Selecting eigenstates to include in adiabatic connection corrections.

[gabrielasd]

For the AC routine one needs to select the states corresponding to the ERPA approximation being used out of all the solutions to the ERPA problem, e.g., for particle-hole ERPA those corresponding to electron excitations.

Currently, we do this selection based solely on the sign of the eigenvalues, picking the positive side of the spectrum (which generally works well).

See: https://github.com/gabrielasd/eomee/blob/03e9efe9ed191f0699f5da66f75bcffa8ae54905/eomee/excitation.py#L317 https://github.com/gabrielasd/eomee/blob/03e9efe9ed191f0699f5da66f75bcffa8ae54905/eomee/spinadapted/particlehole.py#L394 https://github.com/gabrielasd/eomee/blob/03e9efe9ed191f0699f5da66f75bcffa8ae54905/eomee/doubleionization.py#L208 https://github.com/gabrielasd/eomee/blob/03e9efe9ed191f0699f5da66f75bcffa8ae54905/eomee/spinadapted/holehole.py#L275

However, some times, there can be states of different origin mixed in, e.g., for hole-hole ERPA, there can be N+2 states appearing as positive eigenvalues (usually one would expect these states to appear as negative eigenvalues solution to the ERPA problem).

Therefore, an additional selection criteria could be the overlap of the eigenstates, which is positive for the states corresponding to the transition operator being described.

[PaulWAyers]

I'll need you to explain this in a bit more detail. I think I get the main idea. Certainly it always takes energy to remove two electrons, so the double-IPs are reliably positive. The double-EAs (adding two electrons) lower the energy if they are stable, and raise the energy if they are unstable. So you get both signs. I'm not sure exactly which overlap is being referred to, though.