Radial distribution function integration

Radial distribution function integral equations for, 22-27 of a fluid, 20... 

Microscopic theory yields an exact relation between the integral of the radial distribution function g(r) and the compressibility... 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

The s-states have spherical symmetry. The wave functions (probability amplitudes) associated with them depend only on the distance, r from the origin (center of the nucleus). They have no angular dependence. Functionally, they consist of a normalization coefficient, Nj times a radial distribution function. The normalization coefficient ensures that the integral of the probability amplitude from 0 to °° equals unity so the probability that the electron of interest is somewhere in the vicinity of the nucleus is unity. 

A number of approximate integral equations for the radial distribution function g(r) of fluids have been proposed in recent years. Two particularly useful approximations are the Percus-Yevick (PY)1,2 and the Convolution Hypernetted Chain (CHNC)3-4 equations. In this paper an efficient numerical method of solving these equations is described and the results obtained bv applying the method to the PY equation are discussed. A later paper will describe the behavior of the... 

Approximate evaluations of the radial distribution function in dense systems are being obtained as solutions to integral equations derived from firsl principles under well-defined approximations. 

With the above-mentioned radial distribution function, expression for the Einstein frequency after performing the angular integration Eq. (114) reduces to... 

Structural information obtained from the first peak of the radial distribution functions. Distances are given in A and Ns is the coordination number obtained from the integration of the first peak. 

A different approach to mention here because it has some similarity to QM/MM is called RISM-SCF [5], It is based on a QM description of the solute, and makes use of some expressions of the integral equation of liquids (a physical approach that for reasons of space we cannot present here) to obtain in a simpler way the information encoded in the solvent distribution function used by MM and QM/MM methods. Both RISM-SCF and QM/MM use this information to define an effective Hamiltonian for the solute and both proceed step by step in improving the description of the solute electronic distribution and solvent distribution function, which in both methods are two coupled quantities. There is in this book a contribution by Sato dedicated to RISM-SCF to which the reader is referred. Sato also includes a mention of the 3D-RISM approach [6] which introduces important features in the physics of the model. In fact the simulation-based methods we have thus far mentioned use a spherically averaged radial distribution function, p(r) instead of a full position dependent function p(r) expression. For molecules of irregular shape and with groups of different polarity on the molecular periphery the examination of the averaged p(r) may lead to erroneous conclusions which have to be corrected in some way [7], The 3D version we have mentioned partly eliminates these artifacts. 

The structure of liquids can be analyzed by the calculated radial distribution function (RDF), which defines the solvation shells. In Fig. 16.1, the calculated RDF of the liquid Aris shown, and in Table 16.1, the structure is compared with the experimental results. Four solvation shells are well defined. The spherical integration of these peaks defines the coordination number, or the number of atoms in each solvation shell. The first shell that starts at 3.20A has a maximum at 3.75A, and ends at 5.35 A, has an average of 13 Ar atoms. Therefore, in the first solvation shell, there is a reference Ar atom surrounded by other neighboring 13 Ar atoms. All the maxima of the RDF, shown in Table 16.1, are in good agreement with the experimental results obtained by Eisenstein and Gingrich [29], using X-ray diffraction in the liquid Ar in the same condition of temperature and pressure. The calculated... 

The corresponding contribution to the radial distribution function can be obtained by means of a Fourier inversion analogous to the one for the experimental data using the same modification function and upper integration limit. 

The calculations needed for corrections and normalization of solution X-ray diffraction data, for calculation of radial distribution functions, for model calculations, and for least-squares refinements (16) can conveniently be done on a personal computer for which integrated program systems are available (17). 

Within PB theory [2] and on the level of a cell model the cylindrical geometry can be treated exactly in the salt-free case [3, 4]. The Poisson-Boltzmann (PB) solution for the cell model is reviewed in the chapter in this volume on the osmotic coefficient. The PB approach can provide for instance new insights into the phenomenon of Manning condensation [5-7]. For example, the distance up to which counterions can be called condensed can be conveniently found via the inflection point in the log plot of the integrated radial distribution function P(r) of counterions [8, 9], defined as... 

The total nonbonded contribution to the stress ty is then < " = ]C/ (T y>(P) which is the sum of er"fc(/j) over all atoms /i that engage in nonbonded interactions. This sum may be written in an integral form by use of the radial distribution function g(r), where... 

Using the pair-wise additivity of U(R), it is possible to integrate Eq. (18) over the equilibrium configurations of (N — 2) particles. If one then uses the definition of the radial distribution function, an expression for E in terms of g( r) and u(r) is obtained, and it is referred to as the energy equation... 

Let us discuss now a usual theoretical method employed in the calculation of the radial distribution function. The basic equation obeyed by g(r) is the integral equation, introduced by Ornstein and Zernike in 1914 [25,32]... 

To evaluate the volume integrals in (84), the radial distribution function must be known. The pair distribution function affected by the Brownian motion and the relative electrophoretic velocity between a pair of particles is generally nonuniform and nonisotropic. When the particles are sufficiently small so that Brownian motion dominates, one can use a simple distribution function based on hard-sphere potential... 

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