, n|, to the block of electronic states dressed with n ± 1 photons M n±1 is now evident. The M n and M n+1 blocks are coupled through √ n + 1μ. Additionally, the electronic states with the same photon number (i.e., n = m) are connected only via the DSE terms D αβ . Note here that N F is the maximum number of included Fock basis states for the photonic sub-system. Recall for blocks with larger numbers of Fock states, one picks up the additional √ n + 1 term on each of the μ blocks, which effectively increases the effects of the coupling terms μ with increasing numbers of photons. Noting again the block structure in Eq. 112, the computational efficiency of this exact diagonalization can be drastically increased by the use of sparse matrix methods (e.g., Lanczos), which can be heavily relied on for approximate diagonalization of the lowest eigenvalues and eigenvectors without loss of physics but with a large computational speed-up. For this approach to be successful, one is required to treat the number of electronic (i.e., size of M and μ) and photonic basis states (i.e., N F ) as convergence parameters which provides a rigorous approach to solving the QED Hamiltonian, and it is exact for an infinite basis set. However, in the literature, often the electronic system is truncated to only include the ground and first excited molecular states and only the vacuum |0⟩ and |1⟩ photonic states. As we will see in the following section, this will lead to a breakdown of the physics, especially at larger coupling strengths specifically due to the DSE terms connecting blocks of the Hamiltonian far-away in energy.