The Mean Field Model

A) HOW THE MEAN-FIELD MODEL LEADS TO ORBITALS AND CONFIGURATIONS... 

By expressing the mean-field interaction of an electron at r with the N- 1 other electrons in temis of a probability density pyy r ) that is independent of the fact that another electron resides at r, the mean-field models ignore spatial correlations among the electrons. In reality, as shown in figure B3.T5 the conditional probability density for finding one ofA - 1 electrons at r, given that one electron is at r depends on r. The absence of a spatial correlation is a direct consequence of the spin-orbital product nature of the mean-field wavefiinctions... 

The Mean-Field Model, Which Forms the Basis of Chemists Pictures of Electronic Structure of Molecules, Is Not Very Accurate... 

Mean-field models are obviously approximations whose aeeuraey must be determined so seientists ean know to what degree they ean be "trusted". For eleetronie struetures of atoms and moleeules, they require quite substantial eorreetions to bring them into line with experimental faet. Eleetrons in atoms and moleeules undergo dynamieal motions in whieh their eoulomb repulsions eause them to "avoid" one another at every instant of time, not only in the average-repulsion manner that the mean-field models embody. The inelusion of instantaneous spatial eorrelations among eleetrons is neeessary to aehieve a more aeeurate deseription of atomie and moleeular eleetronie strueture. 

In summary, the dynamieal interaetions among eleetrons give rise to instantaneous spatial eorrelations that must be handled to arrive at an aeeurate pieture of atomie and moleeular strueture. The simple, single-eonfiguration pieture provided by the mean-field model is a useful starting point, but improvements are often needed. 

Corrections to the mean-field model are needed to describe the instantaneous Coulombic interactions among the electrons. This is achieved by including more than one Slater determinant in the wavefunction. 

Figure 6.2a shows chronoamperometric transients for CO oxidation recorded on three different stepped electrodes for the same final potential. Clearly, the electrode with the higher step density is more active, as it oxidizes the CO adlayer in a shorter period of time. Figure 6.2b shows a fit of a transient obtained on a Pt(15, 15, 14) electrode (terrace 30 atoms wide) by both the mean field model [(6.5), solid line] and the N G model [(6.6), dashed line]. The mean field model gives a slightly better fit. More importantly, the mean field model gives a good fit of all transients on all electrodes. 

Figure 3a shows the mean-field predictions for the polymer phase diagram for a range of values for Ep/Ec and B/Ec. The corresponding simulation results are shown in Fig. 3b. As can be seen from the figure, the mean-field theory captures the essential features of the polymer phase diagram and provides even fair quantitative agreement with the numerical results. A qualitative flaw of the mean-field model is that it fails to reproduce the crossing of the melting curves at 0 = 0.73. It is likely that this discrepancy is due to the neglect of the concentration dependence of XeS Improved estimates for Xeff at high densities can be obtained from series expansions based on the lattice-cluster theory [68,69].

In each case, the mean-field model forms only a starting point from which one attempts to build a fully correct theory by effecting systematic corrections (e.g., using perturbation theory) to the mean-field model. The ultimate value of any particular mean-field model is related to its accuracy in describing experimental phenomena. If predictions of the mean-field model are far from the experimental observations, then higher-order corrections (which are usually difficult to implement) must be employed to improve its predictions. In such a case, one is motivated to search for a better model to use as a starting point so that lower-order perturbative (or other) corrections can be used to achieve chemical accuracy (e.g., 1 kcal/mole). 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

The characteristic parameters Dapp and qc occurring in the theory can be expressed in the context of the mean-field model for polymer mixtures by... 

Thermal fluctuations can contribute dominantly to the scattering intensity right after the isothermal phase separation starts [70,76], Therefore, conditions 1) and 3) must be fulfilled to ensure that the effect of thermal noise is negligible. The dynamics of phase separation can be adequately described by the mean-field model if condition 2) is satisfied. Condition 2) is a direct consequence of the Landau Ginzburg criterion [75]. Thus, one may establish prerequisites for Eqs. (27) and (33) are the conditions 1) and 3), while Eq. (34) requires conditions 2) and 3). For example, Eq. (27) and as a consequence Eq. (33) cannot be confirmed experimentally not even for small values of q if the quench depth e is too small [70]. Moreover, owing to the effect of thermal fluctuations, Eq. (33) fails at q as qc even if the Landau Ginzburg criterion is fulfilled [70,77]. Thus, in the former case condition 2) is violated whereas in the latter example conditions 1) and 3) are not satisfied. 

It is instructive to consider the predictions of the mean field model for the above type of composite. A frequently used mean field model is that of Kerner [32],... 

The self-consistent mean field models developed so far have been very useful in describing the trends in the mechanical properties of the composite as a function of the volume fraction of inclusions, and in giving some physical insight on how given inclusions modify the mechanical response. However, the mean field models have not been extensively tested against experimental data, particularly at high volume fraction of inclusions. 

In the mean-field models, a first-order phase transition is observed for values of Y which are less than 10% below the value obtained by Ziff, Gulari and Barshad [60,61]. The main failure of the MF model is that it does not predict the second-order phase transition that is observed in the simulations. This is due to the complete neglection of spatial correlations. 

At this point let us make a remark concerning the size of the basis. In order to obtain convergence, one must sometimes include (Briels et al., 1984) basis functions with high values of / and n. High values of l are needed in particular when the orientations of the molecules are fairly well localized. This leads to a rapidly increasing size of the basis. Two measures can be taken to simplify the problem. First, one can adapt the basis of molecule P to the site symmetry at P, which block-diagonalizes the secular problem. If this does not sufficiently reduce the problem, the mean field model Hamiltonian (96) can be further separated by writing... 

The mean field model outlined in the preceding section provides us with a set of single-particle states... 

Fig. 2. Orientational probability distributions of the molecular axes in (a) a-nitrogen and (b) y-nitrogen. Contours of constant probability for the molecule in the origin, calculated in the mean field model, are plotted as functions of the polar angles (0, ) with respect to the crystal axes (Fig. 1). The angle 0 increases linearly with the radius of the plots from 0 (in the center) to tt72 (at the boundary) d> is the phase angle.

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