Pair distribution function specific

For the case of a pure monatomic liquid, in the limit that there are only pair interactions, the pair distribution function provides a complete microscopic specification from which all thermodynamic properties can be calculated 2>. If there are (excess) three molecule interactions, then one must also know the triplet distribution function to complete the microscopic description the extension to still higher order (excess) interactions is obvious. 

The W functions should be calculated on the basis of quantum mechanics. However, at sufficiently high temperatures and for massive systems, classical or semi-classical expressions in terms of interatomic potential functions, F12CR), etc., are useful. Specifically, in the classical limit, we may write the pair distribution function as... 

The pair distribution function g(0) is similar to Eq. 5.36, except that we now have to consider the possibility of like pairs. Specifically, we write... 

In order to evaluate the collisional integrals Pc, qc, and y, explicitly, it is important to know the specific form of the pair distribution function /(2)(vi, ri V2, ry. t). The pair distribution function /(2) may be related to the single-particle velocity distribution function f by introducing a configurational pair-correlation function g(ri, r2). In the following, we first introduce the distribution functions and then derive the expression of /(2) in terms of f by assuming /(1) is Maxwellian and particles are nearly elastic (i.e., 1 — e 1). 

By use of a single isotopic substitution of ion (I = M or X), a first-order difference function Gi(r) can be obtained by direct differencing of all G(r) that contain all ten pair-distribution functions of the solution. Giir) is then a linear combination of only the four radial distribution functions specific to I and can be written as in Eq. (2) (74). 

To define a phase in a cluster is not a straightforward task [4]. We speak about phase-hke forms, with specific pair-distribution functions [5], that help to distinguish between different thermodynamic states in small systems. In what follows we use terms phase change and phase transformations for small systems, preserving the term phase transitions for bulk matter. 

As in the case of the singlet MDF, here we also start with the specific pair distribution function defined in (2.28). To get the generic pair distribution function, consider the list of events and their corresponding probabilities in table 2.2. Note that the probabilities of all the events on the left-hand column of table 2.2 are equal. 

In the foregoing section the distribution function in the complete phase space, /(x, t), was discussed and the general equation of change presented. In the sections to follow we often present results in terms of averages with respect to lower-order distribution functions, specifically averages involving the phase space or the configuration space of one molecule or a pair of molecules. This section is devoted to the definitions of these contracted distributions functions. 

In this section we discuss five assumptions that have traditionally been introduced in order to continue with the development of the kinetic theory of transport phenomena and get useful results. Some of these assumptions are made because we do not at present know enough about the distribution functions that appear in the expressions in Tables 1 and 2, in particular the pair distribution function and the momentum-space distnbution function. Other assumptions are introduced in order to simplify the subsequent problem-solving for specific molecular models. All the assumptions we present here can be challenged some of them should be modified, and some of them may ultimately be eliminated. 

For q values lower than 0.04 A , the scattering curves showed a marked increase in intensity, especially for the systems at n = 0.5. This effect was also observed for other Fe systems (e.g. chloride, nitrate, phosphate) at n = 1.0 and n = 1.5 [48,58,70], which suggests that this feature is not specific to a particular Fe system but can be generally related to a low hydrolysis ratio (n < 1.5) [83]. Nonetheless, neither the exact cause(s) nor the influence of the actual structural features on the increased intensity has been elucidated. For this reason, the low q portion of the scattering curves is generally not considered in the computation ofthe pair distribution functions P r) (Figure 5.7). At n = 0.5, the first peak of the P r) function, which corresponds to the radius of the subunit, is located at almost the same value, i.e. r = 6 A for both Si/Fe ratios. The presence of 6 A clusters can only be explained by formation of Fe monomer-Si04 complexes, whose occurrence as dissolved species can be predicted... 

The starting point is the basic probability density P(X ), which was defined in Chapter 1, Eq. (1.52), for the T, F, ensemble (a similar procedure can be employed for introducing the molecular distribution function in other ensembles). The specific pair distribution function is defined as the probability density of finding molecule 1 at X and molecule 2 at X". This... 

