Theory separation step

In order to connect this variational principle to density functional theory we perform the search defined in equation (4-13) in two separate steps first, we search over the subset of all the infinitely many antisymmetric wave functions Px that upon quadrature yield a particular density px (under the constraint that the density integrates to the correct number of electrons). The result of this search is the wave function vFxin that yields the lowest... 

Various instruments of theoretical chemistry have been widely to describe separate steps of solvent extraction of metal ions. Because of the complexity of solvent extraction systems, there is still no unified theory and no successful approach aimed at merging the extraction steps. It has already been pointed out that the challenging problem for theoreticians dealing with solvent extraction of metals, in particular with thermodynamic calculations, is to evaluate correctly solvent effects by the use of the most accurate explicit solvation models and QM calculations. However, such calculations on extremely large sets consisting of hundreds or even thousands of molecules, necessary to model all aspects of the extraction systems, are still impossible due to both hardware and software limitations. 

A very important point is that crystallization processes are highly selective and can lead to a 100% pure product within one separation step—at least in theory—if the mixture to be separated... 

At low pH values, when additional protons are present, the separation step becomes reversible and one observes homogeneous proton recombination. The reaction under these conditions undergoes a transition from unimolecular (correlated pairs) to a bimolecular (or pseudo-unimolecular) reaction. The rate of this recombination reaction is expected to diminish with increasing concentration of inert salt, which screens the Coulombic attraction between the proton and the anion. In fact, the classical Bronsted-Bjerrum theory of salt effects puts all of the effect in the recombination reaction while predicting zero salt effect on the dissociation direction [7]. 

In order to incorporate both features neglected by the original Flory-Huggins theory into the present approach, we have conceptually subdivided the dilution process into two separate steps as formulated in (18). Such a separation is permissible because the Gibbs energy of dilution represents a function of state. 

Separations in hydrophobic interaction chromatography have been modeled as a function of the ionic strength of the buffer and of the hydrophobicity of the column, and tested using the elution of lysozyme and ovalbumin from octyl-, butyl- and phenyl-Sepharose phases.2 The theoretical framework used preferential interaction analysis, a theory competitive to solvophobic theory. Solvophobic theory views protein-surface interaction as a two-step process. In this model, the protein appears in a cavity in the water formed above the adsorption site and then adsorbs to the phase, with the free energy change... 

To achieve the desired separation of the reactive degree of freedom from the bath modes, we use time-dependent normal form theory [40,99]. As a first step, the phase space is extended through the addition of two auxiliary variables a canonical coordinate x, which takes the same value as time t, and its conjugate momentum PT. The dynamics on the extended phase space is described by the Hamiltonian... 

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