# Avoid the singularity of Coulomb matrix elements in grid-based calculations

Whenever one is doing grid-based calculations for particles interacting via the Coulomb potential, the singularity of the Coulomb potential $w(\mathbf r_1,\mathbf r_2) = \frac1{|\mathbf r_1 - \mathbf r_2|}$complicates the whole thing tremendously. Particularly for three-dimensional cartesian grids, whenever two gridpoint-triple $(x_i,y_j,z_k)$ coincide one gets infinity and that spoils any numerics.

I know of the following methods to avoid this unpleasant behavior (plus I add my experiences):

1. Cutoff parameters: introduce a suitable number $\alpha$ and replace the Coulomb potential by $\frac1{|\mathbf r_1 - \mathbf r_2 + \alpha^2|}$. This is very simple but it's not easy to find the "right" cutoff parameter $\alpha$ (meaning that the singularity is removed while the result is the same as for an exact method).

2. Applying the Poisson equation: the idea here is to evaluate the electrostatic potential of one particle in the given representation, then project on the other. The singularity somehow disappears in this process. This approach is generic, but rather costly (at least O(gridpoints))

3. Use some alternative representation for the Coulomb potential, such as Fourier or Laplace transform, other coordinate systems, projection on other basis functions etc. I have the impression these are often some kind of workarounds where the singularity is usually hidden in an infinite sum which is then truncated, or the like.

I'd like to do some large scale quantum many-body calculations on extensive grids, too large to apply accurate evaluation methods such as 2. or 3., still at the same time I want correct results. Is there any way to accomplish that?

• Probably useful: physics.stackexchange.com/q/126987/25301 – Kyle Kanos Nov 14 '17 at 11:09
• Since there is a lack of details (what the particles are? bosons? fermions? what particle functions are used?), it is hard to answer this question. This problem for electrons is solved in DFT calculations: take a look at grids, used in DFT for integrals, which do not have analytical solution; in particular, at Lebedev grids. – ancient_polaroid Nov 10 '18 at 1:56
• Can you explain 2) a bit more? What other representation? – lalala Nov 10 '18 at 7:36
• @ancient_polaron: the question is meant independently of the particle species. In second quantization, you basically need the electron integrals for either fermions or bosons. But, fair point, for fermions you can use anti-symmetry in certain cases and get rid of the requirement to calculate the wavefunction at gridpoints where particle locations coincide. – davidhigh Nov 10 '18 at 7:46
• @lalala: with representations I meant the "Poisson world" and the "grid world". Nothing fancy, but it's indeed not expressed all too clearly. – davidhigh Nov 10 '18 at 8:35

The simplest way that I've used to avoid such singularities is to define the coordinate axes in such a way that the origin does not coincide with a grid point. In other words, instead of $$x=n \Delta$$, where $$n$$ is an integer representing the grip points and $$\Delta$$ is the grid spacing, define it as $$x=(n+0.5)\Delta$$.