[Numerical tests] particle-particle ERPA/AC

[gabrielasd]

Some results for pp-RPA can be found in the references listed bellow.

(indirect) pp-RPA spectrum: 10.1063/1.4834875: Tables VI and VII in support doc, for BH and CH cation excitation energies. The calculations are ran with cc-pVQZ basis, which we can not do, but alternative results can be found in tables II and III here: 10.1063/1.4994827, this time with aug-cc-pvdz.

pp-RPA correlation energy: 10.1063/1.4820557: Figs 1. and 4, H2 and LiH dissociation curves (with RHF MOs as reference) 10.1063/1.4820556: Table I, ground state energies for atoms and small molecules (cc-pVTZ basis) 10.1103/PhysRevB.99.195149: has binding curves for H2, N2 and HF, but I'd have to run CCSD and MRCISD jobs to be able to compare against their data (also, N2 with cc-pVTZ is off-limits for our current ERPA implementations)

[gabrielasd]

Hi @PaulWAyers

this are the tests I have so far to validate our pp-(E)RPA and pp-AC implementations: numerical_tests_pprpa.docx The jobs were run with the code in the develop branch (updated branch).

let me know what you thinks and whether I should add further clarification to the document. For now all I have a re manual tests, but if these are good, I'll include the content (part of it) in the test module.

[PaulWAyers]

Idon't know what to think. It's not quite what other people are getting, but it's good enough (and often the same) that I can't imagine that we have a bug. Some weird threshold stuff probably, but I don't think it's worth tyring to find a way to match their data.

[gabrielasd]

@PaulWAyers this is a summary of my latest tests for pp-AC on top of RHF.

I think the results in the attached file give a more definitive validation of the implemented pp/hh-AC and ERPA methods. Or at least that our generalized version (spin-adapted) properly reduces to the known RPA form.

We could reproduce the pp-RPA results for H2 and LiH from Scuseria, and N2 from Weitao (suplementary information for 10.1103/PhysRevA.88.030501) To show that both hh and pp channels gave the same answer they results for all systems are included; they correspond to curves labelled hh@RHF and pp@RHF.

Part of the verification done was that the matrix elements of the generalized code matched those of the usual pp-RPA matrix structure (from an ad-hock implementation of the equations in Weitao's paper); the results from this latter implementation correspond to curve hh-plasmon (I used the formula based on the N-2 side of the spectrum since its usually the lower number of eigenvalues).

Graphs are here: ppRPA_implementation_validation.docx

There are still some issues though, particularly related to the reconstruction of the 2-RDM. I don't get a full match between the 2-RDM from the particle-particle resolution vs the hole-hole resolution at points other than at alpha=0 along the adiabatic connection path (and I believe the 2RDM from the hole-hole problem is the correct one). However, at the scale resolution in the graphs here, and from the results I've seem for DOCI, it doesn't appear to be a problem. But I'm let to look further into this.