Linear scaling computation of the Fock matrix

Computation of the Fock matrix is currently the limiting factor in the application of Hartree-Fock and hybrid Hartree-Fock/density functional theories to larger systems. Computation of the Fock matrix is dominated by calculation of the Coulomb and exchange matrices. With conventional Gaussian-based methods, computation of the Fock matrix typically scales as ∼N2.7, where N is the number of basis functions. A hierarchical multipole method is developed for fast computation of the Coulomb matrix. This method, together with a recently described approach to computing the Hartree-Fock exchange matrix of insulators [J. Chem. Phys. 105, 2726 (1900)], leads to a linear scaling algorithm for calculation of the Fock matrix. Linear scaling computation the Fock matrix is demonstrated for a sequence of water clusters at the restricted Hartree-Fock/3-21G level of theory, and corresponding accuracies in converged total energies are shown to be comparable with those obtained from standard quantum chemistry programs. Restri...

[1]  Jan Almlöf,et al.  THE COULOMB OPERATOR IN A GAUSSIAN PRODUCT BASIS , 1995 .

[2]  Benny G. Johnson,et al.  Linear scaling density functional calculations via the continuous fast multipole method , 1996 .

[3]  Peter M. W. Gill,et al.  The prism algorithm for two-electron integrals , 1991 .

[4]  J. Nichols,et al.  A generalized fast multipole approach for Hartree—Fock and density functional computations , 1995 .

[5]  Michael S. Warren,et al.  A portable parallel particle program , 1995 .

[6]  K. Esselink The order of Appel's algorithm , 1992 .

[7]  Chen,et al.  Dual-space approach for density-functional calculations of two- and three-dimensional crystals using Gaussian basis functions. , 1995, Physical review. B, Condensed matter.

[8]  C. Alsenoy,et al.  brabo: a program for ab initio studies on large molecular systems , 1993 .

[9]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[10]  Volker Dyczmons,et al.  No N4-dependence in the calculation of large molecules , 1973 .

[11]  Methods for efficient evaluation of integrals for Gaussian type basis sets , 1974 .

[12]  G. Diercksen,et al.  Methods in Computational Molecular Physics , 1983 .

[13]  B. C. Carlson,et al.  On the expansion of a Coulomb potential in spherical harmonics , 1950, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  A. Becke Density-functional thermochemistry. III. The role of exact exchange , 1993 .

[15]  Clemens C. J. Roothaan,et al.  New Developments in Molecular Orbital Theory , 1951 .

[16]  Benny G. Johnson,et al.  THE CONTINUOUS FAST MULTIPOLE METHOD , 1994 .

[17]  R. Mcweeny Perturbation Theory for the Fock-Dirac Density Matrix , 1962 .

[18]  Cho,et al.  Wavelets in electronic structure calculations. , 1993, Physical review letters.

[19]  Piet Hut,et al.  A hierarchical O(N log N) force-calculation algorithm , 1986, Nature.

[20]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[21]  I. Panas Two-electron integrals and integral derivatives revisited , 1991 .

[22]  Henrik Gordon Petersen,et al.  The very fast multipole method , 1994 .

[23]  Martin Head-Gordon,et al.  Rotating around the quartic angular momentum barrier in fast multipole method calculations , 1996 .

[24]  D. Bassolino,et al.  Conformational isomerism of endothelin in acidic aqueous media: a quantitative NOESY analysis. , 1992, Biochemistry.

[25]  Eric Schwegler,et al.  Linear scaling computation of the Hartree–Fock exchange matrix , 1996 .

[26]  Gregory S. Tschumper,et al.  Structures, thermochemistry, and electron affinities of the PFn and PF−n series, n=1–6 , 1996 .

[27]  Christopher Miller Competition for block of a Ca2+-activated K+ channel by charybdotoxin and tetraethylammonium , 1988, Neuron.

[28]  Eric Schwegler,et al.  Fast assembly of the Coulomb matrix: A quantum chemical tree code , 1996 .

[29]  G. Marius Clore,et al.  Refined solution structure of the oligomerization domain of the tumour suppressor p53 , 1995, Nature Structural Biology.

[30]  Martin Head-Gordon,et al.  Fractional tiers in fast multipole method calculations , 1996 .

[31]  E. Davidson,et al.  One- and two-electron integrals over cartesian gaussian functions , 1978 .

[32]  Shigeru Obara,et al.  Efficient recursive computation of molecular integrals over Cartesian Gaussian functions , 1986 .

[33]  Wei,et al.  Wavelets in self-consistent electronic structure calculations. , 1996, Physical review letters.

