Automatic numerical integration techniques for polyatomic molecules

Abstract

We describe a new algorithm for the generation of 3D grids for the numerical evaluation of multicenter molecular integrals in density functional theory. First, we use the nuclear weight functions method of Becke [A. D. Becke, J. Chern. Phys. 88, 2547] to decompose a multicenter integral f F(r) dr into a sum of atomic-like single-center integrals. Then, we apply automatic numerical integration techniques to evaluate each of these atomic-like integrals, so that the total integral is approximated as f F(r) dr= 2,jw jF(rj). The set of abscissas rj and weights Wj constitutes the 3D grid. The 3D atomic-like integrals are arranged as three successive monodimensional integrals, each of which is computed according to a recently proposed monodimensional automatic numerical integration scheme which is able to determine how many points are needed to achieve a given accuracy. When this monodimensional algorithm is applied to 3D integration, the 3D grids obtained adapt themselves to the shape of the integrand F(r), and have more points in more difficult regions. The function F(r), which, upon numerical integration, yields the 3D grid, is called the generating function of the grid. We have used promolecule densities as generating functions, and have checked that grids generated from promolecule densities are also accurate for other integrands. Our scheme is very reliable in the sense that, given a relative tolerance E, it generates 3D grids which are able to approximate multicenter integrals with relative errors smaller than E for all the molecules tested in this work. Coarser or finer grids can be obtained using greater or smaller tolerances. For a series of 21 molecules, the average number of points per atom for E=2.0·1O-3 , E=2.0·1O-4 , E=2.0·1O-5 , E=2.0·1O-6 , and E=2.0·1O-7 is respectively 3141 parentheses are the maximum errors obtained when integrating the density). It is possible to reduce the number of points in the grid by taking advantage of molecular symmetry. It seems that our method achieves a given accuracy with fewer points than other recently proposed methods.In this paper, we will describe another scheme for the 6520 J.

SG‐0: A small standard grid for DFT quadrature on large systems

“…9 -18 Most of these can be classified as either "standard" models in which the grid points and weights are known beforehand, 9 -16 or "adaptive" schemes in which points and weights are chosen dynamically as the integrand is explored. 17,18 More than a decade ago, we introduced SG-1, a standard grid that aimed to yield moderately accurate results at a low computational cost. 11 It was designed to produce exchange-correlation energies with a computational effort comparable to that required to calculate the Coulomb energy using the linear-scaling Continuous Fast Multipole Method (CFMM) 19 and includes approximately 3700 grid points per atom.…”

Non-Extensive Entropies on Atoms, Molecules and Chemical Processes

“…For this, we calculated the electron density in the position space using the following functionals, B3LYP, BHandH, M062x, MP2, MP3 and TPSS, and with the following ab initio methods, CCS, CCSD, CISD, using a standard quantum chemistry program, Gaussian 09 [13], with the basis set DGDZVP [14,15], to obtain the energy value and the corresponding wave function. The electron density in the position space was calculated with the DGrid program [16] using the wave function obtained through several methodologies, and for the entropy calculations we used the integration algorithm designed by Pérez-Jordá et al [17] with a precision of 1 × 10 −5 .…”

Shannon Informational Entropies and Chemical Reactivity

“…The electron density was calculated with Pérez-Jordá's algorithms [8] and a D-Grid 4.6 [9]. Molecular Electrostatic Potential (MEP) isosurfaces were obtained with Molden 5.0 [22].…”