Diagrammatic perturbation theory

The localized many-body perturbation theory (LMBPT) applies localized HF orbitals which are unitary transforms of the canonical ones in the diagrammatic many-body perturbation theory. The method was elaborated on models of cyclic polyenes in the Pariser-Parr-Pople (PPP) approximation. These systems are considered as not well localized so they are suitable to study the importance of non local effects. The description of LMBPT follows the main points as it was first published in 1984 (Kapuy etal, 1983). 

The following important conclusion can be drown, that the diagrammatic many-body perturbation theory can be used in a localized representation. 

The plan of this paper is as follows - In section 2, the basic experimental data required in the re-evaluation of the empirical correlation energies of the N2 CO, BF and NO molecules are collected. The essential theoretical ingredients of our re-determination are given in section 3 including new fully relativistic calculations including the frequency independent Breit interaction and electron correlation effects described by second order diagrammatic perturbation theory for the Be-like ions B", C, O" ... 

In recent years the diagrammatic technique of the perturbation theory found wide application in solving the stochastic differential equations, e.g., see a review article by Mikhailov and Uporov [68]. 

The presented form of the master equation (2.3.67) permits us to employ the diagrammatic technique of the perturbation theory [44, 108-110, 113]. The free Hamiltonian could be written as... 

The central topic of this chapter is the derivation of diagrammatic perturbation theory, which lays the basis for all the subsequent analysis. 

More recently, Caves and Karplus71 have used diagrammatic techniques to investigate Hartree-Fock perturbation theory. They developed a double perturbation expansion in the perturbing field and the difference between the true electron repulsion potential and the Hartree-Fock potential, V. This is compared with a solution of the coupled Hartree-Fock equations. In their interesting analysis they show that the CPHF equations include all terms first order in V and some types of terms up to infinite order. They propose an alternative iteration procedure which sums an additional set of diagrams and thus should give results more accurate than the CPHF scheme. Calculations on Ha and Be confirmed these conclusions. 

Before the discovery, in 1986, by Bednorz and Muller [1] of high-Tc superconductivity, there was a rather widespread belief that Solid State Theory was satisfactorily built so as to allow the comprehension of any solid state problem, provided the computations to be carried out were feasible. Nourishing this believe there was, on the free quasi-particle side, the Fermi Liquid Theory and tools as Green Functions and Diagrammatic Perturbation Theory, which allowed to study most of the band-theoretic-based solid-state properties. Namely, Diagrammatic Perturbation Theory has... 

Figure 1-2. Schematic diagrammatic representation of the E correction (Brandow skeletons). The horizontal lines represent the denominators, while the vertical bar separates the monomers A and B. The two-electron integral corresponding to the dotted interaction line is a Coulomb integral. The dashed interaction lines represent antisymmetric two-electron integrals of the monomers. Diagram (a) is the intermolecular perturbation theory form of the MP5 contribution s, diagram (d) of qQ(/7), while (b) and (c) are combinations of 7s T and E (I)...

In the past twenty years, there has been increasing interest in the calculation of correlation energies and other properties of atomic and molecular systems by means of diagrammatic many-body perturbation theory techniques3-9 due to Brueckner10 and Goldstone.11 Diagrammatic many-body perturbation theory provides a simple pictorial representation of electron correlation effects in atoms... 

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. 

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