Local approximation

This calculation shows explicitly the correction to the MFA. With the quadratic form considered for and the local approximation, the calculation of k T n det 0A) can be performed exactly [39]. The change in the free energy by unit of volume, A.Fcharging when we switch on the charge is given by... 

For h- = h+ = h the preceding is identical with expre.ssion (7) (see Example 2). Plain calculations of the local approximation error at a point x show that... 

Thus, operator (26) provides the local approximation of order 1 on any irregular pattern with h /i+. 

In Section 1.2 we have found the local approximation error taking now the form... 

Another conclusion can be drawn from this example. Of course, the study of the local approximation is unsufRcient for determination of the order of the difference approximation and proper evaluation of the quality of a difference operator. 

Conditions (5) of Section 1 relating to the second-order local approximation for the conservative scheme (17) acquire the form... 

This means that scheme (2) provides a local approximation of order 2, so that II ij. 11 < Mh, where M = const > 0 is independent of h. 

We claim that in this case the operator A generates the local approximation of zero order... 

In Chapter 2 we came across the necessity of generalizing the notion of approximation in the real situations when a difference scheme cannot provide on the grid local approximations with a desired order in the norm of the space C, but it does the same in one of the negative norms, that is, in a certain sense of summarizing. 

Likewise, it may happen that a scheme on the grid u> cannot provide local approximations in t, but at the final stage the approximation will be achieved once we bring together the residuals over several time layers. The notion of summarized approximation needs certain clarification. It seems worthwhile giving simple examples. 

Here the coefficients and f(x) are so chosen as to provide on a uniform grid a local approximation of order 2. By analogy with Chapter 4, Section 3 it is plain to justify the uniform convergence of scheme (81) with the rate... 

The above geometrical argument may be supplemented by an explicit demonstration that the singularity at the point D is of the focus-focus type (Eq. (7)), when Eq. (51) applies. The procedure is to employ the local approximations... 

There are several good reasons to focus on linear models. Theory may indicate that a linear relation is to be expected, e.g. Lambert-Beer s law of the linear relationship between concentration and absorbance. Even when a linear relation does not hold strictly it can be a sufficiently good local approximation. Finally, one may try and find a transformation of the individual variables (e.g. a logarithmic transformation), in order to obtain an acceptable linear model for the transformed variables. Thus, we simplify eq. (36.1) to... 

Note that the external field vanishes in the normal surface tension calculation. In the fully local approximation there is no surface tension. Thus we can obtain the surface tension y associated with a planar interface of area A by the expression... 

From the early advances in the quantum-chemical description of molecular electron densities [1-9] to modem approaches to the fundamental connections between experimental electron density analysis, such as crystallography [10-13] and density functional theories of electron densities [14-43], patterns of electron densities based on the theory of catastrophes and related methods [44-52], and to advances in combining theoretical and experimental conditions on electron densities [53-68], local approximations have played an important role. Considering either the formal charges in atomic regions or the representation of local electron densities in the structure refinement process, some degree of approximate transferability of at least some of the local structural features has been assumed. 

The practical problems associated with the evaluation of exchange integrals led Slater (1974) many years ago to introduce a localized approximation to the exchange. This has been widely utilized and more recently has been modified to include additional correlation effects. Potentials so obtained are called local-density potentials. ... 

Thus, in a steady state, dNi/dt = 0, one has aN = const., which actually holds over considerable ranges of mass number (thus being usable as a local approximation), but not globally (except if r —> oo) in particular, it breaks down near the magic numbers (see Fig. 6.3). 

Within this local-density approximation one can obtain exact numerical solutions for the electronic density profile [5], but they require a major computational effort. Therefore the variational method is an attractive alternative. For this purpose one needs a local approximation for the kinetic energy. For a one-dimensional model the first two terms of a gradient expansion are ... 

Within a local approximation, the electron density is just a function of the modified potential and the induced electron density 8p(r) given by Eq (85) reduces to [36] ... 

The scheme analyzed so far is, in a way, a simplification of the Hartree-Fock scheme. As such, it is only a model approximation. The most serious drawback is the replacement of a fundamentally quantum mechanical term, whose very nature is to be non local, by a local approximation. Of course, when the system is in an electronic degenerate state, or when the BO approximation is no longer valid, the density functional method cannot be applied. For a discussion of this and other limitations the reader is referred to the paper by Bersuker [117],... 

Figure 5.4 Fundamental vibrations of acetylene in the local approximation.

As mentioned above, LSD yields a reasonable description of the exchange-correlation hole, because it satisfies several exact conditions. However, since the correlation hole satisfies a zero sum rule, the scale of the hole must be set by its value at some value of . The local approximation is most accurate at points near the electron. In fact, while not exact at m = 0, LSD is highly accurate there. Thus the on-top hole provides the missing link between the uniform electron gas and real atoms and molecules [18]. 

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