TY - JOUR

T1 - One-particle many-body Green's function theory

T2 - Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms

AU - Hirata, So

AU - Doran, Alexander E.

AU - Knowles, Peter J.

AU - Ortiz, J. V.

N1 - Funding Information:
S.H. has been supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Grant No. DE-FG02-11ER16211 and by CREST, Japan Science and Technology Agency. A.E.D. has been supported by the SciDAC program of the Department of Energy, Office of Science, Office of Basic Energy Sciences under Grant No. DE-FG02-12ER46875. J.V.O. has been supported by the National Science Foundation through Grant No. CHE-1565760 to Auburn University. Dr. Rodney J. Bartlett, Dr. Karol Kowalski, Dr. Marcel Nooijen, Dr. Yngve hrn, and Dr. Takeshi Yanai are thanked for illuminating discussions and advice.
Publisher Copyright:
© 2017 Author(s).

PY - 2017/7/28

Y1 - 2017/7/28

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85026442079&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85026442079&partnerID=8YFLogxK

U2 - 10.1063/1.4994837

DO - 10.1063/1.4994837

M3 - Article

C2 - 28764347

AN - SCOPUS:85026442079

VL - 147

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 4

M1 - 044108

ER -