Many-body perturbation theory diagrams

Next, we present some observations concerning the connection between the reconstruction process and the iterative solution of either CSE(p) or ICSE(p). The perturbative reconstruction functionals mentioned earlier each constitute a finite-order ladder-type approximation to the 3- and 4-RDMCs [46, 69] examples of the lowest-order corrections of this type are shown in Fig. 3. The hatched squares in these diagrams can be thought of as arising from the 2-RDM, which serves as an effective pair interaction for a form of many-body perturbation theory. Ordinarily, ladder-type perturbation expansions neglect three-electron (and higher) correlations, even when extended to infinite order in the effective pair interaction [46, 69], but iterative solution of the CSEs (or ICSEs) helps to... 

It should perhaps be stated at this point that the use of diagrams in the many-body perturbation theory is not obligatory. The whole of the theoretical apparatus can be set up in entirely algebraic terms. However, the diagrams are both more physical and easier to handle than the algebraic expressions and it is well worth the effort required to familiarize oneself with the diagrammatic rules and conventions. 

The relationship between the coupled-electron pair approximation (c.e.p.a.) and the many-body perturbation theory has been discussed in detail by Ahl-richs.149 All of the methods denoted by c.e.p.a. (x) (x=0, 1, 2, 3) may be related to the summation of certain classes of diagrams in the many-body perturbation theory to infinite order. For example, c.e.p.a. (0), which is Cizek s linear approximation or Hurley s c.p.a. (0) ansatz150 is equivalent to the summation of all double-excitation linked diagrams in the perturbation series. This is also denoted d.e.m.b.p.t. (double excitation many-body perturbation theory) by some workers.151 168... 

Many varieties of diagrams have used throughout the chemical physics literature for many years (e.g., see Refs. 1, 2, 80, 117, and 119). The diagrammatic formalism we have chosen here has been frequently used in work by the Bartlett group among others and is particularly straightforward for conventional coupled cluster and many-body perturbation theories. 

It is possible for groups of three or more lines to be identified as equivalent, though this can happen only in many-body perturbation theory, expectation value coupled cluster theory, or unitary coupled cluster theory. For such diagrams, a prefactor of where n is the number of electron lines, must be included. 

LITERATE MANY-BODY PERTURBATION THEORY PROGRAMMING THIRD-ORDER RING DIAGRAMS... 

A literate program for third-order many-body perturbation theory ring diagram components... 

S. Wilson, Literate many-body perturbation theory programming Third-order ring diagrams, this volume. 

Foundations to the CC methods were laid by Coester and Kuemmel,1 Cizek,2 Hubbard,3 Sinanoglu,4 and Primas,5 while Cizek2 first presented the CC equations in explicit form. Also Hubbard3 called attention to the equivalence of CC methods and infinite-order many-body perturbation theory (MBPT) methods. From this latter viewpoint, the CC method is a device to sum to infinity certain classes of MBPT diagrams or all possible MBPT diagrams when the full set of coupled-cluster equations is solved. The latter possibility would require solving a series of coupled equations involving up to IV-fold excitations for N electrons. Practical applications require the truncation of the cluster operators to low N values. 

Investigation of the connection between the Many-Body Perturbation Theory (MBPT) approach and the Furry representation of Quantum Electrodynamics (QED) has been shown to allow a precise definition of QED effects.82 Every MBPT diagram has a corresponding Feynman diagram, but there are Feynman diagrams that have no MBPT counterpart. 

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