Strength function approximate computation

We presented fully self-consistent separable random-phase-approximation (SRPA) method for description of linear dynamics of different finite Fermi-systems. The method is very general, physically transparent, convenient for the analysis and treatment of the results. SRPA drastically simplifies the calculations. It allows to get a high numerical accuracy with a minimal computational effort. The method is especially effective for systems with a number of particles 10 — 10, where quantum-shell effects in the spectra and responses are significant. In such systems, the familiar macroscopic methods are too rough while the full-scale microscopic methods are too expensive. SRPA seems to be here the best compromise between quality of the results and the computational effort. As the most involved methods, SRPA describes the Landau damping, one of the most important characteristics of the collective motion. SRPA results can be obtained in terms of both separate RPA states and the strength function (linear response to external fields). 

In addition, comparison to solutions of the Hubbard and PPP models including electron correlation shows the VB wave function to be a more accurate initial approximation than the Hartree Fock solution at the correlation strengths likely to be encountered in realistic semiempirical models. In spite of the qualitative superiority of the VB wave function, systematic computational approaches to more accurate treatment of correlation are still most readily achieved when starting from the independent particle limit, but the correlated wave functions thus built up are likely to be interpretable in valence bond terms. 

Artificial neural networks (ANNs) are most often used for function approximation and for an object classification even if only incomplete and noisy data are available (Rafiq et al. 2001). In structural rehability analysis the role of ANN as a universal tool for function approximation is utilized when the limit state function under consideration is complicated and computer-time consuming, cf Hurtado Alvarez (2001), Gomes Awruch (2004). Typical examples are nonlinear problems, e.g. the assessment of post-budding strength of plates or shells. The inevitable FEM calculations of strength are carried out for suitably chosen sets of training and validation patterns. A subsequent reliability analysis then can be performed by the obtained ANN approximation of the strength function. 

The angular velocity and angular momentum acfs themselves are important to any dynamical theory of molecular liquids but are very difficult to extract directly from spectral data. The only reliable method available seems to be spin-rotation nuclear magnetic relaxation. (An approximate method is via Fourier transformation of far-infrared spectra.) The simulated torque-on acfs in this case become considerably more oscillatory, and, which is important, the envelope of its decay becomes longer-lived as the field strength increases. This is dealt with analytically in Section III. In this case, computer simulation is particularly useful because it may be used to complement the analytical theory in its search for the forest among the trees. Results such as these for autocorrelation functions therefore supplement our... 

Fig. 11. The Mott non-metal to metal transition as a function of the average sei>aration between the atoms of a cluster of 91 atoms. Shown are the computed energies in units of t (logarithmic scale) of the excited electronic states relative to the ground state. Metallic behavior requires that there is a quasi-continuum of states. In the lowest approximation, shown as dashed lines, states where an electron has moved have excess energies of U above the covalent states, t measures the strength of the exchange coupling between adjacent atoms and hence decreases exponentially with the spacing between atoms. When tsiU, the exchange coupling can overcome the Coulomb repulsion. See Refs. 229 and 334 for details of the computational method.

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