Extrapolation Techniques and Other Technical Issues

Finite difference methods allow the use of techniques which extrapolate to step size — 0 (i. e., the exact solution) and control the numerical truncation error [494,495]. This can be done for every numerically calculated quantity f if we assume an analytic behavior of f, 

Multigrid methods with control of the numerical truncation error are very useful for the solution of matrix equations since they start with a small number of grid points, use extrapolation techniques similar to Richardson s and reduce the step size until the result is accurate enough. The method of Bu-lirsch and Stoer (cf. [495, p. 718-725] and [502, p. 288-324]), for instance, consists essentially of three ideas. The calculated values for a given step size h are 

Since the SCF equations can only be solved iteratively, a good starting approximation is needed to set up the potentials Ukivir) in the Fock operator. The quality of this approximation determines the convergence behavior of the SCF cycles. There are numerous possibilities to generate an initial set of radial 

Another model potential can be obtained utilizing the statistical theory for the electron distribution in an atom due to the studies done by Thomas and Fermi [266, p. 145-156]. The Thomas-Fermi potential can be seen as the simplest potential possible within the framework of density functional theory in section 8.8. This close connection to DFT shows that exchange and correlation functionals can easily be introduced into the program code despite numerical 

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