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

TL;DR

The paper develops a formally exact Green's function framework for general non‑Hermitian $N$‑body interactions, centering on the CC similarity‑transformed Hamiltonian. By adopting a biorthogonal quantum theory, it defines the single‑particle CC Green's function $\tilde{G}$ and derives a Dyson equation $\tilde{G} = G_0 + G_0\tilde{\Sigma}[\tilde{G}]\tilde{G}$ with a diagrammatic irreducible self‑energy $\tilde{\Sigma}[\tilde{G}]$ that depends on $\tilde{G}$ and on CC‑generated effective interactions up to arbitrary order. The work provides perturbative diagrams up to third order, establishes a self‑consistent renormalized self‑energy via exact equation‑of‑motion, and shows how CC theory yields the ground‑state energy and spectral properties through a CC Dyson supermatrix. It then connects this CC self‑energy framework to IP/EA‑EOM‑CC and $G_0W_0$, introducing CC‑$G_0W_0$ and CC‑BSE formulations, and derives the CC Bethe–Salpeter kernel from functional derivatives of the self‑energy. Overall, the approach unifies coupled‑cluster and Green's function methodologies for non‑Hermitian many‑body Hamiltonians, enabling robust descriptions of ground and excited states in systems where higher‑body interactions and non‑Hermiticity are essential.

Abstract

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, $\tildeΣ[G_0]$. 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, $\tildeΣ[\tilde{G}]$, and demonstrate its relationship to the perturbative expansion of the self-energy, $\tildeΣ[G_0]$. 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.

Figures (4)

• Figure 1: The 2p1h/2h1p excitation character restricted third-order one-particle irreducible coupled-cluster self-energy diagrams expressed with respect to the reference Green's function, $G_0$.
• Figure 2: The 2p1h/2h1p excitation character restricted third-order one-particle irreducible skeleton coupled-cluster self-energy diagrams expressed with respect to the exact single-particle coupled-cluster Green's function, $\tilde{G}$.
• Figure 3: The set of third-order 2p1h/2h1p excitation character restricted 1PI coupled-cluster self-energy diagrams that are obtained from the diagrams in Figure \ref{['fig:sc_cc_se']} by replacing the self-consistent propagators with their lowest-order linearized equivalents: $\tilde{G}\rightarrow G_0 + G_0\tilde{\Sigma}^{\infty}G_0$.
• Figure 4: Diagrammatic representation of the CC-$G_0W_0$ self-energy. The bubble diagrams are summed to infinite-order through the $\mathbf{\tilde{D}}^{\text{CC-$G_0W_0$}}$ supermatrix representation (Eq. \ref{['eq:cc_gowo_super']}) and the two-body antisymmetrized effective interactions are evaluated within the rCCD approximation. The non-interacting Green's function lines appearing in the dynamical self-energy diagrams are not fully antisymmetrized with respect to fermionic exchange.