where we have dropped the R dependency for simplicity. Note that there is a finite coupling between the MH state ϕ ν with n ν photons and the MH state ϕ ϵ with m ϵ photons through the ⟨m ϵ |n ν ⟩ V νϵ term, which is the off-diagonal matrix element of the electronic Hamiltonian, V νϵ , scaled by the overlap, ⟨m ϵ |n ν ⟩, of the PFS. This overlap is non-zero and is simply the overlap of two harmonic oscillator wavefunctions that are shifted from one another by q 0 ϵ -q 0 ν = λ ωc • [µ ϵϵ -µ νν ]. Thus, instead of having an explicit light-matter interaction term ωc 2 λ • μ(â † + â) (and the DSE) as shown in Eq. 111, these interactions are now completely carried through ⟨m ϵ |n ν ⟩ • V νϵ (R). This basis is expected (and has been explicitly shown for models systems ) to efficiently converge the photonic basis, especially when the permanent dipoles µ νν (R) in the MH basis are large. For additional discussion, see Ref. 41. A similar basis will be described in the Sec. 3.2 called the generalized coherent state (GCS) basis. 148