The coherent-state transformation in quantum electrodynamics coupled cluster theory

We analyze the coherent-state (CS) transformation in quantum electrodynamics coupled cluster (QED-CC) theory from the perspective of its non-vanishing commutator with the polaritonic cluster operator. In particular, we show that a QED Hartree-Fock (QED-HF) reference state parameterized by the CS transformation leads to a QED-CC Lagrangian formally determined by CS-representations of polaritonic Hamiltonian, polaritonic cluster, and polaritonic de-excitation operators. The approach proposed herein differs from the original formulation of QED-CC theory in the definition of the photon state basis and exploits photon-added coherent states in contrast to previously considered displaced number states. We find a renormalization of both QED-CC correlation energy and QED-CC ground state induced by the CS transformation, which depends on the mean-field expectation value of the molecular dipole operator and, therefore, breaks origin invariance for charged systems. Electronic contributions to correlation energy and QED-CC ground state are renormalized by CS-transformed mixed excitation and de-excitation operators. In contrast, the CS-transformed single-photon excitation affects only the QED-CC ground state but not directly the correlation energy. The renormalized QED-CC ansatz becomes similar to the original QED-CC formulation for large cavity frequencies, leading to small renormalization corrections. A divergent renormalization energy for molecules with a non-vanishing molecular dipole moment is found in the low-frequency limit, which we discuss with respect to multi-photon excitations in the polaritonic cluster operator and the relevance of the cavity-Born-Oppenheimer framework.