Dyson equation

A simple repetition of the iteration procedure (2.20)-(2.22) results in divergence of higher order solutions. However, a perturbation theory series may be summed up so that all unbound diagrams are taken into account, just as is usually done for derivation of the Dyson equation [120]. As a result P satisfies the integral-differential equation... 

Corrections to the zeroth-order, inverse propagator in equation 26 are gathered together in a term known as the self-energy matrix, E(E). The Dyson equation may be written as... 

Turning first to the metal film, we let G be the Greenian for the infinite, one-dimensional metal, and g its finite counterpart. These two Greenians are connected by the Dyson equation (3.3)... 

The semiconductor support can be treated in a similar manner (Bose and Foo 1974), since the Greenians G2 and g2 of the infinite and semi-infinite solids, respectively, are also linked by the Dyson equation, i.e.,... 

Knowing gi and g2, the Greenian gv2 for the composite metal-support system is given by the Dyson equation... 

In this section, we utilize the Dyson equation (5.15) to develop the surface GF of the composite system. Inserting (5.17) in (5.15), the matrix element... 

It is fairly easy to show, from the Dyson equation (6.45), that the surface... 

The above recursive process, for the 2-atom chain, can now be repeated for the 3-atom case. The Dyson equation analogous to (7.23) is... 

The frontier orbitals responses (or bare Fukui functions) f (r) and the Kohn-Sham Fukui functions (or screened Fukui functions)/, (r) are related by Dyson equations obtained by using the PRF and its inverse [32]. Indeed, by using Equation 24.57 and the chain rule for functional derivatives in Equation 24.36, one obtains... 

Equations 24.75 and 24.76 generalize the Dyson equations derived previously [32], The differences between /1 and the frontier orbitals have nice and simple physical interpretations [24,32] as either the variation of the electronic density induced by effective external potentials 8vj ... 

It is worth noting that an alternative computational scheme to solve the Dyson equations Equations 24.75 and 24.76 has appeared recently [55,56], Simple models of the response s have also been developed and applied to reactivity [57]. 

However, it can be argued that appropriate approximate solutions of the Schwinger-Dyson equations can yield reasonable approximations for Q, or the pressure, see the schematic Figure 1. 

The other difference refers to the way in which the different groups approximate A. Thus, while VCP focus the attention on a renormalization of the approximated 3-RDM which uses the complementary holes matrix, NY again inspire their procedure on the assumption of an analogy with the Dyson equation. Finally, Mazziotti uses a self-consistent algorithm which may be sumarized as The A-RDM is calculated by means of an algorithm which coincides with the one proposed by NY, expressed in a compact form as ... 

Nakatsuji and Yasuda [56, 57] focused on the term appearing in A, which according to the perturbative expansion could be interpreted as a linked diagram of two A elements. In analogy to the Dyson equation, they proposed to estimate the A with a procedure whose main step, expressed in a spin-orbital basis, may be written... 

Next we consider the case where more than one barrier (or perturbation element) appears in the diffusion path, and the particle receives multiple perturbations. Teraoka and Hayakawa [107] assumed that the Green function for this case satifies the following Dyson equation ... 

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