Function topologic

S. Noury, X. Krokidis, F. Fuster, and B. Silvi, Computational tools for the electron localization function topological analysis, Comput. Chem. 23, 597-604 (1999). 

Rangan, V. S., Joshi, A. K. and Smith, S., Mapping the functional topology of the animal fatty acid synthase by mutant complementation in vitro. Biochemistry 40 (2001) 10792-10799. 

Polo V, Andres J, Berskit S, Domingo LR, Silvi B (2008) Understanding reaction mechanisms in organic chemistry from catastrophe theory applied to the electron localization function topology. J Phys Chem A 112 7128... 

Here the transformation of the function topology in a reliability topology is discussed. To describe all components with their reUability parameters (figure 14),... 

This step is dedicated to the extraction of various flaw parameters (topological, geometrical and functional), such as texture, size or shape, which ate essential for the pattern recognition module. 

As mentioned in the introduction, the simplest way of approximately accounting for the geomehic or topological effects of a conical intersection incorporates a phase factor in the nuclear wave function. In this section, we shall consider some specific situations where this approach is used and furthermore give the vector potential that can be derived from the phase factor. 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, dynamic phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schrddinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)... 

It is known that multivalued adiabatic electronic manifolds create topological effects [23,25,45]. Since the newly introduced D matrix contains the information relevant for this manifold (the number of functions that flip sign and their identification) we shall define it as the Topological Matrix. Accordingly, K will be defined as the Topological Number. Since D is dependent on the contour F the same applies to K thus K = f(F),... 

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