Non-hermitian Green's function theory with $N$-body interactions: the coupled-cluster similarity transformation

We present the diagrammatic theory of the irreducible self-energy and Bethe-Salpeter kernel that naturally arises within the Green's function formalism for a general $N$-body non-hermitian interaction. In this work, we focus specifically on the coupled-cluster self-energy generated by the similarity transformation of the electronic structure Hamiltonian. We develop the biorthogonal quantum theory to construct dynamical correlation functions where the time-dependence of operators is governed by a non-hermitian Hamiltonian. We extend the Gell-Mann and Low theorem to include non-hermitian interactions and to generate perturbative expansions of many-body Green's functions. We introduce the single-particle coupled-cluster Green's function and derive the perturbative diagrammatic expansion for the non-hermitian coupled-cluster self-energy in terms of the `non-interacting' reference Green's function, $\tilde{\Sigma}[G_0]$. From the exact equation-of-motion of the single-particle coupled-cluster Green's function, we derive the self-consistent renormalized coupled-cluster self-energy, $\tilde{\Sigma}[\tilde{G}]$, and demonstrate its relationship to the perturbative expansion of the self-energy, $\tilde{\Sigma}[G_0]$. Subsequently, we show that the usual electronic self-energy can be recovered from the coupled-cluster self-energy by neglecting the effects of the similarity transformation. We show how the coupled-cluster ground state energy can be obtained from the coupled-cluster self-energy and provide an overview of the relationship between approximations for the coupled-cluster self-energy, IP/EA-EOM-CC and the $G_0W_0$ approximation. As a result, we introduce the CC-$G_0W_0$ self-energy by leveraging the connections between Green's function and coupled-cluster theory. Finally, we derive the diagrammatic expansion of the coupled-cluster Bethe-Salpeter kernel.