Convergence angle, optimizing

From this observation we infer self-similarity on a cosmic scale, from atoms to galaxies, which implies the same numerical basis for the atomic models of Nagaoka and Bohr, assumed to be self-similar with the rings of Saturn and the planets, respectively. In principle, the periodic accumulation of extranuclear electron density on an atom could also be optimized by specifying an appropriate convergence angle. 

The same periodic function results from optimization on a golden spiral with a variable convergence angle of Art In — 1), which describes a spherical standing wave with nodes at n. Analysis of the wave structure shows that it correctly models the atomic electron distribution for all elements as a function of the golden ratio and the Bohr radius, uq. Normalization of the wave structure into uniform spherical units simulates atomic activation, readily interpreted as the basis of electronegativity and chemical affinity. 

Fig. 2 Atomic shell structure as it emerges from electron-density optimization on a golden spiral. The variable convergence angle of An/ In — 1) manifests in the appearance of 2n — 1 additional cycles s, p, d, /) in each interval between Bohr levels n and n — I, shown here as elementary ripples. In contrast to the Bohr-Schrodinger (BS) model, closed shells in the Ford-circle simulation (FC) invariably coincide with noble-gas configurations...

The optimization approach mentioned above is applied for the optimization of a fish-tail die shown in Fig.3. The die structure control parameters include the convergent angle and the divergent angle which can be seen fi-om Fig.4. By means of the mapped mesh generation... 

Fig. 16. Rotation about the CH-2-C-2-0-2-C(=0) torsion angle of 17 BO. Note the two minima at syn BO and the ring-closed dioxolenium ion CO conformations. Also note the two TSs separating BO from CO. Rotation was made starting at the syn BO minimum and moving in the direction of TS1. The small discontinuity near 150° is a result of this rotation. Missing points failed to converge. The 60th point did, however, return to almost exactly the starting conformation. The structures shown are for the associated stationary points, and the reported energies which differ from the graph reflect the optimization without the bond-rotation constraints.

The two optimal angles 0r and 0 converge to one single angle as the accuracy of the calculation is increased. 

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