Why is my quantum chemistry software not printing as many 2e- integrals as I expect?

Question

Background: In Hartree-Fock theory, the two-electron integral is given by:

$[ij|kl]$ = $\int dx_{1}dx_{2}(\chi_{i}(x_{1})\chi_{j}(x_{1})\frac{1}{r_{12}}\chi_{k}^{*}(x_2)\chi_{l}(x_{2})$

I am interested in writing out the 2e-integral for the Helium atom (for a given basis set) in PySCF. I want to do this with, and without density-fitting (DF)/resolution-of-identity (RI).

Attempt: In my attempt to produce the 2e-integral from PySCF (without DF/RI), I have used the following code:

from pyscf import gto
mol = gto.M(atom='He 0 0 0', basis='6-31g')
eri = mol.intor('int2e', aosym='s8')
print(eri)

>> [1.54393568 0.78544719 0.42702546 0.80395186 0.47618957 0.61593794]

Background: I was expecting a much larger data set. I also imagined that there should be an SCF loop; however, this is not indicated by the PySCF documentation. Why am I getting unexpected results?

Thank you in advance for any help.

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Note: This question is related to the following earlier questions:

• One-center two-electron integrals between 1s STO
• What programs can do electron integrals and SCF calculations with more than one CPU?
• Exchange-Correlation Two-Electron Integrals

Answer 1

I figured it might be worthwhile to address some the misconceptions in the question, as they are probably fairly common.

No SCF prior

The AO two-electron integrals (and actually all the AO integrals used in the SCF procedure) can be generated before doing any cycles. This is because they are just among atomic orbitals, which we know from the beginning of the calculation. In practice, you will probably see two-electron integrals recomputed during the SCF cycle, but this is just to avoid storing ~$N^4$ such values, which becomes exorbitant for even fairly small calculations.

To compute MO integrals, you would need to have completed all the SCF cycles, but as long as you aren't doing any Post-SCF methods, you don't actually need to explicitly transform the AO integrals to MO in order to compute the energy.

Fewer integrals than expected

This just comes down to symmetry. For $\ce{He}$ with the 6-31G basis set, you have two atomic orbitals, which should lead to $2^4=16$ two-electron integrals. However, because the integrals have permutational symmetry (see this question on Chem SE), there are actually only 6 unique integrals, which you have written in your question.

The mol.intor('int2e', aosym='s8') command explicitly asks it to use this permutational symmetry. If you wanted it to simply produce all $n^4$ integrals, you could instead use mol.intor('int2e', aosym='s1').