Self-interaction-free condition

To be fair, there are several well-known exact conditions that the LSD approximation does not get right it is not self-interaction free[37], v (r) does not have the correct — 1/r behavior at large r for finite systems[38], it does not contain the integer discontinuity[39—41], etc. These shortcomings may be overcome by other improvements[42], but not by the gradient corrections discussed in this article. 

This condition then ensures that the functionals are self-interaction free and that each o-spin electron excludes another o-spin electron from its immediate vicinity. From a computational perspective, when the nonsymmetric form of the one-matrix is used, then the nonlinear equations (Equation 1.115) are decoupled in this case, the equations can be solved using Newton s method in one dimension. But when the... 

The restriction j > i to the sum in eq 3.5 is imposed to avoid the untenable condition that each bead in the spring-bead chain interacts with itself. However, this bead self-interaction creeps into eq 3.20 as the integrand corresponding to s = s. Thus eq 3.20 is physically unacceptable, and must be modified so as to be free from self-interaction. Edwards [12] then proposed replacing eq 3.20 by... 

As mentioned above, the PCM is based on representing the electric polarization of the dielectric medium surrounding the solute by a polarization charge density at the solute/solvent boundary. This solvent polarization charge polarizes the solute, and the solute and solvent polarizations are obtained self-consistently by numerical solution of the Poisson equation with boundary conditions on the solute-solvent interface. The free energy of solvation is obtained from the interaction between the polarized solute charge distribution and the self-... 

First, there are constraints among the material quantities, which are rather straightforward for ideal quantum fluids. In case of interacting matter one can recall the self-consistent description of the interacting fermions in the 4 dimensional analogy [13], Second, the particle number density in the center, n(r = 0), is a free initial condition, as it was in the 4 dimensional case. Instead of density, we may use energy density, e(r = 0) = 0 as well. 

In the pH range close to the protein s lEP an interesting phenomenon of non-uniform redistribution of protein molecules among polysaccharide chains occurs (Tolstoguzov et al. 1985). The reason is that in the vicinity of the protein lEP the hydrophobic protein-protein and electrostatic protein-polysaccharide interactions can be energetically comparable with each other. Protein-protein association on the anionic polysaccharide matrix (or self-association of proteins), which is mainly due to hydrophobic interactions, is usually enhanced when the pH approaches the protein lEP. Accordingly, under conditions of a relatively weak protein-polysaccharide interaction, each free site situated near the site on the polysaccharide chain already occupied by a protein molecule becomes thermodynamically preferable for further binding of protein molecules. This leads to cooperative protein adsorption on an anionic polysaccharide. Some parts of polysaccharide chains tend to be completely covered by protein molecules (as in a virus) while other parts are completely free of protein. 

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