[34]  José M. Pérez-Jordá,et al.  A concise redefinition of the solid spherical harmonics and its use in fast multipole methods , 1996 .

[35]  Christopher Miller,et al.  The charybdotoxin receptor of a Shaker K+ channel: Peptide and channel residues mediating molecular recognition , 1994, Neuron.

[36]  B. Honig,et al.  Classical electrostatics in biology and chemistry. , 1995, Science.

[37]  S. Wilson Distributed basis sets ofs-type Gaussian functions for molecular electronic structure calculations: Applications of the Gaussian cell model to one-electron polycentric linear molecular systems , 1996 .

[38]  Scott B. Baden,et al.  Dynamic Partitioning of Non-Uniform Structured Workloads with Spacefilling Curves , 1996, IEEE Trans. Parallel Distributed Syst..

[39]  A. Messiah Quantum Mechanics , 1961 .

[40]  J. Applequist,et al.  Traceless cartesian tensor forms for spherical harmonic functions: new theorems and applications to electrostatics of dielectric media , 1989 .

[41]  Per-Olov Löwdin,et al.  Quantum Theory of Many-Particle Systems. II. Study of the Ordinary Hartree-Fock Approximation , 1955 .

[42]  Charles W. Bauschlicher,et al.  A comparison of the accuracy of different functionals , 1995 .

[43]  V. Barone,et al.  Comparison of convetional and hybrid density functional approaches. Cationic hydrides of first-row transition metals as a case study , 1996 .

[44]  R. K. Nesbet,et al.  Self‐Consistent Orbitals for Radicals , 1954 .

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

[46]  N. Handy,et al.  Dynamical and Nondynamical Correlation , 1996 .

[47]  L. Cohen,et al.  Hartree–Fock density matrix equation , 1976 .

[48]  Stephen Wolfram,et al.  Mathematica: a system for doing mathematics by computer (2nd ed.) , 1991 .

[49]  L. Hernquist,et al.  Performance characteristics of tree codes , 1987 .

[50]  S. Ten-no An efficient algorithm for electron repulsion integrals over contracted Gaussian-type functions , 1993 .

[51]  Y. Saad,et al.  Finite-difference-pseudopotential method: Electronic structure calculations without a basis. , 1994, Physical review letters.

[52]  Martin Head-Gordon,et al.  A method for two-electron Gaussian integral and integral derivative evaluation using recurrence relations , 1988 .

[53]  Benny G. Johnson,et al.  The performance of a family of density functional methods , 1993 .

[54]  Sullivan,et al.  Large-scale electronic-structure calculations with multigrid acceleration. , 1995, Physical review. B, Condensed matter.

[55]  Paul Gibbon,et al.  A 3D hierarchical tree code for dense plasma simulation , 1994 .

[56]  Norman Wagner,et al.  Telescoping fast multipole methods using Chebyshev economization , 1995 .

[57]  Michael J. Frisch,et al.  Achieving Linear Scaling for the Electronic Quantum Coulomb Problem , 1996, Science.

[58]  Axel D. Becke,et al.  Density‐functional thermochemistry. IV. A new dynamical correlation functional and implications for exact‐exchange mixing , 1996 .

[59]  C. Van Alsenoy,et al.  Ab initio calculations on large molecules: The multiplicative integral approximation , 1988 .

[60]  R. Lindh,et al.  An efficient method of implementing the horizontal recurrence relation in the evaluation of electron repulsion integrals using Cartesian Gaussian functios , 1991 .

[61]  K. Esselink A comparison of algorithms for long-range interactions , 1995 .

[62]  Benny G. Johnson,et al.  The efficient transformation of (m0|n0) to (ab|cd) two-electron repulsion integrals , 1993 .

[63]  Henry F. Schaefer,et al.  New variations in two-electron integral evaluation in the context of direct SCF procedures , 1991 .

[64]  Roland Lindh,et al.  The reduced multiplication scheme of the Rys-Gauss quadrature for 1st order integral derivatives , 1993 .

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

[66]  Michael S. Warren,et al.  Fast Parallel Tree Codes for Gravitational and Fluid Dynamical N-Body Problems , 1994, Int. J. High Perform. Comput. Appl..

[67]  Joshua E. Barnes,et al.  Error Analysis of a Tree Code , 1989 .

[68]  K. Schmidt,et al.  Implementing the fast multipole method in three dimensions , 1991 .

[69]  H. Y. Wang,et al.  An efficient fast‐multipole algorithm based on an expansion in the solid harmonics , 1996 .

[70]  P. Löwdin Quantum Theory of Electronic Structure of Molecules , 1960 .

[71]  Martin Head-Gordon,et al.  Derivation and efficient implementation of the fast multipole method , 1994 .