Both approaches can reach an exact limit. For pQED, the exact limit can be achieved when one builds Eq. 1 or 2 in the complete basis of exact adiabatic eigenstates and photonic Fock states. Of course, it is practically impossible to reach this limit for most molecular systems, so a practical variational approach will consider building the pQED Hamiltonian in a truncated basis of many-electron states. In practice, this basis of adiabatic eigenstates is not only incomplete, but the states themselves are inexact because they results from an approximate quantum chemistry method. The approximations inherent in practical quantum chemistry calculations include truncation of the single-particle basis (e.g. using a finite number of Gaussian-type orbital basis functions), and truncation of the many-electron space through, for example, truncation of the excitation rank in a configuration interaction (CI) ansatz. The exact limit of the scQED approach includes a complete single-particle basis for the electronic subsystem, a complete many-electron basis (through, e.g., a FCI ansatz that includes all excited electronic configurations, often represented as Slater determinants or configuration state functions), and a complete photonic Fock space, which is also not practical in general. Our goal in this current paper is to study the convergence properties of both methods towards their exact limits, and to point out quantitative differences that arise outside of the complete basis limit.