Pair correlation function method

The chemical shieldings were then recalculated in this same system using the QM/MM method [10], To this end, each molecule was considered individually. The water molecule of interest and its first solvation shell were treated quantum mechanically, whereas the surrounding water molecules were taken into account with an empirical force field representation (MM molecules). The first solvation shell was defined via a distance criterion on the oxygen-oxygen distance. As a threshold, the first minimum of the O-O pair correlation function was taken this occurs at 3.5 A [93]. All... 

After these transformations the model can be solved effectively by numerical methods. As the initial condition, we have to specify the concentration of adsorbed particles and the pair correlation function. For example, for non-correlated distributed pairs we set F Jr) = 1. 

Here, we report some basic results that are necessary for further developments in this presentation. The merging process of a test particle is based on the concept of cavity function (first adopted to interpret the pair correlation function of a hard-sphere system [75]), and on the potential distribution theorem (PDT) used to determine the excess chemical potential of uniform and nonuniform fluids [73, 74]. The obtaining of the PDT is done with the test-particle method for nonuniform systems assuming that the presence of a test particle is equivalent to placing the fluid in an external field [36]. 

Using the method of isotope substitution, it is possible to obtain partial pair correlation functions experimentally. Two isotopically different samples were prepared at c = 0.1 M, one where all the hydrogen in the butylammonium chains was ordinary II and one where they were entirely deuterated, to make use of the large difference in scattering... 

Why should one go to all this trouble and do all these integrations if there are other, less complex methods available to theorize about ionic solutions The reason is that the correlation function method is open-ended. The equations by which one goes from the gs to properties are not under suspicion. There are no model assumptions in the experimental determination of the g s. This contrasts with the Debye-Htickel theory (limited by the absence of repulsive forces), with Mayer s theory (no misty closure procedures), and even with MD (with its pair potential used as approximations to reality). The correlation function approach can be also used to test any theory in the future because all theories can be made to give g(r) and thereafter, as shown, the properties of ionic solutions. 

We implement a modified version of the reconstruction method developed in a previous work to model two porous carbons produced by the pyrolysis of saccharose and subsequent heat treatment at two different temperatures. We use the Monte Carlo g(r) method to obtain the pair correlation functions of the two materials. We then use the resulting pair correlation functions as target functions in our reconstruction method. Our models present structural features that are missing in the slit-pore model. Structural analyses of our resulting configurations are useful to characterize the materials that we model. 

In this work, we explore the use of the pair correlation function as the target function in our reconstruction method [4]. This speeds up the simulations, allowing us to construct models in much larger simulation boxes. We build models for two saccharose-based carbons treated at different temperatures. We compare the exact pore size distributions and perform Grand Canonical Monte Carlo (GCMC) simulations of nitrogen at 77 K in the resulting models. 

In the Reverse Monte Carlo (RMC) method [5], the pair correlation function or the structure factor is calculated after each random move (Ssim(<]) or gsimfr)) and compared to the respective target function obtained from experimental diffraction data (Sexp(q) or gexp(r)). It is possible to calculate Ssm(q) with full periodicity from the atomic positions. This method is best in principle [10], but the computational cost is much greater than for any of the other available methods. It is also possible to obtain Ssm(q) by first calculating gsm(r) from the atomic positions and then Fourier transform this function and calculate Ssim(q). The disadvantage of this approach is that there is an additional computational cost associated with the Fourier transform of gsm(r) after each move. 

Figure 2. Pair correlation functions of saccharose-based carbons obtained with MCGR (solid line) and reconstruction method (dashed line), a) CS400, b) CSIOOO...

The results summarised here are for pure water at the temperature 25°C and the density 1.000 g cm , and are obtained by solving numerictdly the Ornstein-Zemike (OZ) equation for the pair correlation functions, using a closure that supplements the hypernet-ted chain (HNC) approximation with a bridge function. The bridge function is determined from computer simulations as described below. The numerical method for solving the OZ equation is described by Ichiye and Haymet and by Duh and Haymet. ... 

We propose the study of Lennard-Jones (LJ) mixtures that simulate the carbon dioxide-naphthalene system. The LJ fluid is used only as a model, as real CO2 and CioHg are far from LJ particles. The rationale is that supercritical solubility enhancement is common to all fluids exhibiting critical behavior, irrespective of their specific intermolecular forces. Study of simpler models will bring out the salient features without the complications of details. The accurate HMSA integral equation (Ifl) is employed to calculate the pair correlation functions at various conditions characteristic of supercritical solutions. In closely related work reported elsewhere (Pfund, D. M. Lee, L. L. Cochran, H. D. Int. J. Thermophvs. in press and Fluid Phase Equilib. in preparation) we have explored methods of determining chemical potentials in solutions from molecular distribution functions. 

The most important information to be extracted from either experiment or theory is structural information. In first-principles methods this information is readily available. Experimentally, it may be extracted directly from diffraction techniques. While X-ray diffraction provides information on solids, which means on frozen structures, other techniques like neutron diffraction can provide statistical information like pair correlation functions that are also easily obtained from a molecular dynamics trajectory. 

Particular pair correlation function among a number of pair correlation functions can possibly be determined by isomorphous substitution, that is, by substituting one element by another chemically similar element. The implied assumption is that the structure around the atoms of the given element is not altered. There are also other methods... 

As mentioned above, a set of experimental data does not necessarily correspond to a unique molecular structure. Moreover, even unphysical structures may be consistent with a set of experimental data. It is therefore necessary to carefully choose a set of constraints to limit the number of possible structures. The uniqueness theorem of statistical mechanics [30, 31] provides a guide to the number and type of constraints that should be appfied in the RMC method in order to get a unique structure [32]. For systems in which only two- and three-body forces are important, the uniqueness theorem states that a given set of pair correlation function and three-body correlation function determines aU the higher correlation functions. In other words, assuming that only two- and three-body forces are important, the RMC method must be implemented along with constraints that describe the three-body correlations [27]. 

These are two equations in the unknowns guo and guu, assuming that guu and guu are exactly the same [8, 9]. Thus, the two pair correlation functions may be obtained separately. This method requires that careful and precise experiments be carried out. 

In this section, we describe a theory for calculating observables resulting from incoherent excited state transport among chromophores randomly distributed in low concentration on isolated, flexible polymer chains. The pair correlation function used to describe the distribution of the chromophores is based on a Gaussian chain model. The method for calculating the excitation transfer dynamics is an extension to finite, inhomogeneous systems of a truncated cumulant expansion method developed by Huber for infinite, homogeneous systems (25.26). 

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