Direct correlation function definition

The definition of the terms can be found in Refs. 91, 92. The excess chemical potential, computed from the direct correlation function of the... 

The /i-bond diagrams of (f>(r 2) cannot contain any articulation pairs, and so the diagrams connecting the root point 2 with the n h-bonds connected to root point 1 do not have nodal points. This definition is just of the (n + l)-particle direct correlation function c(r2, r2,..., r +2) and one has... 

To find the long-wavelength limit of A(k) one has to recall the definitions (37), (38), and (48), as well as the properties of direct correlation functions Ca f(k) when k —> 0. For the mixture of charge particles the long-range contributions to the pair potentials are entirely determined by the Coulomb interactions [2], Hence, when k is small, one has,... 

One of the important methods of developing a model for the potential energy of the liquid system is that based on integral equations. These include the Kirkwood integral equation, the Born-Green-Yvon equation, and the Omstein-Zernike equation. The last approach leads to the definition of the direct correlation function c(fi2). It is the approach which is most frequently used and is the one which is considered here. 

The SSOZ equation is an exact integral equation relating the two site-site correlation functions h y(r) and c y(r). Strictly speaking, it is simply a definition of the site-site direct correlation function c,y(r). This fact is made clearest in the derivation by H iye and Stell. Here they consider the Ornstein-Zernike equation for an equimolar mixture of two species a and y. For each atom... 

Attempts to develop a more satisfactory theory within the interaction-site picture have developed along two different lines. The first relies on treating the RISM-OZ relation as providing the definition of the site-site direct correlation functions, c p. In this way the usual RISM closures can be modified by adding terms to c p such that the asymptotic result is satisfied. 

When used in connection with integral equations, it is considered the definition for the direct correlation function Cij. It can be shown by cluster expansion that the direct correlation function has the following closed analytical expression ... 

The term (r ) represents the direct part and the integral term represents the influence of the particle at r on that at r as mediated by a third particle. Equation (41) is called the Ornstein-Zernike equation and is really a definition of the direct correlation function c(r). Note that the Fourier transform of the Ornstein-Zernike equation gives... 

Underpotential deposition of heavy metals on H2 evolving electrodes is a well known problem [133], The existence of a direct correlation between H2 evolution activity and metal work function, makes UPD very likely on high work function electrodes like Pt or Ni. Cathode poisoning for H2 evolution is aggravated by UPD for two reasons. First, deposition potentials of UPD metals are shifted to more anodic values (by definition), and second, UPD favors a monolayer by monolayer growth causing a complete coverage of the cathode [100]. Thus H2 evolution may be poisoned by one monolayer of cadmium for example, the reversible bulk deposition potential of which is cathodic to the H2 evolution potential. 

It follows from the definition of the functionals and Ti that the exchange-correlation functional is invariant under the rotation and translation of the electron density. This follows directly from the fact that the kinetic energy operator f and the two-electron operator W are invariant with respect to these transformations. This also has implications for the exchange-correlation functional. 

Capacity. Capacity is a measure of the quantity of ions, acid, or base removed (adsorbed) by an ion-exchange material. The quantity removed is directly correlated with the number of functional groups. Capacity is reported in several different ways, but requires further definition because the word by itself does not cover all situations. Total capacity is a measure of all the functional groups on a resin and is recorded on a weight as well as a volume basis. 

Atopic dermatitis is the most common itchy dermatosis, with well-documented alteration in the stratum comeum function. Numerous studies have revealed an increase in the basal transepidermal water loss (TEWL) in the stratum corneum of patients with this condition. Of note, this increase in TEWL was also described in the clinically unaffected skin of atopies.2 There have been direct correlations shown between the degree of inflammation and severity of barrier impairment in atopic dermatitis. Despite these findings, to date there have been no definitive reports correlating degree of barrier function with itch variability. 

Mathematical expressions for the functionals which are found in the Kohn-Sham operator are usually derived either from the model of a uniform electron gas or from a fitting procedure to calculated electron densities of noble gas atoms. Two different functionals are then derived. One is the exchange functional Fx and the other the correlation functional Fc, which are related to the exchange and correlation energies in ab initio theory. We point out, however, that the definition of the two terms in DFT is slightly different from ab initio theory, which means that the corresponding energies cannot be directly compared between the two methods. 

Information can be extracted directly from the PDF in a model-independent way because of its definition as the atom-pair correlation function described in Equation (13). 

The first relation, (a) => (b), follows directly from the formal definition of the pair correlation function, assuming pairwise additivity of the total potential energy. The second relation, (b) => (c), follows from the definition of Gap, and the third relation (c) => (d), is obvious. 

The appearance of Avogadro s number in this expression results from the definition (7.2.57) of the constant K. This can be justified as follows. In ordinary space, the correlation function g(r) — 1 differs practically from zero only in a range of the same order as the size R of the molecules, let us say for instance 10 nm. Therefore, the Fourier transform //n(0) is a volume of the order of 103 nm3, but the product //n(0) is a macroscopic volume ( 104 cm3) which can be more directly measured, for instance, by looking at the osmotic pressure. 

Certain features of the self-correlation triangle centered at the origin are of interest and can be interpreted directly in terms of the structural parameters. In view of the definition of the correlation function, Equation (5.159), we find the following ... 

Water residence times on the protein surface are not directly measurable experimentally, bnt can be defined as the relaxation time of time correlation functions of the popnlation of the hydration shell [5], Due to differences in the definition from one investigation to another, the values reported in the literature exhibit considerable variability. Nonetheless, heterogeneity in water dynamics near the protein surface is clearly manifested in distributions of water residence times. The distribution we have constrncted for the N state of HocLA in solution from the residence time of water next to each residue is plotted up to lOOps in Figure 16.1d. The distribution is very broad, ranging from 2.6 to 241 ps, but highly skewed toward shorter residence times. The mean residence time of 23 ps is about 2.5 times longer than the rotational correlation time for hydration water. 

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