quantum chemistry - 3-index integral schemes for density fitting

Density fitting allows us to approximate the 4-index 2-electron integrals using 3-index integrals:

$$(ij|kl) \approx \sum_{Q}^{N_{aux}} (ij|Q)(Q|kl)$$

My question is, what schemes are used for the evaluation of the 3-index integrals? Do we simply modify existing schemes such as Obara-Saika and McMurchie-Davidson or are completely separate schemes given for this purpose?

Answer

I think in the early days one really used just the normal integral schemes Obara-Saika, McMurchie-Davidson, Rys where for one exponent just a Gaussian with exponent zero was used. Later the people examined the schemes for this special purpose and modified them for calculating 3 index coulomb integrals. A quite recent paper on this topic can be found in Gyula Samu and Mihály Kállay, J. Chem. Phys. 2017, 146, DOI: 10.1063/1.4983393.

One minor thing I want to point out is that you presented RI/DF in a slightly unusual way (notation). Using RI/DF the integrals are evaluated as

$$(ij|kl) \approx \sum_{PQ}^{N_{aux}} (ij|P)[V^{-1}]_{PQ} (Q|kl)$$

where

$$V_{PQ} = \left(P|Q\right) = \int \int {{\phi_{P}({r_1}) \frac{1}{r_{12}} \phi_{Q}({r_2})}}d{r_1}d{r_2}$$

Of course you can rewrite it a bit with forming $V_{PQ}^{-1/2}$ to arrive at a similar expression as given by you.