where L g is the associated Laguerre polynomial and ξ = (q 0 νq 0 µ )/ √ 2ω c with q 0 ν = λ ωc • µ νν . 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 in the MH basis are large. For additional discussion on the PFS basis, see Ref. 61. Further, a similar basis has been used in the quantum optics and recently in the polariton communities, which is referred to as the generalized coherent state (GCS) basis, which also relies on the molecular dipole information to define a new photonic basis. The polaritonic Hamiltonian in the PFS basis is, in general, not as sparse as that of the adiabatic-Fock matrix (see Eq. 19) since all shifted Fock states between different MH electronic states φ ν are connected via their shifted harmonic oscillator overlaps, so sparse matrix techniques cannot be as readily applied. The main advantage of the PFS basis is that many of the overlaps will be near zero due to the widely varying magnitudes of the molecular dipole µ νν elements. In this way, the PFS basis, in principle, will allow one to use many fewer shifted Fock states as a basis than the vacuum Fock states used in the adiabatic-Fock basis (see Section 3.1.1) and therefore may perform more efficiently in many cases. More rigorous testing is required for real, ab initio molecular systems where the dipole matrices are extremely non-diagonal in the many-state adiabatic basis compared to many of the tested model systems, such as the harmonic oscillator or the double-well potentials where the dipole matrix is nearly diagonal, hence the adiabatic states are already almost equal to the MH states themselves. In our experience, the photonic basis is more easily converged in ab initio systems compared to the convergence of the electronic basis, which often requires many states to provide a useful convergence (see Sec. 3.2 for additional discussion). Here, only two diabatic states were considered, which are denoted as the ionic state |I⟩, and covalent state |C⟩. These two diabatic states are coupled through a diabatic coupling V IC (R ) (dotted yellow line in Fig. ) that causes a splitting (avoided crossing) near the anti-crossing of the diabatic potentials V C (R ) and V I (R ) (solid red and blue line in Fig. , respectively). The adiabatic electronic states, ground |G(R ) ⟩ and excited |E(R ) ⟩ states can be obtained by diagonalizing the electronic