Many-body basis for the optical model

Frantz, Lee M.; Mills, R. L.

doi:10.1016/0029-5582(60)90278-9

Cited by 78 publications

(24 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These diagrammatic rules are often introduced on the basis of intuitive arguments and justified by somewhat imprecise mathematical logic. For instance, as shown in this study, the removal of reducible diagrams takes place as a result of systematic factorization of denominators, 31,34,91,92 much like the cancellation of unlinked diagrams in MBPT. 11,31,34,[91][92][93][94][95] None of these delicate details seems to have been fully discussed.…”

Section: Introductionmentioning

confidence: 62%

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

Hirata

Doran

Knowles

et al. 2017

The Journal of Chemical Physics

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 62%

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

Hirata

Doran

Knowles

et al. 2017

The Journal of Chemical Physics

View full text Add to dashboard Cite

show abstract

“…In that context, several authors noticed that some diagrams corresponding to different orderings can be added to give a simple sum 1,19,[24][25][26][27] , as we saw in the previous section. These are special cases of the simplification that we shall present which, as far as we know, was never stated in full generality.…”

Section: Permutations and Treesmentioning

confidence: 80%

Tree expansion in time-dependent perturbation theory

Brouder

Mestre

Patras

2010

Journal of Mathematical Physics

View full text Add to dashboard Cite

The computational complexity of time-dependent perturbation theory is well-known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynman-type diagrams...). We show that a very efficient perturbative expansion, both for theoretical and numerical purposes, can be obtained through an original parametrization by trees and generalized iterated integrals. We emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics. These tools are applied to the adiabatic approximation and the effective Hamiltonian. We prove perturbatively and non-perturbatively the convergence of Morita's generalization of the Gell-Mann and Low wavefunction. We show that summing all the terms associated to the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the time-independent equation for the wave operator and we give an explicit non-recursive expression for the term corresponding to an arbitrary tree.

show abstract

“…Since the structure of this spin-adapted theory is exactly the same as in the spin-orbital MBPT, it is not surprising that the Frantz-Mills theorem [12] holds for the disconnected diagrams, and the linked cluster result obtains. Thus generally, the nth order spin-adapted MBPT energy diagrams should be obtained by considering all possible linking of n unoriented V's so that connected diagrams result.…”

Section: A Closed-shell Casementioning

confidence: 89%

A unitary group formulation of many‐body theory: Diagram systematics and use of the spin shifts

Mukhopadhyay

Pickup

1984

Int J of Quantum Chemistry

View full text Add to dashboard Cite

This paper shows that the spin-shift formalism developed in B. T. Pickup and A. Mukhopadhyay [lnt. J. Quantum Chern. 26, 101 (1984)] supports a one-component diagrammatics which has a systematics akin to that in the spin-orbital many-body theory. The diagrams are neither Goldstone nor Yutsis type, and characterize thechain U ( 2 R ) 3 U ( R ) @ S U ( 2 ) on which thespin-shift formalism is based. Accordingly, while the lines in such diagrams are labeled by the orbital indices, the diagram structure adequately reflects the irreducible representation of the group U ( R ) . In this sense the paper presents a unitary group approach to the natural generalization of the usual many-body theory for the spin-adapted cases. A set of very simple rules is derived; their similarity with the corresponding rules in the ordinary many-body theory and practical utility are discussed in connection with (a) matrix elements over many-electron spin states and (b) closed-and open-shell many-body perturbation theory. A possibility of integral-driven many-body perturbation theory for open-shells is indicated. Connections of this formalism with others are also discussed.

show abstract

Many-body basis for the optical model

Cited by 78 publications

References 10 publications

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

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

Tree expansion in time-dependent perturbation theory

A unitary group formulation of many‐body theory: Diagram systematics and use of the spin shifts

Contact Info

Product

Resources

About