# More CG Simplification and Wrapping Up X/y/z Spin Bases

For the first part of this last week, I continued on my work to get sums of Clebsch-Gordan coefficients to simplify. Using the same general logic that I outlined in the last blog entry, besides general cleaning up of the code, most of the work at the beginning of the week was spent on trying to develop a function that could check an expression for CG coefficients matching a set of conditions.

The rationale behind trying to write such a method is that it would make it much easier to identify the times where symmetries could be utilized. With such a method, the process of checking for CG coefficients could be done in a single function and the logic for implementing CG symmetries could be handled in this one function. The current method uses lists of tuples to specify the conditions on CG coefficients. For example, if the j1 value of a CG coefficient needed to be =0, you could pass the tuple (“j1”,0), or if the m1 and m3 values match (“m1”,“m3”). All the conditions for each CG coefficient are combined into a single tuple. The current snag with simplifications of sums of products of CG coefficients. For example: $$\sum_{\alpha,\beta} C_{\alpha\beta}^{c\gamma} C_{\alpha\beta}^{c’\gamma’} = \delta_{cc’} \delta_{\gamma\gamma’}$$

While the current method would be able to check for specific values on the CG coefficients, I have yet to come up with a good way to check that the m1 and m2 values are the same when they can take any value, as in this example. As it stands, this code still seems like quite a hack and will need some work before it is good to go.

What is left with this part of the project is:

• Getting simplification to work with sums of products of terms (as in the example above)
• Applying CG symmetries to perform simplifications
• Simplification of symbolic CG sums
• Fixing up the printing of CG terms
• Final testing/documentation

This part of the project has unfortunately fallen behind the preliminary schedule by a bit, as it was due to be finished up last week. I’ll outline what I’m currently working on finish up next, but hopefully I can finish the CG stuff ASAP so I can move on to working on the spin stuff which is the true meat of the project and try to get back on schedule.

After meeting with my project mentor, Ondrej, on Wednesday, it was decided that the focus would shift to finishing up the work I’d started on x/y/z spin bases and representation of spin states that I’d started before GSoC had officially started.

The first order of business was identifying an error in the Wigner small-d function, which is used extensively in the changing of spin bases. With Ondrej noting that the small-d function was defined only on a small interval and then me discovering the bug in the Rotation.d method, we were able to address this. However, no sooner had this been done than Ondrej is able to work out a better equation for the small-d function, which will likely replace the current implementation.

Other than this, most of the work this week on the x/y/z basis representation was in documentation, testing and generally cleaning up the code to be pulled. The current pull request (my first work to be submitted since the start of GSoC) is still open here. While this pull integrates the current work on basis representation, after this pull there is still some work that will need to be done testing both the Wigner small-d and the D functions, for both symbolic and numerical values, and ensuring they return the correct results. Because the representation code relies so heavily on these functions, it is imperative that these functions evaluate properly. Once these are fully tested, there will also likely need to be more tests to ensure all the representation code returns the right values for as many odd cases as would be necessary to test. Hopefully I can finish this up soon and move on to other work that still needs to be done.