Triplet distribution function

We define an "i-th nearest neighbour complex to be a pair of oppositely charged defects on lattice sites which are i-th nearest neighbours, such that neither of the defects has another defect of opposite charge at the i-th nearest neighbour distance, Rit or closer. This corresponds to what is called the unlike partners only definition. A different definition is that the defects be Rt apart and that neither of them has another defect of either charge at a distance less than or equal to R. This is the like and unlike partners definition. For ionic defects the difference is small at the lowest concentrations the definition to be used depends to some extent on the problem at hand. We shall consider only the first definition. It is required to find the concentration of such complexes in terms of the defect distribution functions. It should be clear that what is required is merely a particular case of the specialized distribution functions of Section IV-D and that the answer involves pair, triplet, and higher correlation functions. In fact this is not the procedure usually employed, as we shall now see. 

Expressions for the medium modifications of the cluster distribution functions can be derived in a quantum statistical approach to the few-body states, starting from a Hamiltonian describing the nucleon-nucleon interaction by the potential V"(12, l/2/) (1 denoting momentum, spin and isospin). We first discuss the two-particle correlations which have been considered extensively in the literature [5,7], Results for different quantities such as the spectral function, the deuteron binding energy and wave function as well as the two-nucleon scattering phase shifts in the isospin singlet and triplet channel have been evaluated for different temperatures and densities. The composition as well as the phase instability was calculated. 

The structure of a liquid is conventionally described by the set of distributions of relative separations of atom pairs, atom triplets, etc. The fundamental basis for X-ray and neutron diffraction studies of liquids is the observation that in the absence of multiple scattering the diffraction pattern is completely determined by the pair distribution function. 

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. 

For some time it has been known that the spectral moments, which are static properties of the absorption spectra, may be written as a virial expansion in powers of density, q", so that the nth virial coefficient describes the n-body contributions (n = 2, 3. ..) [400]. That dynamical properties like the spectral density, J co), may also be expanded in terms of powers of density has been tacitly assumed by a number of authors who have reported low-density absorption spectra as a sum of two components proportional to q2 and q3, respectively [100, 99, 140]. It has recently been shown by Moraldi (1990) that the spectral components proportional to q2 and q3 may indeed be related to the two- and three-body dynamical processes, provided a condition on time is satisfied [318, 297]. The proof resorts to an extension of the static pair and triplet distribution functions to describe the time evolution of the initial configurations these permit an expansion in terms of powers of density that is analogous to that of the static distribution functions [135],... 

At this point, the analogy with the static case is complete and one may take advantage of the results obtained for the static case. Interesting quantal expressions for the dynamic pair and triplet distribution functions have been communicated [297] the latter may be separated into a pairwise additive and an irreducible component, as in the static case. 

Note that the only approximation made in the derivation of Eq. (177) is the use of the Kirkwood superposition approximation for the triplet distribution function of the liquid [21]. In a dense liquid at low temperature (near its triple point), this is not a bad approximation [21],... 

Fig. 7.28. The experimental evidences of a bimodal character of the SC distribution function on the silica surface, (a) Excitation spectra of singlet-singlet (solid line) and triplet-singlet (dashed line) luminescence of SC in the initial silica sample and (1-3) their changes during the oxidation of the sample. Luminescence was detected at 300 K at the wavelength of 285 nm for the singlet-singlet transition and at 455 nm for the triplet-singlet transition (b) EPR spectra of products of addition of H atom to SS (only lines of (=28Si-0-)28Si -H is shown) (1) sample A (superposition of EPR spectra V and 1 ) (2) sample B and (3) sample C (see text for details).

In their work on the insertion of triplet oxygen to the hydrogen molecule, Bader and Gangi138 found it necessary to use the UHF method in order to investigate the changes of polarization of electron spins at various points on the potential energy surface. The spin density distribution function [Pg.43]

Together with an appropriate closure for the pair and triplet distribution functions, one may restrict consideration in this limit to the first two equations in the hierarchy (23). Again, in this approximation, one may follow a course of the Debye-Hiickel approach to obtain the mean field potential, while image forces are accounted for. In this way, the distribution of ions in the system will be known, and interaction forces can be calculated on the basis of this distribution. 

The corresponding temperature variation of y is obtained with use of Equation 17. Since this recursion formula will lead to number densities for sets of four molecules, we summarize only the analogous reduction. Again with use of the mechanical equilibrium condition, here for both pair and triplet distribution functions, vanishing residual terms are found. The condition, similar to that of Equation 22, is that for fixed Rx and R2, with R3 at infinity, (1,2,3) approaches p(2) (1,2)pW (3). 

