Radial Gausslets

Abstract

Gausslets are one of the few examples of basis sets for electronic structure which allow for two-index/diagonal electron-electron interaction terms. A weakness of gausslets is that, because of their 1D origin, they have been tied to Cartesian coordinates. Here we generalize the gausslet construction for the radial coordinate in three dimensions for atomic basis sets. These radial gausslets make a very compact radial basis with a relatively modest number of functions, with diagonal interaction terms. We illustrate the accuracy of this construction with Hartree--Fock and exact diagonalization on atomic systems.

Figures (6)

• Figure 1: Boundary gausslets, where $N_t=6$ extra edge gausslet tails were added to ensure completeness near $x=0$. The gausslets become noticeably similar to their original form in location and shape for $x \ge 4$.
• Figure 2: Radial gausslets, formed from the boundary gausslets shown in Fig. 1 for $N_t=6$ but with the fit to $\delta(x)$ removed. All the functions vanish at $x=0$. The dashed vertical line for each function shows $\bar{x}_i$, while the solid line shows $x_i$. These two moments are exactly equal for the boundary gausslets. For $x \ge 4$, the radial gausslets are very similar to the boundary gausslets and $x_i \approx \bar{x}_i$.
• Figure 3: Radial gausslets including two extra x-gaussians added; otherwise, similar to Fig. 2. The widths of the two Gaussians were optimized to make all peaks have minimal separation between $x_i$ and $\bar{x}_i$
• Figure 4: Error in the energy of the hydrogen atom versus baseline gausslet spacing for several forms of radial gausslets, when IDA is used for the one--particle potential. All basis functions were scaled uniformly by $a$. The red curve shows the odd-even basis without any extra x-gaussians. The green and blue show with one or two optimized x-gaussians added. The black curve is with the exact matrix elements of the basis, i.e. the Galerkin form of the potential.
• Figure 5: Error in the RHF energy of the He atom energy versus number of radial gausslets for $K=6$, with two added x-gaussians. Here $c$ represents the core spacing of the coordinate transformation, and the symbols correspond to different values of $s$, ranging from $0.7$ to $0.05$. Radial functions are omitted if their center is at radius greater than $10$. The actual spacing of functions near $r=0$ is about an order of magnitude smaller than $c$, obtained by shrinking the functions of Fig. 3 by a factor of $c$.