Pair probability function

This result for P2(ri,ra) can only be true in the limit that the motion of electron 1 and hence its probability of being at a certain point in space is totally independent of the motions of electron 2, i.e. there are no forces acting between the two electrons and their motions are totally uncorrelated. The best that can be done in a model which employs an expression of the form of equation (19) for the pair probability function is to determine the m (which in turn determine the one-particle probability function) in such a way that the electron it describes experiences the average field of the other electrons, a situation which is attained in the Hartree-Pock limit of the orbital model. 

The Hartree-Fock wavefunction is not a simple product function, but an antisymmetrized sum of such products. As pointed out above, the effect of the anti-symmetrization is to selectively remove, from the region of space about a given electron, charge density of electrons with the same spin. Thus, the pair probability function for electrons with the same spin is not of the form given in equation (19) but instead is... 

The oft-quoted result of antisynunetrizing the wave function with respect to the permutation of the space-spin coordinates of every pair of electrons is that no two electrons with the same spin can occupy the same point in space. In chemistry, however, one is most interested in the spatial distribution of the electrons. To determine the manner in which the exclusion principle affects the electron distribution and its properties in real space, one must determine how many pairs of electrons, on the average, contribute to the electron density over the region of interest. This information is given by the electron pair density, alternatively called the pair probability function. 

The description of the atomic distribution in noncrystalline materials employs a distribution function, (r), which corresponds to the probability of finding another atom at a distance r from the origin atom taken as the point r = 0. In a system having an average number density p = N/V, the probability of finding another atom at a distance r from an origin atom corresponds to Pq ( ). Whereas the information given by (r), which is called the pair distribution function, is only one-dimensional, it is quantitative information on the noncrystalline systems and as such is one of the most important pieces of information in the study of noncrystalline materials. The interatomic distances cannot be smaller than the atomic core diameters, so = 0. 

Although we work with the pair distribution functions, what we are to solve are essentially the point probability functions fj(rj), i=A and B. 

Great simplification is achieved by introducing the hypothesis of independent reaction times (IRT) that the pairwise reaction times evolve independendy of any other reactions. While the fundamental justification of IRT may not be immediately obvious, one notices its similarity with the molecular pair model of homogeneous diffusion-mediated reactions (Noyes, 1961 Green, 1984). The usefulness of the IRT model depends on the availability of a suitable reaction probability function W(r, a t). For a pair of neutral particles undergoing fully diffusion-con-trolled reactions, Wis given by (a/r) erfc[(r - a)/2(D t)1/2] where If is the mutual diffusion coefficient and erfc is the complement of the error function. 

A quantity of central importance in the study of uniform liquids is the pair correlation function, g r), which is the probability (relative to an ideal gas) of finding a particle at position r given that there is a particle at the origin. All other structural and thermodynamic properties can be obtained from a knowledge of g r). The calculation of g r) for various fluids is one of the long-standing problems in liquid state theory, and several accurate approaches exist. These theories can also be used to obtain the density profile of a fluid at a surface. 

Much more detailed information about the microscopic structure of water at interfaces is provided by the pair correlation function which gives the joint probability of finding an atom of type/r at a position ri, and an atom of type v at a position T2, relative to the probability one would expect from a uniform (ideal gas) distribution. In a bulk homogeneous liquid, gfn, is a function of the radial distance ri2 = Iri - T2I only, but at the interface one must also specify the location zi, zj of the two atoms relative to the surface. We expect the water pair correlation function to give us information about the water structure near the metal, as influenced both by the interaction potential and the surface corrugation, and to reduce to the bulk correlation Inunction when both zi and Z2 are far enough from the surface. 

For visualization purposes we have made plots of pair distribution functions, defining the electron-nuclear radial probability distribution function D(ri) by the formula... 

Compare this with Eq. (1.1.15). In general, the probabilities P k) cannot be factorized into a product of probabilities each pertaining to a single site [as in Eq. (1.1.15)]. We define the pair correlation function by... 

