al.  CORRIGENDUM: Momentum-space formalism for the total energy of solids , 1979 .

[4]  P. Pulay.  Convergence acceleration of iterative sequences. the case of scf iteration , 1980 .

[5]  Leonard Kleinman,et al.  Efficacious Form for Model Pseudopotentials , 1982 .

[6]  Car,et al.  Unified approach for molecular dynamics and density-functional theory. , 1985, Physical review letters.

[7]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[8]  J. Tersoff,et al.  New empirical approach for the structure and energy of covalent systems. , 1988, Physical review. B, Condensed matter.

[9]  Warren E. Pickett,et al.  Pseudopotential methods in condensed matter applications , 1989 .

[10]  Davenport,et al.  First-principles calculation of the activation energy for diffusion in liquid sodium. , 1990, Physical review letters.

[11]  William H. Press,et al.  Numerical recipes , 1990 .

[12]  Renata M. Wentzcovitch,et al.  First principles molecular dynamics of Li: Test of a new algorithm , 1991 .

[13]  Lin,et al.  Real-space implementation of nonlocal pseudopotentials for first-principles total-energy calculations. , 1991, Physical review. B, Condensed matter.

[14]  W. Goddard,et al.  Charge equilibration for molecular dynamics simulations , 1991 .

[15]  Yang,et al.  Direct calculation of electron density in density-functional theory. , 1991, Physical review letters.

[16]  Martins,et al.  Efficient pseudopotentials for plane-wave calculations. II. Operators for fast iterative diagonalization. , 1991, Physical review. B, Condensed matter.

[17]  Galli,et al.  Large scale electronic structure calculations. , 1992, Physical review letters.

[18]  Ab initio calculation of density dependence of liquid-Na properties. , 1992, Physical review. B, Condensed matter.

[19]  T. Arias,et al.  Iterative minimization techniques for ab initio total energy calculations: molecular dynamics and co , 1992 .

[20]  Martins,et al.  Energy versus free-energy conservation in first-principles molecular dynamics. , 1992, Physical review. B, Condensed matter.

[21]  Davenport,et al.  Fractional occupations and density-functional energies and forces. , 1992, Physical review. B, Condensed matter.

[22]  Daw.  Model for energetics of solids based on the density matrix. , 1993, Physical review. B, Condensed matter.

[23]  T. Darden,et al.  Particle mesh Ewald: An N⋅log(N) method for Ewald sums in large systems , 1993 .

[24]  White,et al.  Detonations at nanometer resolution using molecular dynamics. , 1993, Physical review letters.

[25]  Li,et al.  Density-matrix electronic-structure method with linear system-size scaling. , 1993, Physical review. B, Condensed matter.

[26]  Wu,et al.  Higher-order finite-difference pseudopotential method: An application to diatomic molecules. , 1994, Physical review. B, Condensed matter.

[27]  Frederick H. Streitz,et al.  Electrostatic potentials for metal-oxide surfaces and interfaces. , 1994 .

[28]  P. Madden,et al.  The dynamic structure of liquid sodium from ab initio simulation , 1994 .

[29]  Williams,et al.  N-scaling algorithm for density-functional calculations of metals and insulators. , 1994, Physical review. B, Condensed matter.

[30]  A. Nakano,et al.  Multiresolution molecular dynamics algorithm for realistic materials modeling on parallel computers , 1994 .

[31]  Chan,et al.  Density-functional energies and forces with Gaussian-broadened fractional occupations. , 1994, Physical review. B, Condensed matter.

[32]  Steven J. Stuart,et al.  Dynamical fluctuating charge force fields: Application to liquid water , 1994 .

[33]  Rajiv K. Kalia,et al.  Massively parallel algorithms for computational nanoelectronics based on quantum molecular dynamics , 1994 .

[34]  G. M. Stocks,et al.  Order-N multiple scattering approach to electronic structure calculations. , 1995, Physical review letters.

[35]  Görling,et al.  Hardness of molecules and the band gap of solids within the Kohn-Sham formalism: A perturbation-scaling approach. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[36]  Shimojo,et al.  First-principles molecular-dynamics simulation of expanded liquid rubidium. , 1995, Physical review. B, Condensed matter.

