# Four-electron integrals over GTOs

I was looking through the following presentation: http://folk.uio.no/helgaker/talks/SostrupIntegrals_10.pdf

Looking at the section on 4-electron integrals, I couldn't help but notice that the slides wrote that the 4-electron terms (Kij and Jij) were Gabcd x Dcd. [see slide 25]

I realised that Dcd was referring to the dipole-moment integral. Maybe I just missed something here, but what does the dipole moment integral have got to do with the 4-electron integrals?

This is particularly confusing for me since I notice that the guide here (which uses the same notation afaik) uses Dij^2 as the auxiliary kinetic-energy integral.

https://joshuagoings.com/2017/04/28/integrals/

• $J_{ij}$, $K_{ij}$ are certainly not four-electron quantities (which probably are a thing in extended, explicitly correlated theories). We could debate whether those are one- or two-electron quantities. On slide 25 of the first link, $D$ refers to the density matrix (often called $P$ in other literature), compare slide 30. This is also clear from the context (Fock matrix $F$ from $J,K$ and density matrix). The $g_{abcd}$ on slide 25 are two-electron electron integrals, which are typically four-center. – TAR86 Sep 9 '19 at 18:10