Ordered Green functions

The name lesser originates from the time-ordered Green function, the main function in equilibrium theory, which can be calculated by diagrammatic technique... 

Now we are able to define contour or contour-ordered Green function - the useful tool of Keldysh diagrammatic technique. The definition is similar to the previous one... 

The problem can be generally resolved by using the EOM on the Schwinger-Keldysh time contour. Contour-ordered Green function is defined as... 

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). 

Here the superscript 0 represents the trace with respect to the non-interacting density matrix. The zeroth order Green functions are given in Eq. (55). The terms coming from the lead-molecule coupling (V. ) vanish because they are odd in creation and annihilation operators. Substituting Eq. (C34) in Eq. (C25) gives for the phonon contribution... 

Figure 4.1. Graphs of the Green function (a) and of the ordered Green function (6) for Lagrangian 4.2 8 graphs of polymer theory (c) (des Cloizeaux, 1975) [Reprinted with permission from Des Cloizeaux. J. de Phys. 36 (1975) 281-291. Copyright 1975 by EDP Sciences]...

Dcs Cloizeaux also introduces ordered Green functions. .. k M) as a sum... 

The successive terms of this expansion can be found with equality 49. Define tlie ordered Green functions of polymer theory... 

Each graph line corresponds to a polymer chain. To provide the calculation of thi ordered Green functions, an intermediate expression is introduced ... 

Here, we use short-hand notation 1 = (fi, q) etc. W is a screened potential, and the kernel lQri,cr2 is a two-particle propagator. In the case of multiple electron-hole scattering, the kernel (electron-hole propagator) is a product of electron and hole time-ordered Green functions... 

C.T. Tai. Dyadic Green functions in electromagnetic theory. IEEE Press Series on Electromagnetic Waves, Second Edition, ISBN 0-7803-0449-7, IEEE Order Number PC0348-3, 1993. 

Gonis, A., 1992, Green Functions for Ordered and Disordered Systems, North Holland Elsevier Science Publishers B.V., Amsterdam. 

Linear response function approaches were introduced into the chemistry literature about thirty years ago Ref. [1,2]. At that time they were referred to as Green functions or propagator approaches. Soon after the introduction it became apparent that they offered a viable and attractive alternative to the state specific approaches for obtaining molecular properties as excitation energies, transition moments and second order molecular properties. 

Theoretical chemists learn about a number of special functions, the Hermite functions in connection with the quantisation of the harmonic oscillator, Legendre and associated Legendre functions in connection with multipole expansions, Bessel functions in connection with Coulomb Greens functions, the Coulomb wave functions and a few others. All these have in common that they are the solutions of second order linear equations with a parameter. It is usually the case that solutions of boundary value problems for these equations only exist for countable sets of values of the parameter. This is how quantisation crops up in the Schrddinger picture. Quantum chemists are very comfortable with this state of affairs, but rarely venture outside the linear world where everything seems to be ordered. 

From this equation it follows that dg,A Pa is diagonal in the spin indices. We will therefore in the following put density variation 5p (r) determines the potential variation 5vs,(r) only up to a constant (see also [66] ). To find an explicit expression for the above functional derivative we must find an expression for the inverse density response function i A. In order to do this we make the following approximation to the Greens function (see Sharp and Horton [39], Krieger et al. [21]) ... 

Nakatsuji and Yasuda [56, 57] derived the 3- and 4-RDM expansions, in analogy with the Green function perturbation expansion. In their treatment the error played the role of the perturbation term. The algorithm that they obtained for the 3-RDM was analogous to the VCP one, but the matrix was decomposed into two terms one where two A elements are coupled and a higher-order one. Neither of these two terms can be evaluated exactly thus, in a sense, the difference with the VCP is just formal. However, the structure of the linked term suggested a procedure to approximate the A error, as will be seen later on. 

The presence of large neutral (green) regions in the electrostatic potential map for vitamin E suggests that the molecule will be soluble in lipids (as it must be in order to function as a trap for radicals). 

In order to find the single-particle-like excitation energies it is necessary to investigate the structure of the Green function, given by the Dyson Eq. (14). Sham and Kohn here argued that... 

We denote this by in order to indicate explicitly that the Green function is perturbed by M perturbation elements, and then introduce a series of new (hypothetical) Green functions defined by... 

A more precise value than in [63] of the nonlogarithmic correction of order a Za) for the IS -state was obtained in [66, 67], with the help of a specially developed perturbation theory for the Dirac-Coulomb Green function which expressed this function in terms of the nonrelativistic Schrodinger-Coulomb Green function [68, 69]. But the real breakthrough was achieved in [70, 71], where a new very effective method of calculation was suggested and very precise values of the nonlogarithmic corrections of order a Zo) for the IS -and 25-states were obtained. We will briefly discuss the approach of papers [70, 71] in the next subsection. 

Over the years different methods were applied for calculation of the radiative-recoil correction of order a Za). It was first considered in the diagrammatic approach [1, 3, 2]. Later it was reconsidered on the basis of the Braun formula [4]. The Braun formula depends on the total electron Green function... 

As in the case of corrections of order a (Za) m, not only the diagrams in Fig. 7.3 with insertions of polarization operators in one and the same external Coulomb line but also the reducible diagrams Fig. 7.5 with polarization insertions in different external Coulomb lines generate corrections of order Respective contributions were calculated in [18] with the help of the subtracted Coulomb Green function from [20]... 

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