Note here that we have dropped the explicit dependence on the nuclear position R for clarity. In this basis, the diagonal matrix elements are the diagonal MH energiesV νν (Eq. 12) and PFS harmonic oscillator eigenvalues ω c (n ν + 1 2 ), while the off-diagonal contributions are now the MH coupling elements V ν µ reduced in magnitude by the overlaps 1] between two PFS harmonic oscillator states with n ν photons and m µ photons associated with two different electronic MH states φ ν and φ µ . In this compact basis, the light-matter coupling and dipole self-energy are all neatly housed in the MH coupling and PFS overlaps, ⟨n µ |m ν ⟩ V νµ , and thus all interactions between light and matter DOFs are carried through this single term. Note also that the calculation of the PFS overlaps is analytic and can be written as,