In Sec. 3, we discussed the recent progress of ab initio polariton chemistry calculations, where one aims to solve polariton eigenvalue problems with real molecular systems. Particularly, we reviewed two approaches for performing these calculations: the parameterized QED (Sec. 3.1) approach and the self-consistent QED (Sec. 3.2) approach. Along with a brief overview and direct comparison of the two methods (Sec. 3.3.3), in Sec. 3.3, we showcased recent works that have implemented these approaches and demonstrated their ability to calculate chemically relevant properties (Sec. 3.3.2) in both the excited (Sec. 3.3.1) and ground polaritonic states (Sec. 3.3.4).