The coupling vector can be written in terms of the cavity volume, λ = h ε 0 V ê. The light-matter coupling in the Pauli-Fierz Hamiltonian may also be written in terms of the vector potential operator A 0 b † + b ; for readers familiar with this notation, we note that A 0 = h 2ωε 0 V ê, we can write A 0 = 1 2ω λ. Here, we have limited our considerations to a single cavity mode, but the above Hamiltonian is easily generalized for multiple modes (which leads to increased dimensionality) as discussed in prior work. The second term Ĥcav = ω b † b represents the Hamiltonian for the bare cavity mode, which is a harmonic oscillator with fundamental frequency ω. The last two terms are the bilinear coupling, Ĥblc = -ω 2 (λ • μ)( b † + b), and dipole self-energy terms Ĥdse = 1 2 (λ • μ) , respectively. We will assume a Cartesian coordinate system where λ and μ will have x, y, and z components. The molecular dipole operator μ has both electronic and a nuclear contributions, i.e., μ = μe + µ n . In the Born-Oppenheimer approximation, the nuclear contribution is a constant for a given geometry. In the remainder of the paper, we will adopt the notation d = λ • μ for simplicity, and in cases where we require only the electronic part of this operator, we will denote it as de and we will denote the nuclear contribution as d n .