The electron pair distribution function

We have to introduce several useful concepts such as the electron pair distribution function , and the electron hole (in a more formal way than in Chapter 10, p. 515), etc. 

From the -electron wave function we may compute what is called the electron pair correlation function II(ri, r2), in short, a pair function defined as  

The function r2) measures the probability density of finding one electron at the point indicated by r and another at t2, and tells us how the motions of two electrons are correlated. If II were a product of two functions pi(ri) 0 and P2( 2) 0, then this motion is not correlated (because the probability of two events represents a product of the probabilities for each of the events only for independent, i.e. uncorrelated events). 

Function II appears in a natural way, when we compute the mean value of the total electronic repulsion with the Coulomb operator U =  

We will need this result in a moment. We see, that to determine the contribution of the eleetron repulsions to the total energy we need the two-electron function 11. The first Hohenberg-Kohn theorem tells us that it is sufficient to know something simpler, namely, the electronic density p. How to reconcile these two demands  

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In each of these choices, there is a lot of ambiguity. This, however, is restricted by some physical requirements. The requirements are related to the electron pair distribution function n(ri,r2) = N(N — 1) Q,. / pd -3dr4. .. Arpi, which takes into account that the two electrons, shown by r and 1-2, avoid... 

The requirements are related to the electron pair distribution function... 

Analysis of the radial pair distribution function for the electron centroid and solvent center-of-mass computed at different densities reveals some very interesting features. At high densities, the essentially localized electron is surrounded by the solvent resembling the solvation of a classical anion such as Cr or Br. At low densities, however, the electron is sufficiently extended (delocalized) such that its wavefunction tunnels through several neighboring water or ammonia molecules (Figure 16-9). 

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. ... 

The distribution function Diy) for an atom may thus be determined from electron-diffraction data. This was done for argon by Bartell and Brockway in 1953. Bartell and Gavin i pointed out that the inelastic part of the scattering should be sensitive to electron correlation [cf. equation (56)]. Thus it is possible to obtain the electron-pair correlation function, P r), experimentally in favourable cases. The theory was first applied to two-electron systems later the same authors estimated the effect of electron correlation on the total intensity scattered by beryllium. ... 

Figure 23 Predicted structure of hydrogen plasma by HNC-TFW method in comparison with ab initio calculations (A) proton-electron pair distribution function at electron density rj = 1 and temperature T = 10,000 K (B) the proton-proton pair distribution function at electron density Tj = 1 and temperature T = 3000 K. After Hong (2002).

Yet, the correspondences as central atom = interstitial ligands = coordination polyhedron generates a functional paradigm which qualitatively rationalizes the chemical bond, emphasizing on the electronic pairs distributed in geometric-symmetrical way. 

The reduction of the HF pair-distribution function around r = 0 may be viewed as a local exchange hole in the electron density of parallel spins moving together with the electron under consideration. The composite particle bare electron plus exchange hole represents the simplest case of a QP. However, the neglect of Coulomb correlation in the HF approximation of the HEG, together with the... 

Instead of formulating the wave function for a crystal as a sum of functions describing various ways of distributing the electron-pair bonds among the interatomic positions, as was done in the first section of this paper, let us formulate it in terms of two-electron functions describing a single resonating valence bond. A bond between two adjacent atoms ai and cq- may be described by a function < i3-(l, 2) in which 1 and 2 represent two electrons and the function i may have the simple Heitler-London form... 

In this equation g(r) is the equilibrium radial distribution function for a pair of reactants (14), g(r)4irr2dr is the probability that the centers of the pair of reactants are separated by a distance between r and r + dr, and (r) is the (first-order) rate constant for electron transfer at the separation distance r. Intramolecular electron transfer reactions involving "floppy" bridging groups can, of course, also occur over a range of separation distances in this case a different normalizing factor is used. 

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

One of the desirable features of compact wavefunctions is the ability to use them to examine additional features of the electron distribution without the necessity of repeating extensive computations to recreate complicated wavefunctions. We illustrate this point, and also exhibit the similarity of our wavefunctions with those of the 66-configuration study of Thakkar and Smith [15] by looking at the pair distribution functions. It is most instructive to present these as Z-scaled quantities Figure 3 contains the electron-nuclear distributions D r ) and p(ri) for clarity we only plot data for H, He, Li, and Ne. Even after Z scaling, a small but systematic narrowing of the distributions with increasing Z is still in process at Z-10. 

First let us recall the more important electron distribution functions and their origin in terms of corresponding density matrices. The electron probability density ( number density in electrons/unit volume) is the best known distribution function others refer to a pair of electrons, or a cluster of n electrons, simultaneously at given points in space. 

Due to the rapid decrease in the process probability with increase of the distance between the reagents, it should be expected that reaction (13) will result in electron transfer primarily to the particle A which is nearest to the excited donor particle D. In this case, the condition n < N is satisfied for reaction (13), where n is the concentration of the particles D and N is that of the particles A, and with the random initial distribution of the particles, A, the distribution function over the distances in the pairs D A formed, will have the same form [see Chap. 4, eqn. (13)] as with the non-paired random distribution under the conditions when n IV. In such a situation the kinetics of backward recombination of the particles in the pairs D A [reaction (12)] will be described by eqn. (24) of Chap.4 which coincides with eqn. (35) of Chap. 4 for electron tunneling reactions under a non-paired random distribution of the acceptor particles. Therefore, in the case of the pairwise recombination via electron tunneling considered here, the same methods of determining the parameters ve and ae can be applied as those described in the previous section for the case of the non-pair distribution. However, examples of the reliable determination of the parameters ve and ae for the case of the pairwise recombination using this method are still unknown to us. 

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