  Polaritonic chemistry relies on the strong light-matter interaction phenomena
for altering the chemical reaction rates inside optical cavities. To explain
and to understand these processes, the development of reliable theoretical
models is essential. While computationally efficient quantum electrodynamics
self-consistent field (QED-SCF) methods, such as quantum electrodynamics
density functional theory (QEDFT) needs accurate functionals, quantum
electrodynamics coupled cluster (QED-CC) methods provide a systematic increase
in accuracy but at much greater cost. To overcome this computational
bottleneck, herein we introduce and develop the QED-CC-in-QED-SCF
projection-based embedding method that inherits all the favorable properties
from the two worlds, computational efficiency and accuracy. The performance of
the embedding method is assessed by studying some prototypical but relevant
reactions, such as methyl transfer reaction, proton transfer reaction, as well
as protonation reaction in a complex environment. The results obtained with the
new embedding method are in excellent agreement with more expensive QED-CC
results. The analysis performed on these reactions indicate that the strong
light-matter interaction is very local in nature and that only a small region
should be treated at the QED-CC level for capturing important effects due to
cavity. This work sets the stage for future developments of polaritonic quantum
chemistry methods and it will serve as a guideline for development of other
polaritonic embedding models.
