Iterative localization accuracy

Figure 6.9. Sketch of the ellipsoid of accuracy (EOA) used in ISAT to control the local interpolation error. The initial EOA is usually conservative, and thus the EOA is grown in subsequent iterations.

In a straightforward application of the Newton-Raphson approach, Eq. (32) is solved iteratively for S and T until the convergence criteria (26) are fulfilled to the desired accuracy. This process converges nicely if the initial choice of the orbitals and the Cl coefficients are close to the final result. The energy is then in the local region where the second-order approximation is valid. Obviously such situations will not be very common in actual applications. In practice the starting orbitals are often obtained from a preceding SCF calculation, or even estimated from atomic densities , while the Cl... 

The main factors that affect the time-step size are precision and stability. Precision (accuracy) is related to the magnitude of the local error while stability refers to the decrease of the error in the iterative process. 

The (n + 1)-dimensional Krylov space constructed in Eq. (C.5) spans locally over p(t) and the subsequent n actions of A(t). As an orthogonal but incomplete basis set, the Gaussian quadrature accuracy of order would be expected for the n-dimensional Krylov space approximation. It thus allows the time-local evolution, p t + St) exp[A(t)St]p t) be evaluated accurately with a fairly large St, The project-out error can be estimated similarly as that of the short-iterative-Lanczos Hilbert-space propagator [51]. 

Advantages of the method the method is universal and can practically be applied for any system the method is efficient and can be automated to apply in computer code [62]. Disadvantages of the method in the method the criterion of localization is fixed as the Boys localization criterion, therefore there is no opportunity to receive WF as much as possible localized concerning any another criterion the symmetry use is not included, in some cases the number of iterations strongly depends on the initial approximation the use of a sufficiently dense grid of wavevectors is necessary to obtain a good accuracy in the gradient calculation. 

The target accuracy of the inner iterations depends on whether the calculation takes place in the local or the global part of the optimization. In each outer iteration in the local region, we solve the Newton equations (12.3.2) by means of a series of inner iterations. The resulting step vectcr Xapp is not an exact solution to the linear equations (12.3.2), but contains an error described by... 

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