Filtering a distribution simultaneously in real and Fourier space

We present a method to filter a distribution so that it is confined within a sphere of given radius $r_c$ and, simultaneously, whose Fourier transform is optimally confined within a sphere of radius $k_c$. Our procedure may have several applications in the field of electronic structure methods, like the generation of optimized pseudopotentials and localized pseudocore charge distributions. As an example, we describe a particular application within the SIESTA method for density functional calculations, in removing the spurious rippling of the energy surface generated by the integrations in a real-space grid.

Figure 1

Number of polynomials required to obtain a root mean square error of the expansion of $x{j}_0\left(x\right)$ in the interval $0<x<k_c{r}_c$. $j_0\left(x\right)$ is a spherical Bessel function with $l=0$.

Figure 2

First few eigenfunctions (with highest eigenvalues) of the filter kernel $K^2(x,x^{\prime })$ for $\kappa \equiv k_c{r}_c=25$. Divided by $r$, they give the radial part of the distributions, with angular momentum $l$, that are most localized in a real-space sphere of radius $r_c$ and simultaneously in a reciprocal-space sphere of radius $k_c$.

Figure 3

Eigenvalues of the filter kernel $K^2(x,x^{\prime })$ for $\kappa =25$ and $l=0$ (circles and full line), 1 (squares and dashed line), and 2 (diamonds and dotted line).

Figure 4

Filtered (dashed lines) and unfiltered (full lines) oxygen $2p$ pseudo-atomic-orbital, generated with a strict cutoff $r_c=3.94\text{bohr}$ as proposed by Sankey and Niklewski (Ref. 10). It was filtered with a cutoff $\kappa =25$, which corresponds to a plane-wave cutoff $k_c=6.35{\text{bohr}}^{-1}$, or $k_c^2=40\mathrm{Ry}$. Upper panel: real-space shape. Lower panel: tails of their Fourier transforms.

Figure 5

(a) Total energy of an isolated carbon atom, as it is displaced in a large unit cell, using an integration grid with a plane-wave cutoff $k_c^2=50\mathrm{Ry}$, whose points are separated by $\Delta x=\pi ∕k_c=0.44\text{bohr}$. (b) Magnitude of the eggbox effect (peak to peak of total energy) for several isolated atoms with hard pseudopotentials or nonlocal core corrections (in Pb and Fe), as a function of the plane-wave cutoff of the integration mesh. Full lines, without filtering; dashed lines, with filtering.

Figure 6

Vibrational frequencies of the water molecule, calculated from the Hessian matrix, which was obtained by finite differences from the forces on the atoms displaced from their equilibrium positions. The $x$ axis is the plane wave cutoff of the integration grid used in SIESTA. Full lines, without filtering; dashed lines, with filtering.