Comparing parameterized and self-consistent approaches to ab initio cavity quantum electrodynamics for electronic strong coupling

Molecules under strong or ultra-strong light-matter coupling present an intriguing route to modify chemical structure, properties, and reactivity. A rigorous theoretical treatment of such systems requires handling matter and photon degrees of freedom on an equal quantum mechanical footing. In the regime of molecular electronic strong or ultra-strong coupling to one or a few molecules, it is desirable to treat the molecular electronic degrees of freedom using the tools of ab initio quantum chemistry, yielding an approach referred to as ab initio cavity quantum electrodynamics (ai-QED), where the photon degrees of freedom are treated at the level of cavity quantum electrodynamics. In this letter, we analyze two complementary approaches to ai-QED: (1) parameterized CQED (pQED), a two-step approach where the matter degrees of freedom are computed using existing electronic structure theories, enabling the construction of rigorous ai-QED Hamiltonians in a basis of many-electron eigenstates, and (2) self-consistent CQED (scQED), a one-step approach where electronic structure methods are generalized to include coupling between electronic and photon degrees of freedom. Although these approaches are equivalent in their exact limits, we identify a disparity between the projection of the two-body dipole self-energy operator that appears in the pQED approach and its exact counterpart in the scQED approach. We provide a theoretical argument that this disparity resolves only under the limit of a complete orbital basis and a complete many-electron basis for the projection. We present numerical results highlighting this disparity and its resolution in simple molecular systems, where it is possible to approach these two complete basis limits simultaneously. Additionally, we examine and compare the practical issue of computational cost required to converge each approach towards the complete orbital and many-electron bases.