[37]  Weitao Yang,et al.  A density‐matrix divide‐and‐conquer approach for electronic structure calculations of large molecules , 1995 .

[38]  Kim,et al.  Total-energy global optimizations using nonorthogonal localized orbitals. , 1995, Physical review. B, Condensed matter.

[39]  Martin,et al.  Linear system-size scaling methods for electronic-structure calculations. , 1995, Physical review. B, Condensed matter.

[40]  Burke,et al.  Generalized Gradient Approximation Made Simple. , 1996, Physical review letters.

[41]  Sullivan,et al.  Real-space multigrid-based approach to large-scale electronic structure calculations. , 1996, Physical review. B, Condensed matter.

[42]  Kohn,et al.  Density functional and density matrix method scaling linearly with the number of atoms. , 1996, Physical review letters.

[43]  Kresse,et al.  Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. , 1996, Physical review. B, Condensed matter.

[44]  Aiichiro Nakano,et al.  Parallel multilevel preconditioned conjugate-gradient approach to variable-charge molecular dynamics , 1997 .

[45]  S. L. Dixon,et al.  Fast, accurate semiempirical molecular orbital calculations for macromolecules , 1997 .

[46]  Eric Schwegler,et al.  Linear scaling computation of the Fock matrix , 1997 .

[47]  Siegfried Schmauder,et al.  Comput. Mater. Sci. , 1998 .

[48]  Tomoya Ono,et al.  Timesaving Double-Grid Method for Real-Space Electronic-Structure Calculations , 1999 .

[49]  S. Goedecker.  Linear scaling electronic structure methods , 1999 .

[50]  Stefano de Gironcoli,et al.  First-principles determination of elastic anisotropy and wave velocities of MgO at lower mantle conditions , 1999, Science.

[51]  Rajiv K. Kalia,et al.  DYNAMICS OF OXIDATION OF ALUMINUM NANOCLUSTERS USING VARIABLE CHARGE MOLECULAR-DYNAMICS SIMULATIONS ON PARALLEL COMPUTERS , 1999 .

[52]  Peter S. Lomdahl,et al.  Recent advances in large-scale atomistic materials simulations , 1999, Comput. Sci. Eng..

[53]  Philippe Y. Ayala,et al.  Linear scaling coupled cluster and perturbation theories in the atomic orbital basis , 1999 .

[54]  James R. Chelikowsky,et al.  Electronic Structure Methods for Predicting the Properties of Materials: Grids in Space , 2000 .

[55]  Timothy Campbell,et al.  A scalable molecular-dynamics algorithm suite for materials simulations: design-space diagram on 1024 Cray T3E processors , 2000, Future Gener. Comput. Syst..

[56]  Mark E. Tuckerman,et al.  Exploiting multiple levels of parallelism in Molecular Dynamics based calculations via modern techniques and software paradigms on distributed memory computers , 2000 .

[57]  Donald W. Brenner,et al.  The Art and Science of an Analytic Potential , 2000 .

[58]  J. Fattebert,et al.  Towards grid-based OÑNÖ density-functional theory methods: Optimized nonorthogonal orbitals and multigrid acceleration , 2000 .

[59]  T. Beck.  Real-space mesh techniques in density-functional theory , 2000, cond-mat/0006239.

[60]  Rajiv K. Kalia,et al.  Linear-scaling density-functional-theory calculations of electronic structure based on real-space grids: design, analysis, and scalability test of parallel algorithms , 2001 .

[61]  Weidong Yang,et al.  Linearly Polarized Emission from Colloidal Semiconductor Quantum Rods , 2001, Science.

[62]  M Elstner,et al.  Quantum mechanics simulation of protein dynamics on long timescale , 2001, Proteins.

[63]  R.K. Kalia,et al.  Multiscale simulation of nanosystems , 2001, Comput. Sci. Eng..

[64]  Rajiv K. Kalia,et al.  Hybrid finite-element/molecular-dynamics/electronic-density-functional approach to materials simulations on parallel computers , 2001 .

[65]  Hans-Joachim Werner,et al.  Low-order scaling local electron correlation methods. IV. Linear scaling local coupled-cluster (LCCSD) , 2001 .

