Noninteractive model

From the discussion so far, it is clear that the mapping to a system of noninteracting particles under the action of suitable effective potentials provides an efficient means for the calculation of the density and current density variables of the actual system of interacting electrons. The question that often arises is whether there are effective ways to obtain other properties of the interacting system from the calculation of the noninteracting model system. Examples of such properties are the one-particle reduced density matrix, response functions, etc. An excellent overview of response theory within TDDFT has been provided by Casida [15] and also more recently by van Leeuwen [17]. A recent formulation of density matrix-based TD density functional response theory has been provided by Furche [22]. 

Thus all segment correlations of our noninteracting model take the form of Gaussian functions. We stress that for our Gaussian chain model all these results are valid rigorously for all n > 1 or k2 — k] > 1, respectively. 

Evaluating the product in Eq. (4.3) we get back the Greensfunction GVjfp n) of the noninteracting model to zero order in 3e, Writing... 

The only relevant macroscopic parameter of the noninteracting model is the end-to-end distance, which is invariant under this transformation ... 

The locality hypothesis can be tested in a noninteracting model, in which the functional Fs is replaced by 7 . The kinetic energy orbital functional is T = ni 0 If 10... 

In order to calculate the diffusive flux, a suitable mass transfer model must be assumed. T vo categories of models exist (1) interactive models (due to Krishna and Standart [206] and Toor [207]), and (2) noninteractive models, known also as effective diffusivity models. For the interactive models, the diffusion flux j b is... 

This is in agreement with the requirements proposed by Sykes (1973) and defines the shaded region in Fig. 5.51, indicating that both double-S limitations and a switch between limiting substrates are quite common. However, Equ. 5.172 holds only for kinetic models such as the Monod and exponential models approaching saturation asymptotically. To handle this problem, it is desirable to overlay curves of constant jll on Fig. 5.51 for some of the models discussed earlier. Two different philosophies can be distinguished interactive and noninteractive models. 

For strictly sequential substrate utilization, Bader (1978) has proposed a noninteractive model that will seldom be encountered in its purest form. A noninteractive model basically implies that fx is limited by only one substrate at a time. Therefore, the growth rate will be equal to the lowest rate that would be predicted from the separate single-S models. For Monod-type kinetics, this would be written as follows ... 

Comparisons of both approaches for Monod kinetics are shown in Figs. 5.52c and 5.53b. Noninteractive models are, by their nature, discontinuous functions at the transition line from one substrate limitation to another, predicting... 

Figure 5.53. (a) Conceptual representation of the noninteractive model. Systems 1 and 2 operate independently of one another, (b) Plots of lines of constant dimensionless specific growth rate a function of two dimensionless substrate concentrations... 

The dimerized chain is the simplest model of semiconducting polymers, and is applied in particular to trans-polyacetylene. The noninteracting electronic structure of conjugated polymers with more complex unit cells, such as poly(para-phenylene), will be discussed in their relevant chapters. We emphasize that the noninteracting model is a simple model. It is not a realistic description of the electronic states of conjugated polymers, as it neglects two key physical phenomena electron-phonon coupling and electron-electron interactions. Despite these deficiencies it does provide a useful framework for the more complex descriptions to be described in later chapters. 

This equation is the basis of the adhesion model of friction and accounts for the adhesion component. The idea that frictional work could be usefully described by two components, the ploughing and adhesion contributions, was initially developed for metals and has been extended to polymers. Extensive studies have shown that it is often sufficient to consider that the two components are noninteracting and that the friction is just the simple sum of the predictions of the two models. This approach is described as the two term, noninteracting model and should be regarded as a useful first order approximation only. 

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