Numerical Solution of Matrix Equations

We are now prepared for the difficult task of solving the transformed SCF equations. This involves the process of expressing the equations in such a way that it can be dealt with on a computer. In addition, we already know that the equations cannot be solved in one shot because the operators depend on the solution and a self-consistent-field approach is envisaged, for which tailored convergence accelerators are needed. 

Finite-difference methods operating on a grid consisting of equidistant points ( Xi, Xi = ih + Xq) are known to be one of the most accurate techniques available [496]. Additionally, on an equidistant grid all discretized operators appear in a simple form. The uniform step size h allows us to use the Richardson extrapolation method [494,497] for the control of the numerical truncation error. Many methods are available for the discretization of differential equations on equidistant grids and for the integration (quadrature) of functions needed for the calculation of expectation values. 

Simple discretization schemes use derivatives of Lagrangian interpolation polynomials that approximate the function known only at the grid points x,. These schemes consist of tabulated numbers multiplied by the function s values at m contiguous grid points and are referred to as m-point-formulae by Bickley [498] (cf. Ref. [468, p.914]). For an acceptable truncation error 0(h ), f = 4 or higher, m is larger than t which leads to an extended amount of computation. 

After choosing an appropriate variable transformation function s(r) and the number of inner grid points n the differential equations have to be discretized at the chosen points. Because of the normalization of the range of s to the unit interval, these grid points Sfc and the corresponding step size h are given by the general formula 

In the following sections, implications resulting from the general treatment of the variable transformations are formulated for a couple of finite difference methods of an acceptable numerical truncation error of order h. The accuracy of the data produced by a proper computer implementation may be checked and improved by the Richardson extrapolation. 

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