Pair distribution function calculation from simulation

Figure 7. Water oxygen-exocyclic methylene carbon pair distribution function, calculated from a molecular dynamics simulation of a-D-glucopyranose in aqueous solution, giving the normalized probability of finding a water oxygen atom a given distance r from the C6 carbon atom. (Reproduced from Ref. 32. Copyright 1989 American Chemical Society.)...

Figure 4.5 From the result of a molecular dynamics simulation, the x-ray scattering intensity was calculated, and from it (r), given in the broken curve, was derived, by using exactly the same procedure as was used to treat experimental x-ray scattering intensities. The solid curve is the C-C atom pair distribution function calculated directly from the simulation result. (From Mondello et al.13)...

Calculating the pair distribution function in a simulation is straightforward all we need to do is count the number of atom pairs separated by a distance in the range from r to r -I- Sr, and then normalize it. Usually, g r) is normalized by the number of pairs Nig(r,r + Sr) that would be observed in an ideal gas of the same density, so that, in the limit of large distances, r oo, where correlations disappear, g r) 1. A typical FORTRAN code to calculate g(r) from 0 to rmax with the resolution del tar is given below. The separation distance histogram is calculated for nbins = rmax/ deltar bins, for a model of N particles in a box with sides boxx. 

FIG. 62 The structure of the monolayers of polyst)rene latex particles adsorbed on mica expressed in terms of the pair-correlation function g (a) / = 10 M, = 0.24 (b) / = 10 M, = 0.24. The broken lines represent the pair-correlation function calculated from the Boltzmann distribution g = the continuous lines show the RSA simulations (smoothened) the insets show the adsorbed particles forming a two-dimensional liquid phase. 

Figure 2 Ion-ion radial distribution function calculated from a 100 ps simulation of a l.O M aqueous NaCI solution. The simulation box contained 29 ion pairs and 1531 water molecules. Discontinuities at the 16 A cut-off limit are marked with an arrow...

The simulations are repeated several times, starting from different matrix configurations. We have found that about 10 rephcas of the matrix usually assure good statistics for the determination of the local fluid density. However, the evaluation of the nonuniform pair distribution functions requires much longer runs at least 100 matrix replicas are needed to calculate the pair correlation functions for particles parallel to the pore walls. However, even as many as 500 replicas do not ensure the convergence of the simulation results for perpendicular configurations. 

In practice, the value of the reaction coordinate r is determined from the gas-phase potential energy surface of the complex. Then we use the pair-distribution function for the system (for example, determined by a Monte Carlo simulation) and the intramolecular potential energy Vjatra to calculate the relation between the two rate constants. Alternatively, one may determine the potential of mean force directly in a Monte Carlo simulation. With the example in Fig. 10.2.6 and a reaction coordinate at rj, we see that the potential of mean force is negative, which implies that the rate constant in solution is larger than in the gas phase. Physically, this means that the transition state is more stabilized (has a lower energy) than in the gas phase. If the reaction coordinate is at r, then the potential of mean force is positive and the rate constant in solution is smaller than in the gas phase. 

In spite of the great success of the computer simulation methods in the determination of the microscopic properties of the solutions, the capacity of the traditional MD and MC simulations is always limited by the choice of the suitable potential functions to describe the interatomic interactions. The potentials are most often checked by comparison of the structural properties calculated from the simulation with those determined experimentally. The reverse Monte Carlo (RMC) method, developed by McGreevy and Pusztai [41] does not rely upon knowledge of any interaction potential, instead it generates a large set of atomic configurations on the condition that the difference between the experimental and calculated structure functions (or pair-distribution functions) should be minimum. The same structural... 

There have been numerous other earlier attempts to extract more detailed representations of the pair distribution function from computer simulations. These include calculations of radial functions along vectors (directions) away from the molecule [15], the accumulation of two-dimensional slices of the local density around a molecule [16], and the projection of the full three-dimensional structure onto a two-dimensional (planar) representation [17,18]. These approaches have had some success in providing more detailed structural information and often appeared to represent necessary compromises required by limiting (at that time) computational resources. 

Figure 3. The counterion-counterion pair-distribution function as obtained by theory (lines) and MC simulations (symbols). The calculations apply to cp = 0.0001 and Ce = 0.005 mol dm 3. The results apply, from bottom to top, to zp zc ratios —20 +1, —30 +1,-40 +1 and-50 +1.

In the next two figures we discuss the pair-correlation functions as obtained from the two-density theory and computer simulations. First, in Fig. 3 we compare the counterion-counterion pair-distribution function as obtained theoretically (lines) and from simulations (symbols). The numerical calculations apply to cp = 0.0001 and ce = 0.005 mol dm-3 the results show that the theory underestimates the counterion-counterion correlation. Next, in Fig. 4 the macroion-counterion pair-distribution is shown for the same set of parameters. Finally, in Fig. 5 the macroion-macroion pair-distribution functions are calculated by both theoretical approaches at cp = 0.0001 mol dm-3 solution and for zp = —10 and —30. 

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. 

The differences in the CMC and EMC predictions can be traced to the different pair probability densities estimated by these methods from the given time series. In fig. 9.4, we show a contour plot of the pair distribution function p Xi, Xg) as calculated by a semi-nonparametric (SNP) method [10] with the time-series simulations for these two species superimposed. In comparison, we show in fig. 9.5 a Gaussian pair probability distribution, as is consistent with CMC, with the same means and variances as those of the EMC distribution. The deviations of the EMC from the Gaussian distribution show that higher than second moments contribute. Since the information entropy for... 

The IRT model has been developed in detail in a series of papers of Green, Pimblott and coworkers and has been validated by comparison with full random flight simulations [47,49,51]. The IRT treatment of the radiation chemistry relies upon the generation of random reaction times from initial coordinate positions from pair reaction time distribution functions. A simulation, such as a random flight calculation, starts with the initial spatial distribution of the reactants. The separations between all the pairs of particles are evaluated... 

Computer-simulation calculations describe the solvation structure of Br as ceaseless motion of solvent molecules distributed around the anion at equilibrium r and N Diffraction techniques can provide the distribution functions for these values, though they require rather high concentration samples. Concentrated salt solution may have different solvation structures from that of dilute ones, not only because of the smaller number of free solvent molecules but also the higher possibility of ion-pair formation especially in organic solvents. ... 

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