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Many software packages provide means to calculate the Coulomb and exchange integrals, for example, the exchange

$$\int \phi^*(1) \, \psi^*(2) \frac{1}{|\mathbf{r_1} - \mathbf{r_2}|} \, \psi(1) \, \phi(2) \, d\mathbf{r_1} \, d\mathbf{r_2} $$

using gaussian type orbitals, either in cartesian or spherical coordinates.

I would like to know how this integral is evaluated in practice, since the Coulomb term $1/|\mathbf{r_1}-\mathbf{r_2}|$ should diverge whenever two different electron densities' volume elements overlap. I understand that when you use spherical coordinates, you can get rid of $1/r_{12}$ because the spherical volume element becomes something like $r^2 \cos{\theta} \, dr \, d\theta \, d\phi$, so you cancel $r^2$ in the numerator with $r$ in the denominator. But I'm having trouble to see how can this be done for cartesian GTO in cartesian coordinates.

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    $\begingroup$ There are a number of methods, and a number of introductions to them - a quick search shows up esqc.org/lectures/WK4.pdf which looks like a relatively straightforward introduction to the evaluation of integrals with GTOs, and the material in onlinelibrary.wiley.com/doi/book/10.1002/9781119019572 is good as well. Szabo and Ostlund in the appendix show how to evaluate the ssss term, which in practice is all you need as the rest are derived from recurrences starting at this term. $\endgroup$
    – Ian Bush
    Jan 22, 2022 at 9:52
  • $\begingroup$ @IanBush, great, thanks for the references. Yep, old and faithful integral transforms are the answer. Szabo and Ostlund perform a three-dimensional Fourier transform, and the slides from Wim Klopper present an integral transform of $1/r$ that results in the Boys function, now I understand that better. Unfortunately, I don't have access here to the book on Molecular Electronic‐Structure Theory, but will check it later. $\endgroup$
    – Arc
    Jan 22, 2022 at 15:45
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    $\begingroup$ The Molecular Electronic‐Structure Theory book is very good but not cheap - but your institution might give you online access depending where you are, try onlinelibrary.wiley.com/doi/pdf/10.1002/9781119019572.ch9 . And as well as integral transforms the Gaussian Product Theorem is the other tool you need to do this. Later will see f I have time/can write an answer, unfortunately lots of maths needed and my Math jax-fu is not strong. $\endgroup$
    – Ian Bush
    Jan 22, 2022 at 16:13
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    $\begingroup$ Check out joshuagoings.com/2017/04/28/integrals for as simple an explanation as one will find, along with clear Python code. Say hello to the Boys function. $\endgroup$
    – B. Kelly
    Jan 24, 2022 at 1:19
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    $\begingroup$ @IanBush, my MathJax is ok, so I can help you, if you care to write an answer in the following days, I can fix the math for you. I think its worth to have a fine answer, as this question attracted some upvotes. $\endgroup$
    – Arc
    Jan 24, 2022 at 6:01

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