Here we are interested in the following question What is the joint probability of finding a specific unit i in state a, and a second specific unit J (j i) in state Again, we first derive the general expression for the pair distribution function for the Ising model with any number of states and then we shall particularize to the case of the two-state units. 

Since all of these terms are equivalent, we simply choose one specific term and multiply it by M. This gives the expression which is identical to the definition of the pair distribution function, Eq. (4.3.36). 

Intramolecular correlations are handled in different approximate manners in the various BGY approaches. Taylor and Lipson treat pair correlations on the same chain as input to the theory in a manner similar to PRISM theory. In contrast, the formulations of Eu and Gan, " and also Attard, yield closed integral equations for both the intra- and intermolecular pair distribution functions. Thus, in a sense the intra- and intermolecular pair correlations are treated on an equal footing, and a self-consistent integral equation theory is naturally obtained. Eu and Gan have recently presented a comparison between their BGY approach and self-consistent and non-self-consistent PRISM theory, in both general conceptual terms and within the context of numerical predictions for specific model hard-core systems. For the jointed hardcore chain model studies, the theory of Eu and Gan appears quantitatively superior to PRISM predictions, particularly for the equation of state. ... 

Fig. 3.8. Pair distribution function of an individual chain in a semi-dilute solution, exhibiting different regions with specific power laws 47rr g (r/Rp) Rf denotes the Flory-radius in the dilute state. The dotted line gives the function 4nr (cm). The dashed line indicates the pair distribution function for all monomers, 47rr g, which deviates from 47rr g for r > s-...

Unsatisfied by the low specificity of most of the derived residue-pair potentials, especially those obtained considering distances between C or atoms, some authors use a somewhat more detailed description of the protein conformation, in which each residue is represented not by one, but by two, interactions sites one for the backbone (Ca) and one representing the side-chain atoms most involved in the relevant interactions. When this is combined with the use of the radial pair distribution functions, as described above, residue-pair potentials displaying more specific hydrophilic interactions are obtained. 

The pair distribution function gives the probability of finding a second particle in a given volume element at position r from a first particle. Specifically, in volume... 

An interesting question then arises as to why the dynamics of proton transfer for the benzophenone-i V, /V-dimethylaniline contact radical IP falls within the nonadiabatic regime while that for the napthol photoacids-carboxylic base pairs in water falls in the adiabatic regime given that both systems are intermolecular. For the benzophenone-A, A-dimethylaniline contact radical IP, the presumed structure of the complex is that of a 7t-stacked system that constrains the distance between the two heavy atoms involved in the proton transfer, C and O, to a distance of 3.3A (Scheme 2.10) [20]. Conversely, for the napthol photoacids-carboxylic base pairs no such constraints are imposed so that there can be close approach of the two heavy atoms. The distance associated with the crossover between nonadiabatic and adiabatic proton transfer has yet to be clearly defined and will be system specific. However, from model calculations, distances in excess of 2.5 A appear to lead to the realm of nonadiabatic proton transfer. Thus, a factor determining whether a bimolecular proton-transfer process falls within the adiabatic or nonadiabatic regimes lies in the rate expression Eq. (6) where 4>(R), the distribution function for molecular species with distance, and k(R), the rate constant as a function of distance, determine the mode of transfer. 

By employing a similar procedure as in Section 7.2, we can compute all the distribution functions and the corresponding correlation functions of this system. As an example we derive here the expression for the 1, / pair distribution, and the corresponding pair correlation. The procedure is the same as in Section 7.2, with the additional complexity that we now have two occupied states rather than one as in Section 7.2. [On the other hand, expressions (7.2.24) and (7.2.25) are more general in the sense that they apply to any state a and p. Here, we are interested in specific states, i.e., states such as site i is occupied, for calculating cooperativi-ties.] The probability of finding site 1 occupied and site I occupied is ... 

The solvation structure around a molecule is commonly described by a pair correlation function (PCF) or radial distribution function g(r). This function represents the probability of finding a specific particle (atom) at a distance r from the atom being studied. Figure 4.32 shows the PCF of oxygen-oxygen and hydrogen-oxygen in liquid water. 

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