Investigating cavity QED dynamics often requires a truncation of electronic states applied to the QED Hamiltonians. This is because these matter electronic states are often difficult to obtain, and in a lower energy regime, one can project the QED Hamiltonian to a few physically relevant electronic states without losing significant accuracy. Consider a finite subset of electronic states {|α⟩} where there is a total of N matter states, Eq. 26 can be rewritten to define the projection operator