The use of this parameter fixes the collective Rabi splitting Ω R for any number of molecules N since Ω R ∝ A 0 √ N for N molecules resonantly coupled to the cavity. For all simulations of the current work, we restrict the cavity parameters to Ã0 = 0.01 a.u., ω c = 1.00 a.u., ⟨E eg ⟩ N = 1.00 a.u., and µ (A) eg = 1.00 a.u. In this case, the average of the molecular transition frequencies are resonant with the cavity frequency, ⟨∆E eg ⟩ N = ω c = 1.00, and all molecular dipole moments are fixed and equal. As an example, for N = 3 and the above parameters, the working Hamiltonian becomes, As we will see later, the transition frequencies of the molecular DOFs, ∆E eg , will be sampled from various distributions, Gaussian, Rectangular, and Lorentzian, which represent the primary results of this work. Also, the high degree of symmetry and sparsity in this Hamiltonian lends itself well to approximate approaches that depend on the efficient implementation of matrix-vector multiplication discussed in the following section. The definition of strong coupling arises through the definition of a unitless coupling parameter η = Ω R /(2ω c ) (with ω c = 1.0 a.u. in this work). For a set of identical molecules perfectly resonant with the cavity, strong coupling is defined as η = Ω R /2 = 0.01, ultra-strong coupling as η = Ω R /2 = 0.1, and deep-strong coupling as η = Ω R /2 = 1.0. Note that the Rabi splitting depends on the molecular disorder,