A Sparse Self-Consistent Field Algorithm and Its Parallel Implementation: Application to Density-Functional-Based Tight Binding.

We present an algorithm and its parallel implementation for solving a self-consistent problem as encountered in Hartree-Fock or density functional theory. The algorithm takes advantage of the sparsity of matrices through the use of local molecular orbitals. The implementation allows one to exploit efficiently modern symmetric multiprocessing (SMP) computer architectures. As a first application, the algorithm is used within the density-functional-based tight binding method, for which most of the computational time is spent in the linear algebra routines (diagonalization of the Fock/Kohn-Sham matrix). We show that with this algorithm (i) single point calculations on very large systems (millions of atoms) can be performed on large SMP machines, (ii) calculations involving intermediate size systems (1000-100 000 atoms) are also strongly accelerated and can run efficiently on standard servers, and (iii) the error on the total energy due to the use of a cutoff in the molecular orbital coefficients can be controlled such that it remains smaller than the SCF convergence criterion.

[1]  Seifert,et al.  Construction of tight-binding-like potentials on the basis of density-functional theory: Application to carbon. , 1995, Physical review. B, Condensed matter.

[2]  Linda Hung,et al.  Accurate simulations of metals at the mesoscale: Explicit treatment of 1 million atoms with quantum mechanics , 2009 .

[3]  Matt Challacombe,et al.  A simplified density matrix minimization for linear scaling self-consistent field theory , 1999 .

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

[5]  Anders M. N. Niklasson,et al.  Trace resetting density matrix purification in O(N) self-consistent-field theory , 2003 .

[6]  J. MacQueen.  Some methods for classification and analysis of multivariate observations , 1967 .

[7]  T. Frauenheim,et al.  DFTB+, a sparse matrix-based implementation of the DFTB method. , 2007, The journal of physical chemistry. A.

[8]  Celestino Angeli,et al.  A novel perturbation-based complete active space-self-consistent-field algorithm: Application to the direct calculation of localized orbitals , 2002 .

[9]  Joost VandeVondele,et al.  Linear Scaling Self-Consistent Field Calculations with Millions of Atoms in the Condensed Phase. , 2012, Journal of chemical theory and computation.

[10]  G. Seifert,et al.  Calculations of molecules, clusters, and solids with a simplified LCAO-DFT-LDA scheme , 1996 .

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

[12]  Gustavo E. Scuseria,et al.  Linear scaling conjugate gradient density matrix search as an alternative to diagonalization for first principles electronic structure calculations , 1997 .

[13]  J. C. Slater,et al.  Simplified LCAO Method for the Periodic Potential Problem , 1954 .

[14]  Sándor Suhai,et al.  A Self‐Consistent Charge Density‐Functional Based Tight‐Binding Method for Predictive Materials Simulations in Physics, Chemistry and Biology , 2000 .

[15]  William Jalby,et al.  Quantum Monte Carlo for large chemical systems: Implementing efficient strategies for petascale platforms and beyond , 2012, J. Comput. Chem..

[16]  David R. Kincaid,et al.  Algorithm 586: ITPACK 2C: A FORTRAN Package for Solving Large Sparse Linear Systems by Adaptive Accelerated Iterative Methods , 1982, TOMS.

[17]  Rami Melhem,et al.  Toward Efficient Implementation of Preconditioned Conjugate Gradient Methods On Vector Supercomputers , 1987 .

[18]  Stefano Evangelisti,et al.  Direct generation of local orbitals for multireference treatment and subsequent uses for the calculation of the correlation energy , 2002 .

[19]  M. Elstner,et al.  Validation of the density-functional based tight-binding approximation method for the calculation of reaction energies and other data. , 2005, The Journal of chemical physics.

[20]  Hélio A. Duarte,et al.  Density-functional based tight-binding: an approximate DFT method , 2009 .

[21]  Anthony Skjellum,et al.  A High-Performance, Portable Implementation of the MPI Message Passing Interface Standard , 1996, Parallel Comput..

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

[23]  David E. Manolopoulos,et al.  Canonical purification of the density matrix in electronic-structure theory , 1998 .

[24]  Emanuel H. Rubensson,et al.  Kohn-Sham Density Functional Theory Electronic Structure Calculations with Linearly Scaling Computational Time and Memory Usage. , 2011, Journal of chemical theory and computation.

[25]  L. Dagum,et al.  OpenMP: an industry standard API for shared-memory programming , 1998 .

[26]  Gustavo E. Scuseria,et al.  Semiempirical methods with conjugate gradient density matrix search to replace diagonalization for molecular systems containing thousands of atoms , 1997 .

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

[28]  James J. P. Stewart,et al.  Fast semiempirical calculations , 1982 .

[29]  M. Elstner.  The SCC-DFTB method and its application to biological systems , 2006 .

[30]  David A Mazziotti,et al.  Comparison of two genres for linear scaling in density functional theory: purification and density matrix minimization methods. , 2005, The Journal of chemical physics.

[31]  David R. Bowler,et al.  Density matrices in O(N) electronic structure calculations: theory and applications , 1998 .

[32]  Sándor Suhai,et al.  Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties , 1998 .

[33]  Michael J. Frisch,et al.  Density matrix search using direct inversion in the iterative subspace as a linear scaling alternative to diagonalization in electronic structure calculations , 2003 .

[34]  A. Köster,et al.  Double asymptotic expansion of three-center electronic repulsion integrals. , 2013, The Journal of chemical physics.

[35]  Thomas Frauenheim,et al.  Atomistic simulations of complex materials: ground-state and excited-state properties , 2002 .

[36]  R. Mcweeny.  Some Recent Advances in Density Matrix Theory , 1960 .

[37]  Nonorthogonal density-matrix perturbation theory. , 2005, The Journal of chemical physics.

[38]  Emanuel H. Rubensson,et al.  Assessment of density matrix methods for linear scaling electronic structure calculations , 2011, Journal of physics. Condensed matter : an Institute of Physics journal.

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

[40]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[41]  R. Baer,et al.  Augmented Lagrangian method for order-N electronic structure , 2001 .

[42]  Tai-Sung Lee,et al.  A variational linear-scaling framework to build practical, efficient next-generation orbital-based quantum force fields. , 2013, Journal of chemical theory and computation.