In the above expression, we did not specify the choice of P, which could be either adiabatic or diabatic. Under the limiting case when A 0 = 0 or μ • Â = 0, both the -∇ j μ Â and Pj terms become 0, and Û † = Û → P ⊗ 1R ⊗ 1ph . Thus, under a such limit, Ĥp•A → ĤM + Ĥph ; hence, the matter and the cavity becomes decoupled. When using adiabatic states for the truncation, one can show that 66,68 P p2 j P = (p j -iℏ α,β d j αβ |α⟩⟨β|) , where d j αβ ≡ ⟨α|∇ j |β⟩ is the well known derivative couplings. Besides these adiabatic derivative couplings, the light-matter interaction also induced additional "derivative"-type couplings, -∇ j μ Â and Pj , regardless of the electronic representation used in constructing P. When using the Mulliken-Hush diabatic states which are the eigenstates of the μ ≡ P μ P operator, such that μ = ϕ µ ϕ |ϕ⟩⟨ϕ|, one can prove that Pj = 0 for all nuclei. This is because that ∇ j μ = ϕ ∇ j µ ϕ |ϕ⟩⟨ϕ|, thus both μ Â and [ μ Â, pj ] become purely diagonal matrices, hence all of the higher order commutators in Û † pj Û become zero, resulting in Pj = 0 for j ∈ R.