Here, N el and N F are treated as convergence parameters such that the polaritonic observables are adequately converged (see more details in Section 3.1.3). The obvious first choice is the convergence of the lowest-energy eigenvalues of Eq. 19. In our experience with realistic ab initio systems, the convergence with respect to the number of Fock states is rapid, only requiring N F ∼ 5 to obtain ∼10 meV accuracy. However, due to the complicated and highly off-diagonal dipole matrix in real molecules, the convergence with respect to electronic states is slow, possibly requiring N el ∼ 100 or more states to achieve ∼10 meV accuracy. In Sec. 3.2, we will briefly address other approaches, namely self-consistent schemes, to converge complicated systems which contain many strongly dipole-connected states as well as systems with large light-matter coupling that necessitate a more advanced and rigorous approach toward obtaining the polaritonic eigenstates in a self-consistent way. showing the two lowest-energy excited states with ground-to-excited transition dipole along the C-O bond direction (which is parallel to the cavity polarization direction, ê). The bare electronic states (Fig. ) show two local minima in the lower-energy excited state as well as an avoided crossing (see inset). Upon coupling to the cavity, the local minima near C-O bond length of 1.50 Å can be removed with a negatively detuned cavity (ω c = 6.5 eV, Fig. , colorbar indicates the average photonic character ⟨ â † â ⟩). Alternatively, the avoided crossing can be systematically reduced via a positively detuned cavity (ω c = 9.5 eV, Fig. , see inset). An important feature of this change in the avoided crossing is that the character of both states involved is retained (i.e., mainly electronic excitation character) where both states exhibit negligible amounts of photonic contributions. Direct control over the relative energy of electronic states while maintaining their original character is a useful concept and design principle in processes controlled by non-adiabatic coupling between the excited electronic states. Fig. presents the matter-projected polaritonic transition density in real space,