Current approximation function

In this chapter, the current approximation function Q, defined in Chap. 3, (3.25), will be used extensively. Note also that since this function is a linear combination of the array argument (for example, C as in Q(C,n. H)), the function of a weighted sum of two arrays, such as the arrays u and v (to be met later), the following holds (a being some scalar factor) ... 

In this chapter, the current approximation function Q, defined in Chap. 3, Eq. (3.25), will be used extensively. Note also that since this function is a linear... 

Following on the work of Kohn and Sham, the approximate functionals employed by current DFT methods partition the electronic energy into several terms ... 

Equation (15.23) displays the feature of locality that the blending functions should possess in order to be computationally advantageous that is, during the process of matrix inversion, one wishes the calculation to proceed quickly. As mentioned earlier, the use of linear approximation functions results in at most five terms on the left side of the equation analogous to (15.23), yielding a much crader approximation, but one more easily calculated. The current choice of Bezier functions, on the other hand, is rapidly convergent for methods such as relaxation, possesses excellent continuity properties (the solution is guaranteed to look and behave reasonably), and does not require substantial computation. 

In the previous section we presented the semi-classical electron-electron interaction we treated the electrons quantum mechanically but assumed that they interact via classical electromagnetic fields. The Breit retardation is only an approximate treatment of retardation and we shall now consider a more consistent treatment of the electron-electron interaction operator that also provides a bridge to relativistic DFT, which is current-density functional theory. For the correct description we have to take the quantization of electromagnetic fields into account (however, we will discuss only old, i.e., pre-1940 quantum electrodynamics). This means the two moving electrons interact via exchanged virtual photons with a specific angular frequency u>... 

One can now approximate the current autocorrelation function in the diffusive limit. When the time integration is performed, the above expression reduces to... 

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