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Physical Review A

Coupled-cluster Green's function: Analysis of properties originating in the exponential parametrization of the ground-state wave function

Bo Peng, Karol Kowalski

Published Dec 23, 2016• 32 citations
Publisher

Abstract

In this paper we derive basic properties of the Green's-function matrix elements stemming from the exponential coupled-cluster (CC) parametrization of the ground-state wave function. We demonstrate that all intermediates used to express the retarded (or, equivalently, ionized) part of the Green's function in the $\ensuremath{\omega}$ representation can be expressed only through connected diagrams. Similar properties are also shared by the first-order $\ensuremath{\omega}$ derivative of the retarded part of the CC Green's function. Moreover, the first-order $\ensuremath{\omega}$ derivative of the CC Green's function can be evaluated analytically. This result can be generalized to any order of $\ensuremath{\omega}$ derivatives. Through the Dyson equation, derivatives of the corresponding CC self-energy operator can be evaluated analytically. In analogy to the CC Green's function, the corresponding CC self-energy operator can be represented by connected terms. Our analysis can easily be generalized to the advanced part of the CC Green's function.

DOI
10.1103/physreva.94.062512

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In this paper we derive basic properties of the Green's-function matrix elements stemming from the exponential coupled-cluster (CC) parametrization of the ground-state wave function. We demonstrate that all intermediates used to express the retarded (or, equivalently, ionized) part of the Green's function in the \ensuremathω\ensuremath{\omega}\ensuremathω representation can be expressed only through connected diagrams. Similar properties are also shared by the first-order \ensuremathω\ensuremath{\omega}\ensuremathω derivative of the retarded part of the CC Green's function. Moreover, the first-order \ensuremathω\ensuremath{\omega}\ensuremathω derivative of the CC Green's function can be evaluated analytically. This result can be generalized to any order of \ensuremathω\ensuremath{\omega}\ensuremathω derivatives. Through the Dyson equation, derivatives of the corresponding CC self-energy operator can be evaluated analytically. In analogy to the CC Green's function, the corresponding CC self-energy operator can be represented by connected terms. Our analysis can easily be generalized to the advanced part of the CC Green's function.
We present an extension of the analysis previously applied to the retarded part of the coupled cluster (CC) Green's function to its advanced part. In analogy to our earlier studies for the retarded part, we demonstrate that the advanced CC Green's function is expressed in terms of connected diagrams only, which is a direct consequence of algebraic form of equations satisfied by CC amplitudes. We also demonstrate that ω-derivatives of the advanced CC Green's function can be calculated analytically and can be expressed in terms of connected diagrams only. We analyse the structure of connected diagrams and the role of intermediate operators which satisfy electron affinity equation-of-motion CC-type conditions.
Coupled-cluster and Green's function theories are highly successful in treating many-body electron correlation, and there has been significant interest in identifying and leveraging connections between them. Here we present a diagrammatic definition of the irreducible coupled-cluster self-energy that directly embeds coupled-cluster (CC) theory within the framework of many-body field theory. The equation-of-motion coupled-cluster (EOM-CC) treatment emerges naturally from our definition via the Dyson equation and the Bethe-Salpeter equation (BSE), providing a unified description of the random phase approximation (RPA), G W -BSE, and CC theory for ground state and excitation energies. This clarifies the origin of previously established connections between RPA, G W -BSE, and CC theory, and it exposes the relationship between vertex corrections and the coupled-cluster amplitude equations.
In this paper we discuss a new formalism for producing an analytic coupled-cluster (CC) Green's function for an N-electron system by shifting the poles of similarity transformed Hamiltonians represented in N - 1 and N + 1 electron Hilbert spaces. Simple criteria are derived for the states in N - 1 and N + 1 electron spaces that are then corrected in the spectral resolution of the corresponding matrix representations of the similarity transformed Hamiltonian. The accurate description of excited state processes within a Green's function formalism would be of significant importance to a number of scientific communities ranging from physics and chemistry to engineering and the biological sciences. This is because the Green's function methodology provides a direct path for not only calculating properties whose underlying origins come from coupled many-body interactions but also provides a straightforward path for calculating electron transport, response, and correlation functions that allows for a direct link with experiment. As a special case of this general formulation, we discuss the application of this technique for Green's function defined by the CC with singles and doubles representation of the ground-state wave function.
Connection between GWGWGW and Extended Coupled Cluster
Coupled-cluster (CC) theory and Green's function many-body perturbation theory (MBPT) have long evolved as distinct yet complementary frameworks for describing electronic correlation. While CC methods employ exponential wavefunction parametrizations that guarantee size extensivity and systematic improvability, Green's function approaches such as the GWGWGW approximation describe quasiparticle and optical excitations through diagrammatic resummations. Recent analyses have established a formal correspondence between these frameworks: the GWGWGW approximation is equivalent to an equation-of-motion (EOM) treatment of the direct-ring coupled-cluster doubles (drCCD) method. Within this context, the extended CC (ECC) ansatz offers a unified framework connecting CC and MBPT. This formulation bridges CC-based and Green's function-based methods, providing novel avenues for incorporating vertex corrections within a CC framework that preserves a sum-over-state representation of the self-energy and lead to potentially systematically improvable Green's function approaches.
Coupled-cluster (CC) theory and Green’s function many-body perturbation theory (MBPT) have long evolved as distinct yet complementary frameworks for describing electronic correlation. While CC methods employ exponential wave function parametrizations that guarantee size extensivity and systematic improvability, Green’s function approaches, such as the GW approximation, describe quasiparticle and optical excitations through diagrammatic resummations. Recent analyses have established a formal correspondence between these frameworks: the GW approximation is equivalent to an equation-of-motion (EOM) treatment of the direct-ring coupled-cluster doubles (drCCD) method. Within this context, the extended CC (ECC) ansatz offers a unified framework connecting CC and MBPT. This formulation bridges CC-based and Green’s function-based methods, providing novel avenues for incorporating vertex corrections within a CC framework that preserves a sum-over-state representation of the self-energy and leads to potentially systematically improvable Green’s function approaches.
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 (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, Σ ̃ [ 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, Σ ̃ [ G ̃ ] , and demonstrate its relationship to the perturbative expansion of the self-energy, Σ ̃ [ 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, ionization potential/electron affinity equation-of-motion coupled-cluster theory, and the G 0 W 0 approximation. As a result, we introduce the CC- G 0 W 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.