with | E {Γ} ⟩ = τ{Γ} | E HF 0 ⟩, where {Γ} = {α, n, α ñ } is the set of possible excitations in the cluster operator T (Eq. 35) leading to the set of projection equations { L Γ } . These projections lead to the equations for the excitation amplitudes t {Γ} and are usually solved in a self-consistent manner. A similar CC projection formalism can be found in any electronic structure textbook. There are many different notations for the methods developed by changing the highest level of excitation for each term in the cluster operator. The most straightforward notation is CCSD-n-jm, which implies that the electronic DOFs are treated up to double (SD) excitations in the cluster operator, the photonic excitation is limited to n levels, and the mixed excitation is set to j electronic and m photonic. As per usual CC theory, the cutoff value of the excitation levels leads to effects that include even higher excitations through the exponential treatment of the cluster operator T, thus effectively outperforming similar methods such as CI with the same excitation level cutoff. However, due to the O (N 6 * N N mode F