One-particle many-body Green’s function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms

• Published in: The Journal of chemical physics Vol. 147; no. 4; pp. 044108 - 44138
• Main Authors: Hirata, So, Doran, Alexander E., Knowles, Peter J., Ortiz, J. V.
• Format: Journal Article
• Language: English
• Published: United States American Institute of Physics 28.07.2017; American Institute of Physics (AIP)
• Subjects: Algebra; Algorithms; Convergence; Equivalence; INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; Numerical differentiation; Perturbation methods; Physics; Recursive functions; Theorems
• ISSN: 0021-9606; 1089-7690; 1520-9032; 1089-7690
• DOI: 10.1063/1.4994837

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 Δ MPn 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 Δ MPn, 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 Δ MPn 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 Δ MPn 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.