[REQUEST] Arbitary-order Overlap Integrals (and evaluations enabled thereby)

[PaulWAyers]

It is useful in several contexts to have arbitrary-order overlap integrals. For example, this allows us to compute very-high-dimensional shared electron delocalization indicators (SEDI), powers of the electron density, intracules, and extracules.

3- and 4-Gaussian overlaps from PySCF can be used for testing. 1-Gaussian overlaps and 2-Gaussian overlaps are built-in already and can be used for testing.

An algorithm (two algorithms, in fact) for doing this are included in the attached notes, which were developed in the context of the intracule and extracule. The attached notes are old and have Fortran-style pseudocode (they are that old). [To be clear, I would not suggest implementing the direct algorithms for the intracule/extracule. It is better to pass through the multi-Gaussian overlaps.]

Note that multi-Gaussian overlaps are extremely sparse, so a sparse tensor structure should be used. The main operation one needs are tensor contractions (usually pairing indices to one or more copies of the 1- or 2-electron density matrix), which will determine the type of structure one uses to store the tensor. One may need to use pytorch functionality for this.

If we had the arbitrary-order overlaps coded, I would argue that we should support some additional evaluations, notably:

1. intracule on a grid.
2. extracule on a grid.
3. integrals of powers of the electron density.
4. (maybe) integrals of moments of powers of the electron density. These integrals show up in some of the moment density functional development work that Parr and Nagy did a couple decades ago. That specific topic is not very active right now, though, so this could definitely be deferred to the (far) future.

IntraculeExtracule.pdf