Self-consistent approximation

Quite recently, Pini et al. [56] have reported a new, thermodynamically self-consistent approximation to the OZ relation for a fluid of spherical particles for a pair potential given by a hard-core repulsion and a Yukawa attractive tail (Eq. (6)). The closure to the OZ equation they have proposed has the form... 

In this scheme, thermodynamically self-consistent approximations [3] can be derived by truncating the expansion of , which amounts to a resummation of whole classes of diagrams in perturbation theory. 

F. Colmenero and C. Valdemoro, Self-consistent approximate solution of the 2nd-order contracted Schrodinger equation. Int. J. Quantum Chem. 51, 369 (1994). 

In the perturbative approach the first order (or higher order) expressions for the self-energy and the polarization operator are used. The other possibility is to summarize further the diagrams and obtain the self-consistent approximations (Figs. 18,19), which include, however, a new unknown function, called vertex function. We shall write these expressions analytically, including the Hartree-Fock part into unperturbed Green function Gq(1, 2). 

In the limit of small Ajj, a self-consistent approximation method may be used to give upper and lower bounds for Z (5),... 

The approximate ground state of the form in eq. (1.231) in the ImP subspace is sought in the self consistent approximate form... 

The parameter F mimics tire short time, microscopic dynamics, and depends on structural and hydrodynamic correlations. The memory function describes stress fiuctuations which become more sluggish together with density fiuctuations, because slow structural rearrangements dominate all quantities. A self consistent approximation closing the equations of motion is made mimicking (14a). In the... 

Formula (15.5.4) can also be derived by using the self-consistent approximation introduced by de Gennes.25... 

B. Weiner and O. Goscinski The Self-Consistent Approximation to the Polarization Propagator Int. J. Quantum Chem. 18, 1109 (1980). 

The Harris functional provides not only relatively accurate non-self-consistent estimates of energies and structures, it also provides a basis from which the success of other non-self-consistent approximate methods can be understood. One of the more widely used of these, the tight binding method, is discussed in the next section. 

One particnlar self-consistent approximation to the Schrodinger equation yields a veiy popular class of PES s called pseudopotentials [90Rapl]. The approximation is called density functional theory (DFT) and expresses the total energy of tte system as a sum of two terms ... 

FIGURE 5.9 Comparison of the Li-radial density given with the electron localization function for the simplified self-consistent approximation for Li atomic structure after Putz et al. (2006). 

The periodic potential V(r) determines the properties of the Bloch function. It includes the interaction between all electrons and ions. One applies the Hartree-Fock procedure in self-consistent approximation in order to find the potential V(r). Consequently, the many-partide problem is reduced to the one-electron problem by means of the averaged field V(r). 

This expression still involves processes of electron scattering with respect to each other in the self-consistent approximation (see Figure 6). If these processes are neglected, retaining only the interaction of the electrons with the phonon particles (Figure 7), i.e., if the random phase approximation is used in the self-consistent field method, we obtain for Wif... 

In addition to destroying the isotropy of space, the introduction of the SCF field implies a Markovian (albeit self-consistent) approximation to the inherently non-Markovian process described by (6.16). Thus, if EjiCK... 

Both the Anderson and the Kondo (or Coqblin-Schrieffer) model have been solved exactly for thermodynamic properties such as the 4f-electron valence, specific heat, static magnetic and charge susceptibilities, and the magnetization as a function of temperature and magnetic field B by means of the Bethe ansatz (see Schlottmann 1989, and references therein). This method also allows one to calculate the zero-temperature resistivity as a function of B. Non-equilibrium properties, such as the finite temperature resistivity, thermopower, heat conductivity or dynamic susceptibility, could be calculated in a self-consistent approximation (the non-crossing approximation), which works well and is based on an /N expansion where N is the degeneracy of the 4f level. 

Eq.(2.27) together with the Dyson equation (2.1 ) (with t defines so-called "0-derivable" approximations. 0 derivability guarantees then a self-consistent approximation in many-body perturbation theory which, for a given order of approximation, satisfies various conservation laws. For example the particle-hole attraction in eq.(2.9) is given by (Ward identity) 6 0/6g6g. 

Fig. 6. The difference Compton profile for Ti and TiH experimental and theoretical for several models. The line A shows the experimental line profile difference normalised to two electrons before applying deconvolution for instrumental broadening and smoothing on row data of Ti and TiH. The solid line B shows the difference profile of Bandstructure calculations according to APW self-consistent approximation after a convolution with a Gaussian resolution function of a=0.30 a.u. The remaining lines C-F are the theoretical difference profiles of indicated model after convolution with the previous resolution function. 

A more accurate solvation potential approach was then developed by Chandler and co-workers in the context of a quantum electron in dilute (classical) solution that led to new and more accurate self-consistent approximations for the effective pair potential in This... 

GAP obtain a class of self-consistent approximations for the Green function that converge rapidly and that can be used to calculate all transport properties of interest. The results are in excellent agreement with time dependent fluorescence depolarization experiments on dye molecules in solution [7]. Loring, Andersen, and Payer [5] subsequently extended the GAP method to the case of a two component homogeneous mixture of randomly distributed donor and trap chromophores The LAP theory is also in excellent agreement with trapping experiments performed in solution [44]. 

Figure 8. Diagonal portion of the Green function for the randomly tagged chain at high dilution. The universal curves labeled 3R, 2 and 2C are, respectively, the three-body rational self-consistent approximation, the two body self consistent approximation and the two-particle cumulant. Taken from Figure 2 of reference [13].

TT R p. This problem was solved earlier by Gochanour, Andersen and Fayer [4]. Their three-body self consistent approximation agrees extremely well with Equation (60). 

In Figure 8 we have plotted Equation (62) as well as the two body self-consistent approximation and the three-body rational self consistent approximation for this problem. The three-body rational approximation is believed to describe excitation transport on an ideal chain quite accurately [12]. Although the agreement between curves 3R and 2C is not as good as was observed for the case of randomly distributed chromophores in an infinite volume [13], the two body cumulant is bounded by the two-body and three- ody curves. Thus, it should give a quantitative approximation to G (t). It should be obvious that combination of Equation (57) for C with Equation (62) should lead to a straightforward determination of from measurement of the transient fluorescence anisotropy. 

Inclusion solutions are applicable only for very low porosity or low concentration of inclusions. A method to extend these to some higher concentrations is the self-consistent approximation. This is done by mathematically replacing stepwise the solid material with the resulting effective medium (self-consistent). The method was developed and promoted mainly by Budiansky and O Connell (1976). 

Berryman (1995 see also Mavko et al., 1998) gives a more general form of the self-consistent approximation for n components (i is the index of the individual component) ... 

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