# what is the issue in evaluating two electron integral?

What sort of issue is there in two electron integral in quantum chemistry ? Is it a problem of convergence when we do numerically? Can someone suggest any review article or reference articles which has the numerical algorithm to evaluate and to understand it fully ?

• There is no issue, we can calculate such an integral all right. It is just that we have an awful lot of them. – Ivan Neretin Nov 1 '18 at 9:35

$$$$I_{ijkl} = \langle ij|r^{-1}_{12}|kl\rangle$$$$
where $$i$$, $$j$$, $$k$$ and $$l$$ run over all $$N$$ atomic orbitals. So we need to calculate $$\mathcal{O}(N^4)$$ integrals, which makes this a bottleneck (in terms of memory and computing time).
For Gaussian Type Orbitals we can evaluate the integrals analytically, which is much faster than numerical approaches. But this does not help with the $$N^4$$ scaling.
Evaluation for Gaussian Type Orbitals is explained in Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory by Szabo and Ostlund for s orbitals ($$l=0$$). More general evaluation schemes for higher angular momentum are explained in detail in Molecular Electronic-Structure Theory by Helgaker, Jorgenson and Olsen.