Exchange-Correlation Two-Electron Integrals

Question

Do any freely available or proprietary electronic structure packages explicitly compute two-electron exchange correlation integrals?

I have found a few derivations to calculate excited state properties/spectroscopic observables using DFT,^[1] and they tend to write expressions including integrals of the form

$$\langle\mu\nu|w|\lambda\sigma\rangle$$

where $w$ is the exchange-correlation kernel. However, I have yet to find a package that specifically generates these integrals. It seems that in general they do some sort of prior contraction with the density to avoid the need to generate these terms explicitly when calculating response properties. However, I had wanted to make a small standalone script that used these quantities. Its presumably much less efficient than the approach used in these packages, but the implementation is much clearer for me. Are there any packages that have an existing option to do this? Or is it something that would require some modification of either these packages themselves or some existing API?

1. S. Hirata, M. Head-Gordon, R.J. Bartlett Configuration interaction singles, time-dependent Hartree–Fock, and time-dependent density functional theory for the electronic excited states of extended systems J. Chem. Phys. 111, 10774 (1999); DOI: 10.1063/1.480443

Answer 1

I assume you're referring to eq 51 of the Hirata-Head-Gordon-Bartlett paper.

One should note that these are not two-electron integrals, since there is only one spatial position; these are rather weighted four-center one-electron integrals.

As always, the problem when you have four indices is that there is a huge number of integrals that come out, and you might not have storage for them.

Another issue is that the set of four-products is linearly dependent to a ridiculous degree. If you start out with a atomic basis set, in usual electron repulsion integrals you get basis function products. Most of these will be linearly dependent, and you get out a linearly growing number of independent functions (this is why the Cholesky decomposition is so powerful in repulsion integrals) and the rest $O(N^2)$ are linearly dependent.

Now, instead of basis function products, you have products of basis function products. Again, you only get a linearly growing number of independent functions, with a prefactor higher than in the two-electron case, meaning that the number of linearly dependent functions grows as $O(N^4)$. So, you'd like to get a large number of integrals that are mostly linearly dependent.

Other than this issue, there's nothing that would prevent you from evaluating the integrals in the same way as is done in the paper. That is, you just need numerical quadrature to do it.

If you had an optimized version of the procedure, one could extract the integrals by $O(N^2)$ calls to eq (54). But this is going to be so costly that I doubt that you could do it in anything except the smallest basis set...