Cavity distribution functions

The most difficult and least satisfactory part of the theories is the calculation of the cavity distribution functions for the hard sphere interaction site fluids. If the site-site formalism is being used, then the only suitable... 

The former is obtained through a relation with the cavity pair distribution function, yAA Hs- between two non-polar solutes, assuming hard-sphere interaction among them and also between solute and water molecules. We discuss the evaluation of cavity distribution functions in Appendix 15.A. 

Now the problem reduces to the description of repulsive spherical particles in liquid water. We can get closer to the solution by recognizing that the problem is closely related to the statistical mechanics of hard spheres dissolved in water. One can now use the cavity distribution function defined by... 

We note that the characterization of the degree of reaction in terms of the cavity distribution function is also one of the results of Wertheim s theory of association. 

Likewise, the function, X2) may be interpreted as the relative probability of finding the two solvatons at Xi, X2. This function is sometimes referred to as a cavity-cavity distribution function. This term might be misleading for two reasons. First, our solvatons exert an external field of force which has both attractive and repulsive parts. It is only for hard-sphere solvatons that the external field is equivalent to a cavity. Second, even for hard-sphere particles that are in some sense equivalent to suitable cavities, one must be careful when reference is made to the probability of finding a single or a pair of cavities. The reason for that has been discussed in section 5.10. 

Fig. 4. Radial distribution functions between the centre of a test cavity and the (jxygen atom of the surrounding water. The curves correspond to the different barrier heights for the softcore interaction illustrated in Fig. 3...

The probability of cavity formation in bulk water, able to accommodate a solute molecule, by exclusion of a given number of solvent molecules, was inferred from easily available information about the solvent, such as the density of bulk water and the oxygen-oxygen radial distribution function [65,79]. 

If the distribution function of electrons in the cavity f(e,t) were not allowed to fluctuate, the contacts would be independent generators of current noise whose zero-frequency energy-resolved cumulants ((/ R))e could be obtained from a quantum-mechanical formula... 

If the voltage is high enough, the noise of isolated contacts can be considered as white at frequencies at which the distribution function / fluctuates. This allows us to consider the contacts as independent generators of white noise, whose intensity is determined by the instantaneous distribution function of electrons in the cavity. Based on this time-scale separation, we perform a recursive expansion of higher cumulants of current in terms of its lower cumulants. In the low-frequency limit, the expressions for the third and fourth cumulants coincide with those obtained by quantum-mechanical methods for arbitrary ratio of conductances Gl/Gr and transparencies Pl,r [9]. Very recently, the same recursive relations were obtained as a saddle-point expansion of a stochastic path integral [10]. 

In this study [15,29], we have carried out the 3D-RISM calculation for a hen egg-white lysozyme immersed in water and obtained the 3D-distribution function of oxygen and hydrogen of water molecules around and inside the protein. The native 3D structure of the protein is taken from the protein data bank (PDB). The protein is known to have a cavity composed of the residues from Y53 to 158 and from A82 to S91, in which four water molecules have been determined by means of the X-ray diffraction measurement [30]. In our calculation, those water molecules are not included explicitly. 

It should be noted that one peak of the 3D-distribution function does not necessarily correspond to one molecule. If a water molecule transfers back and forth between two sites in the equilibrium state, two peaks correspondingly appear in the 3D-distribution function. In fact, the number of water molecules within the cavity calculated from the 3D-distribution function is 3.6. It is less than the number of water-binding sites and includes decimal fractions. To explain that, we carried out molecular dynamics (MD) simulation using the same parameters and under the same thermodynamic conditions as... 

Fig. 10.9. Isosurface representation of the 3D distribution function of water oxygen around the low-pressure (3 MPa) and high-pressure (300 MPa) structures of ubiqui-tin. The dark gray surfaces show the area where the distribution function is larger than 2. This is a top-view representation, in which the upper parts (the front parts in the figure) are clipped to bring the internal cavity (marked by dashed circle) into...

In Ftirth s theory of cavities in liquids, there is a distribution function for the probability of the hole size. It is... 

It is also possible to prepare crystalline electrides in which a trapped electron acts in effect as the anion. The bnUc of the excess electron density in electrides resides in the X-ray empty cavities and in the intercoimecting chaimels. Stmctures of electri-dides [Li(2,l,l-crypt)]+ e [K(2,2,2-crypt)]+ e , [Rb(2,2,2-crypt)]+ e, [Cs(18-crown-6)2]+ e, [Cs(15-crown-5)2]" e and mixed-sandwich electride [Cs(18-crown-6)(15-crown-5)+e ]6 18-crown-6 are known. Silica-zeolites with pore diameters of vA have been used to prepare silica-based electrides. The potassium species contains weakly bound electron pairs which appear to be delocalized, whereas the cesium species have optical and magnetic properties indicative of electron locahzation in cavities with little interaction between the electrons or between them and the cation. The structural model of the stable cesium electride synthesized by intercalating cesium in zeohte ITQ-4 has been coirfirmed by the atomic pair distribution function (PDF) analysis. The synthetic methods, structures, spectroscopic properties, and magnetic behavior of some electrides have been reviewed. Theoretical study on structural and electronic properties of inorganic electrides has also been addressed recently. ... 

One of the typical minimized clusters 1 (methane) 10 (waters) is presented in Figure la,b. They show that the methane molecule is enclosed in a cavity formed by water molecules. The two spheres centered on a methane molecule, with radii of 3.6 and 5.35 A, correspond to the first maximum and the first minimum in the radial distribution function goo = goo(roc) in dilute mixtures of methane in water. It is worth noting that... 

We perform concrete calculations in the complex P-representation [Drummond 1980 McNeil 1983] in the frame of both probability distribution functions and stochastic equations for the complex c-number variables. We follow the standard procedures of quantum optics to eliminate the reservoir operators and to obtain a master equation for the density operator of the modes. The master equation is then transformed into a Fokker-Planck equation for the P-quasiprobability distribution function. In particular, for an ordinary NOPO and in the case of high cavity losses for the pump mode (73 7), if in the operational regime the pump depletion effects are involved, this approach yields... 

The fundamental distribution function in the SPT is P0(r), the probability that no molecule has its center within the spherical region of radius r centered at some fixed point Rq in the fluid. Let P0(r+ dr) be the probability that a cavity of radius (r + dr) is empty. (In all the following, a cavity is always assumed to be centered at some fixed point Rq, even when this is not mentioned explicitly.) This probability may be written as a product of two factors... 

In a first step, the simulated ion distribution functions shall be compared with the PB prediction as well as with an extended PB theory using the Debye-Hiickel hole-cavity correction function from Ref. 6. Figure 22 shows the corresponding distribution functions for the systems from Table 2. Qualitatively the PB prediction is already a good description of those systems, if one graciously disregards the inevitable problems at the cell... 

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