Efficient molecular numerical integration schemes

Abstract

New grids for three-dimensional numerical integration are introduced. They include a new mapping for radial integration of the Gauss–Chebyshev type which seems to surpass in accuracy the existing integration schemes as proposed by Becke [J. Chem. Phys. 88, 2547 (1988)], Murray et al. [Mol. Phys. 78, 997 (1993)], or Gill et al. [Chem. Phys. Lett. 209, 506 (1993)]. Lebedev grids are employed for spherical integration. Open ended quadrature schemes are presented using the efficient Lobatto formula for the θ integration. These grids are employed for self-consistent density functional calculations using local approximation and nonlocal corrections and are implemented into the program package turbomole. The results of grid tests and demonstrative applications of energy and especially analytical gradient calculations are given.

Polychlorinated trityl radicals for dynamic nuclear polarization: the role of chlorine nuclei

“…Pyruvic acid forms a glass and, in its enriched form, gives enough sensitivity to carry out 15 frequency sweeps by detecting the carbon signal in solid state. For transfer experiments, 1-13 C pyruvate sodium salt, 2-13 C acetone, or 13 C urea (99% 13 angle. Data were analyzed by using VNMRJ Varian and MestreNova software.…”

“…0 program 12 using grid 6 55 (Lebedev with 590 angular points) for the SCF iterations and grid 7 (Lebedev with 770 angular points) for the final point, and the Gauss-Chebyshev radial integration method. 13 The number of radial points is atom dependent, as specified in the user manual. In every case the tightest convergente option was 60 chosen for the SCF convergence.…”

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

“…Such profiles for hydrogen, carbon, nitrogen, and oxygen are shown in Figure 1 and reveal that ⌬E oscillates when R is too small or too large, but that the amplitudes of these oscillations diminish as N increases. Such oscillations have been noted by other researchers, 12,13 and Mura and Knowles argue that they arise for small R because the resulting grids are not sufficiently diffuse to treat the outer regions of the electron density. 13 Treutler and Ahlrichs suggest that the oscillations when R is large arise from the close approach of the grid points to a neighboring nucleus.…”

“…Instead, a number of schemes have been proposed to map the radial interval onto a finite one, usually [0,1], and then apply an appropriate quadrature. In 1993, Gill et al chose to base SG-1 on the Euler-Maclaurin radial grid, 10 but it is has been argued subsequently that the latter is not as effective as some other grids, including those of Becke, 9 Treutler and Ahlrichs, 12 and Mura and Knowles. 13 In 2003, we introduced a new radial quadrature 15 that combines the logarithmic transformation r ϭ Ϫ Rlnx (3)…”

“…13 Treutler and Ahlrichs suggest that the oscillations when R is large arise from the close approach of the grid points to a neighboring nucleus. 12 Mura and Knowles also propose that the amplitudes of these oscillations can be used as an indicator of the inherent numerical inaccuracy of the radial grids. It therefore follows that an optimal radial grid will have an R value within the range where both the frequency and the amplitudes are small.…”