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I want to compute 2-electron coulomb integrals. When the Primitive Gaussian is not centered at 0, my quadrature scheme fails unless I increase the number of quadrature points. I am currently using a quadrature scheme for the 4-indexed integral:

  • one integral involving the Boys function (1 coordinate)
  • three integrals over $x_1 \in \mathbb{R}^3$

The integrals over the second coordinates $x_2$ are evaluated exactly at quadrature points.

Their respective numerical schemes are as follow:

  • Gauss Chebyshev of the second type for one coordinate
  • a tensor product of Gauss Chebyshev quadrature points for a 3D integration

My problem is that my integration over $\mathbb{R}^3$ scales pretty badly as I increase the number of quadrature points. So maybe using another scheme would be better, like Lebedev? Or are there other methods to simplify the 2-electron integrals?

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Integral calculation is discussed extensively in the book Molecular Electronic-Structure Theory by Helgaker, Jørgensen, and Olsen.

2-electron integrals are usually computed analytically with the McMurchie-Davidson or Obara-Saika methods and further developments thereof, or with Rys quadrature which solves for the exact quadrature nodes and weights. Later on, schemes based on approximate quadrature have also been suggested.

You do not discuss what you are actually doing, but it seems you should first study the existing literature. I also point out that there exist efficient open source libraries for evaluating two-electron integrals, such as LIBINT by Ed Valeev and collaborators, and libcint by Qiming Sun.

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