Queerness/Issue observed with RI-basis set

[evgeniy]

Hello,

I encountered the following, at first glance counter-intuitive result for an rimp2 calculation for a deprotonated (anionic) closed-shell molecule.
Since it is anion diffuse functions could be important. So, I set up a mp2 calculation using the aug-cc-pVDZ basis set.
For some reason I changed only the main basis to aug-cc-pVDZ but the auxiliary basis (cbas), which was set to cc-pVDZ. This is case 1.
When I saw the inconsistency, I repeated the calculation with the right, i.e. aug-cc-pVDZ, auxiliary basis set. This is case 2.
When I compared the mp2 energies in case 1 and 2 I found out that the mp2 energy in case 1 (inconsistent cbas basis set)  is slightly
(~0.2 kcal/mol) lower than the energy in case 2 (the main and cbas basis sets are consistent). This was unexpected as I thought that
the larger the cbas basis set, the more closer the result to that obtained without the use of RI. In fact it turned out that the mp2 energy
obtained with the smaller cbas basis set, i.e. case 1, is slightly lower than the mp2 energy obtained without the use of RI.

I wounder whether the above described queerness is known and what the reason of it is (can be)?

Best regards,
Evgeniy

[JakubV]

Hello,

I would think that absolute value of difference between MP2 without RI and RI-MP2 shall converge (?monotonously) with the size of the auxiliary basis set to zero, sure.

But, on the other hand side, is there any theorem such as Ritz variational theorem? MP2 method alone is not variational. Perturbational methods could give energy below FCI and also converge with the size of basis set "from below" to their (atomic) basis set limits.

With auxbasis - could it be, that there is no clear "from below"/"from above" bound and the difference might even be sort of random? Thus perhaps the convergence towards (non-RI-)MP2 even non-monotonous?

Best regards,
Jakub

P.S.:  I am working with elements which does not have auxbases in the standard library (such as uranium), so I would be interested to know more on this topic too.

[christof.haettig]

Just two remarks:
   - once the (main) basis set describes the HF wavefunction good enough the MP2 energy does converge monotonically the from above to the
      MP2 basis set limit
   - the difference between the RI-MP2 and the MP2 energy as coulomb and exchange correlation contributions and is not a positive definite quantity.
     Thus it can fluctuate around 0 and accidentially be for a given compound and structure close to 0 for a small auxiliary basis. There is nothing like
     a Ritz variational theorem for it. But for systematically optimized basis sets the errors go on average systematically down.
     With the "consistent" combinations of main and auxiliary basis sets the RI error in the MP2 energy should be small compared to the remaining
     (orbital) basis set error in the MP2 energy.