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 ?

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    $\begingroup$ There is no issue, we can calculate such an integral all right. It is just that we have an awful lot of them. $\endgroup$ Commented Nov 1, 2018 at 9:35

1 Answer 1


The main problem of 2-electron integrals is their amount. They are given by

\begin{equation} I_{ijkl} = \langle ij|r^{-1}_{12}|kl\rangle \end{equation}

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.

  • $\begingroup$ Thanks a lot.Can you also please tell whether monte carlo integration technique useful in evaluating this numerically ? $\endgroup$
    – user135580
    Commented Nov 1, 2018 at 13:40
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    $\begingroup$ Monte Carlo would be a variant of numerical integration, which has pros and cons. Anyway there is no point in applying numerical methods when you have an analytic one. Unless you want to use Slater Type Orbitals. There is software available using them, but I don't know how they evaluate their integrals. $\endgroup$
    – Feodoran
    Commented Nov 1, 2018 at 13:47

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