I have a question about the actual calculation of classical Coulomb interaction term in density functional theory (DFT) softwares.

Of course I understand the form Coulomb energy as:

$\int\int \frac{\rho(r_i)\rho(r_j)}{|r_i - r_j|}dr_idr_j$,

but in the actual calculation, this term seems to be computationally expensive due to the distances between all grid points, $r_i$ and $r_j$, set in DFT softwares.

In reality, in a practical calculation, such as that of PySCF, even a small molecule has a large number (e.g. over 100,000) of grid points, so the calculation of all grid distances appears to be very expensive.

Does the actual calculations rely on specific technique for efficiently calculating the grid distances?

  • 2
    $\begingroup$ First note the equation you have written is for the Coulomb energy, not the potential (sort of). Secondly are you particularly interested in one kind of basis set? Plane-waves? Gaussians? $\endgroup$
    – Ian Bush
    Aug 3, 2023 at 13:03
  • $\begingroup$ Thank you for the comment. I have fixed. I consider the calculation of organic molecules and Gaussian or PySCF. $\endgroup$
    – neco
    Aug 4, 2023 at 2:21
  • $\begingroup$ For Gaussian basis sets, real codes use cutoffs for distances between the centers of basis sets $\endgroup$
    – B. Kelly
    Aug 4, 2023 at 6:12
  • $\begingroup$ Poisson equation can be used or you mean the exact exchange. $\endgroup$
    – M06-2x
    Aug 4, 2023 at 17:44
  • $\begingroup$ Also are you interested only in molecular systems, or ones with periodic boundary conditions as well? $\endgroup$
    – Ian Bush
    Aug 5, 2023 at 11:23


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