FIGURE 21. Summary of electronic distribution in triplet aniline (a) Bond distances (A), NBO charges [bracket, in au] and Wiberg indices (parentheses, in au). (b) Topology of the electron density determined from atom-in-molecule calculations p(r) = electron density, L = Laplacian of the density defined as Z,(r) = —V2p(r) and = ellipticity of the bond critical point, (c) Laplacian map of the density, (d) Isosurfaces of the electron localization function, ELF = 0.87 the values are the populations of the valence basins, (e) Spin densities in the molecular (CCN) plane... 

The arguments leading to (3.14) are the same as those for (3.6). The new element which enters here is the triplet distribution function. Similarly, we can write formal relations for average quantities which depend on larger numbers of particles. The result would be integrals involving successively higher order molecular distribution functions. Unfortunately, even (3.14) is rarely useful since we do not have sufficient information on p(3). 

Calculation of the second-order term in Eq. (3.5.4) and the first-order term in Eq. (3.5.5) requires knowledge of the triplet distribution function in the reference fluid which is usually replaced by the Kirkwood superposition approximation. Following Smith, we will refer to the approach as a whole as the reference averaged Mayer (RAM) function theory. Another choice of reference system based upon a division of the Mayer function is that of hard spheres with a diameter chosen so that the first-order term in the free energy vanishes. This gives rise to the so called blip function theory. ... 

Fig. 1. Radial distribution functions at 2rz for the two natural orbitals with largest occupation numbers in the lowest triplet state of helium, the ls2s 3S state. Values of the radius are measured in atomic units.

In order to refine these theories, we need to include the second equations in the corresponding hierarchy of the integral equations for the distribution functions. For the PY g(r) and the CHNC g(r), this procedure is equivalent to the inclusion of more complicated rooted graph integrals in calculating the g( ) 16,17 por the BGY hierarchy, one possible way of the procedure is to truncate the quadruplet distribution function 1234 occurring in the second equation describing the triplet correlation function [= ( "12. is> 23) ra = ki - -il] by a proper combination of [= and g,. , s. The... 

A detailed study of the N2 emission rates has been carried out by Morrill et al (1998). In this study, the quasistatic electric field model (Pasko et al, 1997) was used to calculate the electric fields, and the solution to Boltzmann s equation was used to calculate the electron energy distribution function as a fimction of altitude. Results for excitation of seven triplet states of N2 are shown in Fig. 12 at A = 65 and 75 km. The temporal diuation of the excitations may be understood in terms of the faster relaxation (higher conductivity see Fig. 9) of the E field at the higher altitude. 

The pair distribution function, g(r), gives the probability of finding an atom (or molecule, if simulating a molecular fluid) a distance r from another atom (or molecule) compared to the ideal gas distribution. g(r) is thus dimensionless. Higher radial distribution functions (e.g. the triplet radial distribution function) can also be defined but are rarely calculated and so references to the radial distribution function are usually taken to mean the pairwise version. In a crystal, the radial distribution function has an infinite number of sharp peaks whose separations and heights are characteristic of the lattice structure. 

After studying some of these simplest models—all are formally equivalent—we proceed to introduce the concept of molecular distribution functions. As in Chapter 3, explicit and exact expressions for the molecular distribution functions in terms of molecular parameters can be derived for such systems. In subsequent sections we generalize the Ising model in different directions Multistate units, triplet interactions, continuous systems, and so on. Finally, we apply these methods to some specific problems such as phase transition, the helix-coil transition, and one aspect of liquid water. 

By a straightforward generalization of the arguments given above, we can compute any higher-order distribution function. For instance, for a consecutive triplet of units, we have... 

This factoring of the triplet distribution function into pair and single distribution functions is a characteristic feature of the one-dimensional model. In terms of conditional probability, the same result can be stated as follows ... 

A polymeric complication of all BGY approaches is the need for several different types of three-point correlation functions. For the purely in/ermolecular distribution function the standard Kirkwood superposition approximation is invoked, that is, a real space product of the three corresponding intermolecular pair functions. However, for the intramolecular triplet distribution functions (involving one, two, or three sites on the same polymer), there are many alternative schemes invoked by the different authors. These can be viewed as different closure approximations. Eu and Gan " have focused on analyzing the Kirkwood hierarchy based on Kirkwood-like and Markov-like approxi-... 

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