Figure 7. Water oxygen-exocyclic methylene carbon pair distribution function, calculated from a molecular dynamics simulation of a-D-glucopyranose in aqueous solution, giving the normalized probability of finding a water oxygen atom a given distance r from the C6 carbon atom. (Reproduced from Ref. 32. Copyright 1989 American Chemical Society.)...

This function is the integrally normalized probability for each water molecule being oriented such that it makes an angle B between its OH bond vectors and the vector from the water oxygen to the carbon atom. This function is calculated for those molecules within 4.9 A of the carbon atom (nearest neighbors), as this distance marks the first minimum in the pair distribution function for that atom. The curve in Figure 10 is typical for hydrophobic hydration (22). 

The importance of N-representability for pair-density functional theory was not fully appreciated probably because most research on pair-density theories has been performed by people from the density functional theory community, and there is no W-representability problem in conventional density functional theory. Perhaps this also explains why most work on the pair density has been performed in the first-quantized spatial representation (p2(xi,X2) = r2(xi,X2 xi,X2)) instead of the second-quantized orbital representation... 

Probability box A kind of uncertain number representing both incertitude and variability. A probability box can be specified by a pair of functions serving as bounds about an imprecisely known cumulative distribution function. The probability box is identified with the class of distribution functions that would be consistent with (i.e., bounded by) these distributions. 

Thus, simple deproteinization of plasma with trichloroacetic acid, perchloric acid, phosphoric acid, or acetonitrile, followed by centrifugation and direct injection of the supernatants, yielded low recoveries of malachite green and leuco-malachite green, probably due to insufficient debinding of the analytes (495). Acidification or alkalinization of plasma and subsequent extraction with ethyl acetate also resulted in poor recoveries. In contrast, protein denaturation with a mixture of either acetonitrile or methanol and citric acid could substantially improve Ute recovery of the analytes, possibly due to the pairing-ion function of Ure citrate ions. 

The probability that two atoms 1,2 are separated by the distance R is given by the pair distribution function, g(R), defined as... 

Exercise. If an event has been observed at ta the probability density for some other event (not necessarily the next one) to occur at th is/2(ta, tb)//i(0- One defines the pair distribution function by... 

The classical theory makes especially clear the inherent ambiguity of data analysis with the optical model, and this ambiguity carries over into the quantum model. If we wish to use experimental differential cross sections to gain information about V0(r) and P(b) or T(r), we must assume a reasonable parametric form for V0(r) that determines the shape of the cross section in the absence of reaction. The value P(b) is then determined [or T(r) chosen] by what is essentially an extrapolation of this parametric form. In the classical picture a V0(r) with a less steep repulsive wall yields a lower reaction probability from the same experimental cross-section data. The pair of functions V0 r), P b) or VQ(r), T(r) is thus underdetermined. The ambiguity may be relieved somewhat (to what extent is not yet known) by fitting several sets of data at different collision energies and, especially, by fitting other types of data such as total elastic and/or reactive cross sections simultaneously. 

The probability function /(t) gives the unity probability when the pair is within the interval —a/2integrating Equation 6.135 between the limits x = al2 with t 0. The diffusion coefficient D is the sum of the individual diffusion coefficients 1), knTI(mr)Vj (Einstein equation), where rt is the radius of the component i of the radical-ion pair. Integration of Equation 6.135 between the limits — °° < x < all and all < x < oo gives the probability of a diffusively separated pair (Equation 6.136). 

For an amorphous mass of particles there is no correlation between particles that are far apart. The joint probability of finding particles at r and ri is simply the product of the individual probabilities. Let us define a configurational pair-correlation function g(ri, r2) as... 

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. 

In a solution containing n atomic species, p, q, etc., the number of different pair interactions is n(n + l)/2. The pair correlation function, gm(r), measures the probability of finding an atom q at a distance r... 

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