Adding a function to the SymPy physics.wigner module (pull request #24035)

[Phil L]

Hello everyone,

My name is Phil and I just joined GitHub. I am currently in the process of trying to add a function in the physics.wigner module that computes the integral of 3 real spherical harmonics, which is called the real Gaunt coefficient. The real Gaunt coefficient arises primarily in quantum chemistry calculations when dealing with angular momentum, similarly to the standard Gaunt coefficient. One of the primary motivations for using the real spherical harmonics as basis functions to represent the angular degrees of freedom instead of the standard spherical harmonics, is that the standard spherical harmonics require twice the amount of memory due to them being complex. This is a significant advantage as many calculations involve many millions of atomic entities (e.g. atoms), requiring enormous amounts of memory. As a side note, the real spherical harmonics are also easier to use when visualizing atomic data because they are completely real.

The tradeoff in working with the real spherical harmonics, however, is that the real Gaunt coefficient has many different cases to evaluate that make it a nontrivial calculation. The algorithm I have proposed (outlined in a paper by Homeier and Steinborn 1996), reduces this calculation to just three cases. With the generous aid of the reviewer of my pull request (Christopher Smith), we have reduced the runtime by more than 3X and trimmed down the number of lines from the function. I have also included detailed documentation on the function for reference.

The question I then want to pose to everyone is: who else thinks the real Gaunt coefficient function would be useful? I think it would be useful for quantum chemistry calculations and I personally used it substantially for my own master's research. I would love to hear your thoughts.

All the best,

Phil