A general proof of the size consistency of the many-body Green's-function and the related Rayleigh-Schr\"odinger perturbation theory methods is provided, on the basis of the linked-cluster theorem, for infinite polymer systems. It is assessed by means of calculations of the ionization potentials, and their corresponding pole strengths, on increasing-size model oligomers achieved using the equation-of-motion--algebraic superoperator approach. The orbital picture for the ionization process is shown to be partially preserved in infinite systems. The dependence of the correlation and relaxation effects with the size of the system and the inner character of the ionized orbitals, is interpreted within the framework of the band-structure formalism. The hierarchy observed among the many-body corrections obtained at several levels of approximation used to expand the one-particle propagator are also explained by diagrammatic considerations.
A general proof of the size consistency of the many-body Green's-function and the related Rayleigh-Schr"odinger perturbation theory methods is provided, on the basis of the linked-cluster theorem, for infinite polymer systems. It is assessed by means of calculations of the ionization potentials, and their corresponding pole strengths, on increasing-size model oligomers achieved using the equation-of-motion--algebraic superoperator approach. The orbital picture for the ionization process is shown to be partially preserved in infinite systems. The dependence of the correlation and relaxation effects with the size of the system and the inner character of the ionized orbitals, is interpreted within the framework of the band-structure formalism. The hierarchy observed among the many-body corrections obtained at several levels of approximation used to expand the one-particle propagator are also explained by diagrammatic considerations.
Recently the possible divergence of many-body perturbation theory or many-body green function (propagator) methods with lattice summations in extended systems has been raised. The convergence of these methods with lattice summations is not only the key to establishing their correct scaling properties in inhomogeneous systems, but is also a necessity if numerical calculations are to be meaningful. In this report, it is rigorously shown that many-body perturbation theory (MBPT), coupled cluster theory (CC), and many-body green function (MBGF) methods all converge uniformly with lattice summations, although the integrand for the integration over the reciprocal lattice vector, k, could become infinite at special k values. Our proof is given not only for infinite polymers but also for crystals. In our proof, only the continuity of the zeroth-order band structure and Bloch orbitals with k is used. We show that MBPT, CC, and MBGF methods converge with the radius of the lattice summation range R at least as fast as 1/R1/n where n is the dimensionality of the system. It could be much faster if the zeroth-order band structure and Bloch orbitals have continuous first- or even higher-order derivatives with k. In practical numerical calculations, one should not pursue the convergence of the integrand, but the convergence of the correction considered with lattice summations.
Many functions of physical interest are defined by means of recursion relations where a function of a given rank can be expressed in terms of the functions of lower rank. It is also usually possible to define approximation schemes which, with natural modifications, can be applied to evaluate any function of the hierarchy. Typical examples are density functions, propagators, and Green's functions, and the approximation approaches to them are often based on perturbation theory. The attempt of this work is to clarify the connection between the approximation schemes for propagators and the hierarchy of linking equations. The concept of scheme consistency is introduced and used to extend available approximation schemes. As a specific example the theory is worked out for the particle-particle propagator, a quantity describing the simultaneous removal of two particles from a many-particle system. It is expected that the derived working equations are of importance in the description of double-ionization processes where each of the two removed electrons originates from a different part of the system.
Connection between GW 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 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 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.
A thorough analytical and numerical characterization of the whole perturbation series of one-particle many-body Green's function (MBGF) theory is presented in a pedagogical manner. Three distinct but equivalent algebraic (first-quantized) recursive definitions of the perturbation series of the Green's function are derived, which can be combined with the well-known recursion for the self-energy. Six general-order algorithms of MBGF are developed, each implementing one of the three recursions, the ΔMPn method (where n is the perturbation order) [S. Hirata et al., J. Chem. Theory Comput. 11, 1595 (2015)], the automatic generation and interpretation of diagrams, or the numerical differentiation of the exact Green's function with a perturbation-scaled Hamiltonian. They all display the identical, nondivergent perturbation series except ΔMPn, which agrees with MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 but converges at the full-configuration-interaction (FCI) limit at n=∞ (unless it diverges). Numerical data of the perturbation series are presented for Koopmans and non-Koopmans states to quantify the rate of convergence towards the FCI limit and the impact of the diagonal, frequency-independent, or ΔMPn approximation. The diagrammatic linkedness and thus size-consistency of the one-particle Green's function and self-energy are demonstrated at any perturbation order on the basis of the algebraic recursions in an entirely time-independent (frequency-domain) framework. The trimming of external lines in a one-particle Green's function to expose a self-energy diagram and the removal of reducible diagrams are also justified mathematically using the factorization theorem of Frantz and Mills. Equivalence of ΔMPn and MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 is algebraically proven, also ascribing the differences at n = 4 to the so-called semi-reducible and linked-disconnected diagrams.
Several widely used methods for the calculation of band structures and photo emission spectra, such as the GW approximation, rely on many-body perturbation theory. They can be obtained by iterating a set of functional differential equations (DEs) relating the one-particle Green's function (GF) to its functional derivative with respect to an external perturbing potential. In this work, we apply a linear response expansion in order to obtain insights into various approximations for GF calculations. The expansion leads to an effective screening while keeping the effects of the interaction to all orders. In order to study various aspects of the resulting equations, we discretize them and retain only one point in space, spin and time for all variables. Within this one-point model we obtain an explicit solution for the GF, which allows us to explore the structure of the general family of solutions and to determine the specific solution that corresponds to the physical one. Moreover, we analyze the performances of established approaches like GW over the whole range of interaction strength, and we explore alternative approximations. Finally, we link certain approximations for the exact solution to the corresponding manipulations of the DE which produces them. This link is crucial in view of a generalization of our findings to the real (multidimensional functional) case where only the DE is known.