Implement fitted SAP guess

[susilehtola]

Description

This is an implementation of the fitted version of the SAP guess, described in J. Chem. Phys. 152, 144105 (2020).

The repulsive potential on every atom is fit in terms of error function potentials, which means that the guess can be evaluated with few two-electron integrals; namely

$$ {\bf F}^\text{SAP} = {\bf H} + {\bf V}^\text{SAP} $$

$$ V_{\mu \nu}^\text{SAP} = \sum_{A} (A|\mu \nu) $$

where $A$ is a fully contracted $1s$ function on each atom.

I have no idea how the integrals work in Psi4. I followed a Psi4Numpy example @loriab hinted me at.

The appears to compile; the only problem being the final assembly, i.e. the sum over the 3-center integrals. If anyone can tell me what I should be doing, help would be appreciated...

Closes #2054 .

User API & Changelog headlines

• RN 1
• RN 2

Dev notes & details

• Implement scheme and get something that matches ERKALE
• Use optimal storage

Questions

• Is the basis build call correct? Should I also add a new basis role somewhere?

Checklist

• Tests added for any new features
• All or relevant fraction of full tests run

Status

• Ready for review
• Ready for merge

[susilehtola]

The implementation in #3039 is cleaner. Closing this ticket.