Radial distribution function solution

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... 

After this computer experiment, a great number of papers followed. Some of them attempted to simulate with the ab-initio data the properties of the ion in solution at room temperature [76,77], others [78] attempted to determine, via Monte Carlo simulations, the free energy, enthalpy and entropy for the reaction (24). The discrepancy between experimental and simulated data was rationalized in terms of the inadequacy of a two-body potential to represent correctly the n-body system. In addition, the radial distribution function for the Li+(H20)6 cluster showed [78] only one maximum, pointing out that the six water molecules are in the first hydration shell of the ion. The Monte Carlo simulation [77] for the system Li+(H20)2oo predicted five water molecules in the first hydration shell. A subsequent MD simulation [79] of a system composed of one Li+ ion and 343 water molecules at T=298 K, with periodic boundary conditions, yielded... 

Thus, effects of the surfaces can be studied in detail, separately from effects of counterions or solutes. In addition, individual layers of interfacial water can be analyzed as a function of distance from the surface and directional anisotropy in various properties can be studied. Finally, one computer experiment can often yield information on several water properties, some of which would be time-consuming or even impossible to obtain by experimentation. Examples of interfacial water properties which can be computed via the MD simulations but not via experiment include the number of hydrogen bonds per molecule, velocity autocorrelation functions, and radial distribution functions. 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

The structure of the adsorbed ion coordination shell is determined by the competition between the water-ion and the metal-ion interactions, and by the constraints imposed on the water by the metal surface. This structure can be characterized by water-ion radial distribution functions and water-ion orientational probability distribution functions. Much is known about this structure from X-ray and neutron scattering measurements performed in bulk solutions, and these are generally in agreement with computer simulations. The goal of molecular dynamics simulations of ions at the metal/water interface has been to examine to what degree the structure of the ion solvation shell is modified at the interface. 

The probability of cavity formation in bulk water, able to accommodate a solute molecule, by exclusion of a given number of solvent molecules, was inferred from easily available information about the solvent, such as the density of bulk water and the oxygen-oxygen radial distribution function [65,79]. 

Fig. 17. Gd-aqueous proton radial distribution function for the aqueous solution of the Gd(III)(DOTP) complex (after Borel, A. Helm, L. Merbach, A.E. Chemistry - A European Journal 2001, 7, 600-610).

Fig. 5. VET rate constants of benzene in scC02 as a function of reduced density (filled circles). The solid line represents calculations of the local density at the position of the first maximum of the radial distribution function around an attractive solute in a Lennard-Jones fluid (see Fig. 7 and text for details). Experimental conditions pred = 2.1 (500bar, 318K), prei = 1.6 (150 bar, 318K), pred= 1.2 (lOObar, 318K), pred= 0.7 (lOObar, 328K).

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. 

For the calculation of the binary collision term the radial distribution function g(r) and the static structure factor of the solute S(q) is required. 

Note that the binary HMSA [60] scheme gives the solute-solvent radial distribution function only in a limited range of solute-solvent size ratio. It fails to provide a proper description for such a large variation in size. Thus, here the solute-solvent radial distribution function has been calculated by employing the well-known Weeks-Chandler-Anderson (WCA) perturbation scheme [118], which requires the solution of the Percus-Yevick equation for the binary mixtures [119]. 

It is found that as the solute size is increased, keeping all other parameters fixed, the peak in the solute-solvent radial distribution function slowly disappears and approaches the value 1. This implies that the probability of a solvent particle, provided that there is a solute at the origin, is same everywhere. The solute-solvent static structure factor Si2(q), which can be obtained from g 12(f), will also have no structure and will have a uniform value that is, Si2(q) = 1 for all wavenumbers. 

The reason for the early crossover can be understood from the following discussion. When the interaction energy between the solute and the solvent is increased, the peak of the radial distribution function does not disappear. Thus c 2(q) 0 for all wavenumbers. Hence the density mode contribution does not become zero as happens in the case where the size of the solute is only increased. Hence Dmicroi along with the binary term, also contains the contribution from the density mode. This results in faster decrease of Dmjcro > leading to an early crossover. 

The conduction electrons are scattered by the alkali atoms, the coherence implicit in the radial distribution function. Unlike the case of the scattering of a single electron in a plane wave state by a liquid, discussed previously, in this case the structure factor S(k) must be known up to the Fermi energy (which is 0.5 e.v. — 1 e.v. in saturated metal ammonia solutions). 

In short, our S-MC/QM methodology uses structures generated by MC simulation to perform QM supermolecular calculations of the solute and all the solvent molecules up to a certain solvation shell. As the wave-function is properly anti-symmetrized over the entire system, CIS calculations include the dispersive interaction[35]. The solvation shells are obtained from the MC simulation using the radial distribution function. This has been used to treat solvatochromic shifts of several systems, such as benzene in CCI4, cyclohexane, water and liquid benzene[29, 37] formaldehyde in water(28, 38] pyrimidine in water and in CCl4(31] acetone in water[39] methyl-acetamide in water[40] etc. 

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