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

Abstract

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 basismore » 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« less

Authors:
ORCiD logo [1];  [1]; ORCiD logo [2];  [3]
  1. Univ. of Illinois, Urbana-Champaign, IL (United States). Dept. of Chemistry
  2. Cardiff Univ., Park Place, Cardiff (United Kingdom). School of Chemistry
  3. Auburn Univ., AL (United States). Dept. of Chemistry and Biochemistry
Publication Date:
Research Org.:
Univ. of Illinois, Urbana-Champaign, IL (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22); National Science Foundation (NSF); CREST, Japan Science and Technology Agency
OSTI Identifier:
1473852
Alternate Identifier(s):
OSTI ID: 1372946
Grant/Contract Number:  
FG02-11ER16211; SC0006028; FG02-12ER46875
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 147; Journal Issue: 4; Journal ID: ISSN 0021-9606
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY

Citation Formats

Hirata, So, Doran, Alexander E., Knowles, Peter J., and Ortiz, J. V. One-particle many-body Green’s function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms. United States: N. p., 2017. Web. doi:10.1063/1.4994837.
Hirata, So, Doran, Alexander E., Knowles, Peter J., & Ortiz, J. V. One-particle many-body Green’s function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms. United States. doi:10.1063/1.4994837.
Hirata, So, Doran, Alexander E., Knowles, Peter J., and Ortiz, J. V. Thu . "One-particle many-body Green’s function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms". United States. doi:10.1063/1.4994837. https://www.osti.gov/servlets/purl/1473852.
@article{osti_1473852,
title = {One-particle many-body Green’s function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms},
author = {Hirata, So and Doran, Alexander E. and Knowles, Peter J. and Ortiz, J. V.},
abstractNote = {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},
doi = {10.1063/1.4994837},
journal = {Journal of Chemical Physics},
number = 4,
volume = 147,
place = {United States},
year = {2017},
month = {7}
}

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