Coupling polaritonic Hamiltonians such as Eq. 55 with realistic, ab initio calculations for molecular systems has generated much recent work. Most of the molecular ab initio polariton chemistry works are based on the singlemode light-matter interaction Hamiltonian in Eq. 57, which is equivalent to Eq. 56 as explained in Sec. 2.3.3. Here, for consistency, we choose to use the PF Hamiltonian in Eq. 56 to describe the ab initio methods. In particular, we express Eq. 56 as follows