Self energy and Dyson equation

B Dyson equation and self energy C Static and dynamic self energy... 

Alternatively, by means of the Dyson equation and the second-order truncation approximation, taking Hartree-like self-energies r = U ns) (= E M), we can also get the retarded GFs as follows... 

The importance of the particle spectral function is to exhibit where the unoccupied sp strength is located in energy. It is obtained by solving the Dyson equation and is related to the self-energy by... 

In this section, we describe our model, and give a brief, self-contained account on the equations of the non-equilibrium Green function formalism. This is closely related to the electron and particle-hole propagators, which have been at the heart of Jens electronic structure research [7,8]. For more detailed and more general analysis, see some of the many excellent references [9-15]. We restrict ourselves to the study of stationary transport, and work in energy representation. We assume the existence of a well-defined self-energy. The aim is to solve the Dyson and the Keldysh equations for the electronic Green functions ... 

The GF of d-electrons takes into account Coulomb interaction on a site, therefore it has a self-energy X j P and obeys the Dyson equation. It can be written in momentum representation as ... 

We now have the tools for constructing the spectral function A4p(E). We begin by studying the energy dependence of the self-energy and the solution to the Dyson equation, keeping the static monopole and the dynamic dipole terms and neglecting Fermi sea correlations... 

Fig. 25. Real and imaginary parts of the 4 p self-energy (E) and straight lines E-Ej(zlSCF), i = 4 pi/2.3/2. giving graphical solutions of the Dyson equation (Eq. (15))...

The inclusion of higher order corrections is performed as in the 4p-case. Graphical representations of the real and imaginary parts of the 4 s self-energy and solutions of the Dyson equation are shown in Fig. 27, and the corresponding quasi-particle parameters are given in Table 2. 

Fig. 44. (a) Schematic picture of the real part of the self-energy for a valence hole, together with graphical solutions , and of the Dyson equation (Eq. (15)). (b) is meant to represent a typical outer-valence hole spectrum while (c) and (d) describe the possible behaviour of inner-valence holes. (bHd) are connected with the solutions -( ) resp. Note that in principle the self-energy is different for different valence holes, contrary to what is suggested in (a)... 

In the subsequent discussion we shall need the irreducible 2- and 3-point functions i.e. the electron self energy Sy(p), the vacuum polarisation /7k,pv( ) and the full vertex function rv, PuP2% as knowledge of I vip) and determines the corresponding propagators Gy(p) and Dk,mv( ) completely and fy. iPuPz) represents the perturbative corrections to the free vertex The connection between these quantities is established by the Dyson equations (see e.g. [26])... 

The single-particle Green s function is related to the irreducible self energy by Dyson s equation (38) and to the two-particle Green s function by the equation of motion (37). It can also be related to the xc potential of DFT by the Sham-Schliiter equation [50]... 

These are given in Eqs. (C29) and (C38). The self-energy expressions (C38) and (C40) are calculated perturbatively to second order in the electron-phonon coupling in terms of the zeroth order Green functions (Eq. (55)). The simplest expression for current is obtained by substituting Eqs. (55), (C29) and (C38) in Eq. (51). This zeroth order result can be improved by using the renormalized Green functions obtained from the self-consistent solution of the Dyson equation (44). 

Computations were performed also for the nucleotide base stacks and base pair stacks.31-34 In the case of the cytosine (C) stack we have investigated the DOS and hopping conductivity as a function of the basis set and correlation corrections (inverse Dyson equation with MP2 self energy see above). In Figure 3 we show how the DOS changes if one uses instead of an STO-3G a... 

Methods. -3.1.1 Inverse Dyson Equation with MP2 Self Energy. - In the previous section this method was already formulated for molecules [dimers see equations (11>—(15)]. To apply it to periodic polymers one has only to substitute the level indices /, j, a, b in these equations by capital letters /, /, A, B. These capital letters stand for combined indices I = i,, J = j, ky A = a, ka and — b, . Here i etc. stands for the i-th band and k etc. for a particular quasi momentum in the band. Of course the quasi momenta have to fulfil the conservation law... 

This equation is the main result of the present considerations. In order to define the two-particle self energy (w) and for establishing the connection to the familiar form of Dyson s equation we adopt a perturbation theoretical view where a convenient single-particle description (e. g. the Hartree-Fock approximation) defines the zeroth order. We will see later that the coupling blocks and vanish in a single-particle approximation. Consequently the extended Green s function is the proper resolvent of the zeroth order primary block which can be understood as an operator in the physical two-particle space ... 

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