# Understanding two-electron integrals in Gaussian 09

I was comparing the two-electron integrals (electron repulsion integrals, ERI) printed by various well-established packages (Gaussian, GAMESS, Molpro etc.) and observed that Gaussian is the oddball in including the factors coming from permutational symmetry ("redundancy factors").

Here I use water as an example, with the Huzinaga minimal basis set taken from bse.pnl.gov for consistency (built-ins in some packages had different primitive terms or number of decimal places).

$$\begin{array}{|c|c|}\hline O&0.0&0.0&0.1173\\\hline H&0.0&0.7572&-0.4692\\\hline H&0.0&-0.7572&-0.4692\\\hline \end{array}$$

I assumed no spatial symmetry (xyz coordinates) and used cartesian functions (not that it makes a difference with Huzinaga, but for later changes).

After the runs, the easiest one to understand was the output from GAMESS (option NPRINT=4) which left the integrals uncorrected for redundancy. Molpro (INT,SPRI=2) printed some numbers that are off by factors of 2, 4, 8 but otherwise agree with GAMESS.
But I fail to understand the output from Gaussian (SCF(Conventional) IOp(3/33=6) ExtraLinks=L316 Symmetry=None). Some of the numbers are off by factors of 2 or 4 (but not in the same entries as in Molpro) and some other numbers seem completely unrelated, even having opposite signs.

Q: Does anyone know the reason for this and how to read it correctly (to confirm agreement with other packages)?

Here are my outputs: GAMESS (line ~292), Molpro (line ~187), Gaussian 09 (line ~4810)
The indices of $$[ij|k\ell]$$ are in "canonical" order, ie. $$i\ge j$$, $$k \ge \ell$$, and $${i\choose2} + j \ge {k\choose 2} + \ell$$.

• @Tyberius: They're still alive from where I am ... Here's the link without the # reference to the lines.
– DC Y
Apr 25, 2019 at 11:02
• I think for some reason they didn't work right on a Mac, but they are working on mobile. I believe I may have found the issue with the Gaussian integrals. By default, I believe it prints the so called Raffenetti form of the two electron integrals, which are particular combinations of the ordinary two electron integrals. You can see an example of this on the page for Gaussian's matrix element files Apr 25, 2019 at 12:39
• This would fit well on the new Matter Modeling SE. Jul 16, 2020 at 16:12
• @DCY, did you figure it out? Jun 17 at 23:29