The Fourier transform Coulomb method: Efficient and accurate calculation of the Coulomb operator in a Gaussian basis

We describe a method for calculating the matrix elements of the Coulomb operator for Gaussian basis sets using an intermediate discrete Fourier transform of the density. Our goals are the same as those of the Gaussian and augmented-plane-wave method of Parrinello and co-workers [M. Krack and M. Parrinello, Phys. Chem. Chem. Phys. 2, 2105 (2000)], but our techniques are quite different. In particular, we aim at much higher numerical accuracy than typical programs using plane wave expansions. Our method is free of the effects of periodic images and yields full precision. Other low-scaling methods for the Coulomb operator are compared to the Fourier transform method with regard to numerical precision, asymptotic scaling with molecular size, asymptotic scaling with basis set size, onset point (the size of the calculation where the method outperforms traditional Gaussian integral techniques by a factor of 2), and the ability to calculate the Hartree–Fock exchange operator. The Fourier transform method is super...

[1]  Parr,et al.  Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. , 1988, Physical review. B, Condensed matter.

[2]  Gustavo E. Scuseria,et al.  What is the Best Alternative to Diagonalization of the Hamiltonian in Large Scale Semiempirical Calculations , 1999 .

[3]  Mark E. Tuckerman,et al.  A reciprocal space based method for treating long range interactions in ab initio and force-field-based calculations in clusters , 1999 .

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

[5]  Warren J. Hehre,et al.  AB INITIO Molecular Orbital Theory , 1986 .

[6]  Dunlap.  Three-center Gaussian-type-orbital integral evaluation using solid spherical harmonics. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[7]  Eric J. Bylaska,et al.  Parallel Implementation of the Projector Augmented Plane Wave Method for Charged Systems , 2002 .

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

[9]  Richard A. Friesner,et al.  Pseudospectral Hartree–Fock theory: Applications and algorithmic improvements , 1990 .

[10]  J. Pople,et al.  Self—Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian—Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules , 1972 .

[11]  A. Becke,et al.  Density-functional exchange-energy approximation with correct asymptotic behavior. , 1988, Physical review. A, General physics.

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

[13]  Walter Kohn,et al.  Density functional theory for systems of very many atoms , 1995 .

[14]  D. Vanderbilt,et al.  Soft self-consistent pseudopotentials in a generalized eigenvalue formalism. , 1990, Physical review. B, Condensed matter.

[15]  B. Delley.  An all‐electron numerical method for solving the local density functional for polyatomic molecules , 1990 .

[16]  J. Almlöf,et al.  Principles for a direct SCF approach to LICAO–MOab‐initio calculations , 1982 .

[17]  A. Becke.  A multicenter numerical integration scheme for polyatomic molecules , 1988 .

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

[19]  Jon Baker,et al.  Assessment of the Handy–Cohen optimized exchange density functional for organic reactions , 2002 .

[20]  A. Becke,et al.  Numerical solution of Poisson’s equation in polyatomic molecules , 1988 .

[21]  N. Handy,et al.  Left-right correlation energy , 2001 .

[22]  Matthias Krack,et al.  All-electron ab-initio molecular dynamics , 2000 .

[23]  P. Blöchl,et al.  Electrostatic decoupling of periodic images of plane‐wave‐expanded densities and derived atomic point charges , 1995 .

[24]  J. Pople,et al.  Self‐consistent molecular orbital methods. XX. A basis set for correlated wave functions , 1980 .

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

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

[27]  Florian Weigend,et al.  Auxiliary basis sets for main row atoms and transition metals and their use to approximate Coulomb potentials , 1997 .

[28]  Peter Pulay,et al.  Accurate molecular integrals and energies using combined plane wave and Gaussian basis sets in molecular electronic structure theory , 2002 .

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

[30]  Nicholas C. Handy,et al.  A Kohn-Sham method involving the direct determination of the Coulomb potential on a numerical grid , 1994 .

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

[32]  Peter Pulay,et al.  Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules , 1969 .

[33]  Yihan Shao,et al.  An improved J matrix engine for density functional theory calculations , 2000 .

[34]  Michael J. Frisch,et al.  Achieving linear scaling in exchange-correlation density functional quadratures , 1996 .

[35]  Jon Baker,et al.  Molecular energies and properties from density functional theory: Exploring basis set dependence of Kohn—Sham equation using several density functionals , 1997 .

[36]  Michele Parrinello,et al.  The Gaussian and augmented-plane-wave density functional method for ab initio molecular dynamics simulations , 1999 .