Abstract：Depending on how much on G-equivariant atoms we have introduced in the first class, we may add more details on that and provide some applications in the realm of G-equivariant birational geometry. Then, we explain how one can use data coming from the classical Chen-Ruan cohomology to enhance the invariants associated with G-equivariant atoms. We present applications that can only be obtained from this refinement. We discuss how an ongoing project with L. Katzarkov and M. Kontsevich can use Chen-Ruan atoms, enhanced with gerbes and discrete torsion, to provide a Mendeleev table for classifying different birational types of G-varieties.