Department of Chemistry, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.
School of Chemistry, Cardiff University, Park Place, Cardiff CF10 3AT, United Kingdom.
J Chem Phys. 2017 Jul 28;147(4):044108. 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 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.
本文以通俗易懂的方式对单粒子多体格林函数（MBGF）理论的整体微扰级数进行了深入的分析和数值描述。推导出了格林函数微扰级数的三个截然不同但等价的代数（第一量子化）递归定义，它们可以与熟知的自能递归公式相结合。本文还开发了六种一般阶的 MBGF 算法，每种算法都实现了这三个递归中的一个，即ΔMPn 方法（其中 n 是微扰阶数）[S. Hirata 等人，J. Chem. Theory Comput. 11, 1595 (2015)]、图的自动生成和解释，或使用具有微扰标度哈密顿量的精确格林函数进行数值微分。除了ΔMPn 之外，所有这些方法都显示出相同的、无发散的微扰级数，而ΔMPn 仅在对角和非频依赖近似下与 MBGF 在 1≤n≤3 时一致，但在 n=∞时与全组态相互作用（FCI）极限一致（除非它发散）。本文还给出了微扰级数的数值数据，以量化向 FCI 极限收敛的速度以及对角、非频依赖或ΔMPn 近似的影响。在任何微扰阶数下，基于代数递归，本文在完全时间独立（频域）框架中证明了单粒子格林函数和自能的链接性和因此的大小一致性。通过 Frantz 和 Mills 的因式分解定理，本文还从数学上证明了在单粒子格林函数中修剪外部线以暴露自能图以及去除可约图的合理性。在 1≤n≤3 时，ΔMPn 和 MBGF 在对角和非频依赖近似下的等价性也通过代数证明，同时还将 n=4 时的差异归因于所谓的半可约和链接不连通图。