[66]  A. V. Duin,et al.  ReaxFF: A Reactive Force Field for Hydrocarbons , 2001 .

[67]  J. Grossman,et al.  Linear-scaling quantum Monte Carlo calculations. , 2001, Physical review letters.

[68]  A. P. Alivisatos,et al.  Band Gap Variation of Size- and Shape-Controlled Colloidal CdSe Quantum Rods , 2001 .

[69]  N. Bernstein,et al.  Linear scaling nonorthogonal tight-binding molecular dynamics for nonperiodic systems , 2001 .

[70]  Rajiv K. Kalia,et al.  Hybrid quantum mechanical/molecular dynamics simulation on parallel computers: density functional theory on real-space multigrids , 2002 .

[71]  Robert Walkup,et al.  Simulating materials failure by using up to one billion atoms and the world's fastest computer: Brittle fracture , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[72]  Subhash Saini,et al.  Scalable atomistic simulation algorithms for materials research , 2002 .

[73]  Guanhua Chen,et al.  Linear-scaling time-dependent density-functional theory , 2003 .

[74]  A. V. van Duin,et al.  Shock waves in high-energy materials: the initial chemical events in nitramine RDX. , 2003, Physical review letters.

[75]  Rajiv K. Kalia,et al.  Scalable and portable implementation of the fast multipole method on parallel computers , 2003 .

[76]  Million-atom molecular dynamics simulation by order-N electronic structure theory and parallel computation , 2003, cond-mat/0306461.

[77]  Rajiv K. Kalia,et al.  Short- and intermediate-range structural correlations in amorphous silicon carbide: a molecular dynamics study , 2004 .

[78]  Jean-Luc Fattebert,et al.  Linear scaling first-principles molecular dynamics with controlled accuracy , 2004, Comput. Phys. Commun..

[79]  Rajiv K. Kalia,et al.  Environmental effects of H2O on fracture initiation in silicon: A hybrid electronic-density-functional/molecular-dynamics study , 2004 .

[80]  James R. Chelikowsky,et al.  Surface passivation method for semiconductor nanostructures , 2005 .

[81]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .

[82]  Rajiv K. Kalia,et al.  Embedded divide-and-conquer algorithm on hierarchical real-space grids: parallel molecular dynamics simulation based on linear-scaling density functional theory , 2005, Comput. Phys. Commun..

[83]  W. Kohn,et al.  Nearsightedness of electronic matter. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[84]  Nicola Marzari,et al.  First-principles determination of the structural, vibrational and thermodynamic properties of diamond, graphite, and derivatives , 2004, cond-mat/0412643.

[85]  T. Tanaka,et al.  Analysis of phonons in graphene sheets by means of HREELS measurement and ab initio calculation , 2005 .

[86]  Taisuke Ozaki.  O(N) Krylov-subspace method for large-scale ab initio electronic structure calculations , 2006 .

[87]  James R Chelikowsky,et al.  Self-purification in semiconductor nanocrystals. , 2006, Physical review letters.

[88]  Martin Head-Gordon,et al.  A near linear-scaling smooth local coupled cluster algorithm for electronic structure. , 2006, The Journal of chemical physics.

[89]  Pressure-induced structural transformations in cadmium selenide nanorods , 2006 .

[90]  Arash A. Mostofi,et al.  Implementation of linear‐scaling plane wave density functional theory on parallel computers , 2006 .

[91]  David R. Bowler,et al.  Recent progress with large‐scale ab initio calculations: the CONQUEST code , 2006 .

[92]  A. V. van Duin,et al.  Dynamic transition in the structure of an energetic crystal during chemical reactions at shock front prior to detonation. , 2007, Physical review letters.

[93]  Eiji Tsuchida.  Augmented Orbital Minimization Method for Linear Scaling Electronic Structure Calculations(Condensed matter: electronic structure and electrical, magnetic, and optical properties) , 2007 .

[94]  A. V. Duin,et al.  A Divide-and-Conquer/Cellular-Decomposition Framework for Million-to-Billion Atom Simulations of Chemical Reactions , 2007 .

[95]  Instability of the electron gas in an expanding metal. , 2007, Physical review letters.