Non-Hermitian Green’s function theory with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> -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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>N</a:mi> </a:math> -body non-Hermitian interaction. In this work, we focus specifically on the coupled-cluster (CC) self-energy generated by the similarity transformation of the electronic structure Hamiltonian. The main result of this work is a Green’s function formalism for non-Hermitian many-body quantum systems. 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 “noninteracting” reference Green’s function, <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mover accent="true"> <b:mi mathvariant="normal">Σ</b:mi> <b:mo>̃</b:mo> </b:mover> <b:mrow> <b:mo>[</b:mo> <b:msub> <b:mi>G</b:mi> <b:mn>0</b:mn> </b:msub> <b:mo>]</b:mo> </b:mrow> </b:mrow> </b:math> . 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, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mover accent="true"> <e:mi mathvariant="normal">Σ</e:mi> <e:mo>̃</e:mo> </e:mover> <e:mrow> <e:mo>[</e:mo> <e:mover accent="true"> <e:mi>G</e:mi> <e:mo>̃</e:mo> </e:mover> <e:mo>]</e:mo> </e:mrow> </e:mrow> </e:math> , and demonstrate its relationship to the perturbative expansion of the self-energy, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mover accent="true"> <i:mi mathvariant="normal">Σ</i:mi> <i:mo>̃</i:mo> </i:mover> <i:mrow> <i:mo>[</i:mo> <i:msub> <i:mi>G</i:mi> <i:mn>0</i:mn> </i:msub> <i:mo>]</i:mo> </i:mrow> </i:mrow> </i:math> . 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, ionization potential/electron affinity equation-of-motion coupled-cluster theory, and the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:msub> <l:mi>G</l:mi> <l:mn>0</l:mn> </l:msub> <l:msub> <l:mi>W</l:mi> <l:mn>0</l:mn> </l:msub> </l:mrow> </l:math> approximation. As a result, we introduce the CC- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> <m:msub> <m:mi>W</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:math> 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.