An overlap fitted chain of spheres exchange method.

The "chain of spheres" (COS) algorithm, as part of the RIJCOSX SCF procedure, approximates the exchange term by performing analytic integration with respect to the coordinates of only one of the two electrons, whereas for the remaining coordinates, integration is carried out numerically. In the present work, we attempt to enhance the efficiency of the method by minimizing numerical errors in the COS procedure. The main idea is based on the work of Friesner and consists of finding a fitting matrix, Q, which leads the numerical and analytically evaluated overlap matrices to coincide. Using Q, the evaluation of exchange integrals can indeed be improved. Improved results and timings are obtained with the present default grid setup for both single point calculations and geometry optimizations. The fitting procedure results in a reduction of grid sizes necessary for achieving chemical accuracy. We demonstrate this by testing a number of grids and comparing results to the fully analytic and the earlier COS approximations. This turns out to be favourable for total and reaction energies, for which chemical accuracy can now be reached with a corresponding ~30% speedup over the original RIJCOSX procedure for single point energies. Results are slightly less favourable for the accuracy of geometry optimizations, but the procedure is still shown to yield geometries with errors well below the method inherent errors of the employed theoretical framework.

[1]  Frank Neese,et al.  Efficient Structure Optimization with Second-Order Many-Body Perturbation Theory: The RIJCOSX-MP2 Method. , 2010, Journal of chemical theory and computation.

[2]  Roland Lindh,et al.  Density fitting with auxiliary basis sets from Cholesky decompositions , 2009 .

[3]  Frank Neese,et al.  Accurate theoretical chemistry with coupled pair models. , 2009, Accounts of chemical research.

[4]  F. Neese,et al.  Efficient, approximate and parallel Hartree–Fock and hybrid DFT calculations. A ‘chain-of-spheres’ algorithm for the Hartree–Fock exchange , 2009 .

[5]  R. Lindh,et al.  Low-cost evaluation of the exchange Fock matrix from Cholesky and density fitting representations of the electron repulsion integrals. , 2007, The Journal of chemical physics.

[6]  S. Grimme,et al.  How to compute isomerization energies of organic molecules with quantum chemical methods. , 2007, The Journal of organic chemistry.

[7]  Wenwu Cao,et al.  Applied Numerical Methods Using MATLAB®: Yang/Applied Numerical MATLAB , 2005 .

[8]  Frederick R. Manby,et al.  Fast Hartree–Fock theory using local density fitting approximations , 2004 .

[9]  Seiichiro Ten-no,et al.  Explicitly correlated second order perturbation theory: introduction of a rational generator and numerical quadratures. , 2004, The Journal of chemical physics.

[10]  F. Weigend,et al.  Gaussian basis sets of quadruple zeta valence quality for atoms H–Kr , 2003 .

[11]  Frank Neese,et al.  An improvement of the resolution of the identity approximation for the formation of the Coulomb matrix , 2003, J. Comput. Chem..

[12]  Florian Weigend,et al.  A fully direct RI-HF algorithm: Implementation, optimised auxiliary basis sets, demonstration of accuracy and efficiency , 2002 .

[13]  Emily A. Carter,et al.  Automated selection of optimal Gaussian fits to arbitrary functions in electronic structure theory , 1998, J. Comput. Chem..

[14]  Christian Ochsenfeld,et al.  Linear and sublinear scaling formation of Hartree-Fock-type exchange matrices , 1998 .

[15]  Matthias Krack,et al.  AN ADAPTIVE NUMERICAL INTEGRATOR FOR MOLECULAR INTEGRALS , 1998 .

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

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

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

[19]  C. Wüllen.  A hybrid method for the evaluation of the matrix elements of the Coulomb potential , 1995 .

[20]  Emily A. Carter,et al.  PSEUDOSPECTRAL METHODS APPLIED TO THE ELECTRON CORRELATION PROBLEM , 1995 .

[21]  Marco Häser,et al.  Auxiliary basis sets to approximate Coulomb potentials (Chem. Phys. Letters 240 (1995) 283-290) , 1995 .

[22]  Jan Almlöf,et al.  DIRECT METHODS IN ELECTRONIC STRUCTURE THEORY , 1995 .

[23]  D R Yarkony,et al.  Modern electronic structure theory , 1995 .

[24]  J. Tromp,et al.  Variational discrete variable representation , 1995 .

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

[26]  Jon Baker,et al.  The effect of grid quality and weight derivatives in density functional calculations , 1994 .

[27]  A. Schäfer,et al.  Fully optimized contracted Gaussian basis sets of triple zeta valence quality for atoms Li to Kr , 1994 .

[28]  J. Almlöf,et al.  Integral approximations for LCAO-SCF calculations , 1993 .

[29]  Benny G. Johnson,et al.  A standard grid for density functional calculations , 1993 .

[30]  Hans W. Horn,et al.  Fully optimized contracted Gaussian basis sets for atoms Li to Kr , 1992 .

[31]  Itai Panas,et al.  A fragment multipole approach to long-range Coulomb interactions in Hartree-Fock calculations on large systems , 1992 .

[32]  Itai Panas,et al.  ABINITIO METHODS FOR LARGE SYSTEMS , 1991 .

[33]  R. Friesner,et al.  Pseudospectral Hartree-Fock gradient calculations , 1991 .

[34]  Weitao Yang,et al.  The collocation method for bound solutions of the Schrödinger equation , 1988 .

[35]  Richard A. Friesner,et al.  An Automatic Grid Generation Scheme for Pseudospectral Self-Consistent Field Calculations on Polyatomic Molecules , 1988 .

[36]  Richard A. Friesner,et al.  Solution of the Hartree–Fock equations for polyatomic molecules by a pseudospectral method , 1987 .

[37]  J. Light,et al.  Highly excited vibrational levels of "floppy" triatomic molecules: A discrete variable representation - Distributed Gaussian basis approach , 1986 .

[38]  Richard A. Friesner,et al.  Solution of the Hartree–Fock equations by a pseudospectral method: Application to diatomic molecules , 1986 .

[39]  Richard A. Friesner,et al.  Solution of self-consistent field electronic structure equations by a pseudospectral method , 1985 .

[40]  J. Light,et al.  Generalized discrete variable approximation in quantum mechanics , 1985 .

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

[42]  N. H. Beebe,et al.  Simplifications in the generation and transformation of two‐electron integrals in molecular calculations , 1977 .

[43]  Evert Jan Baerends,et al.  Self-consistent molecular Hartree—Fock—Slater calculations I. The computational procedure , 1973 .

[44]  J. L. Whitten,et al.  Coulombic potential energy integrals and approximations , 1973 .

[45]  D. K. Harriss,et al.  Carlson's Sum over Points and Local Energy , 1969 .

[46]  C. M. Carlson.  Use of Linear Nonsymmetric Eigenvalue Equations for a Sum‐over‐Points Approach: Application to H2+ , 1967 .

[47]  Roland Lindh,et al.  Cholesky Decomposition Techniques in Electronic Structure Theory , 2011 .

[48]  Robert Zaleśny,et al.  Linear-Scaling Techniques in Computational Chemistry and Physics , 2011 .

[49]  Wŏn-yŏng Yang,et al.  Applied Numerical Methods Using MATLAB , 2005 .