Lennard-Jones spheres

Kotelyanskii, M. J. Hentschke, R., Gibbs-ensemble molecular dynamics liquid-gas equlibria for Lennard-Jones spheres and n-hexane, Mol. Simul. 1996,17, 95-112... 

The essential simplification of the model is to replace the alkyl substituents, from mediyl to tert-butyl, by Lennard-Jones spheres of increasing diameters. In this model we cannot calculate the exact values of k and 2 (in fact, these cannot be calculated even if we had the true rotational potential the main missing information is the solvation Gibbs energies of the molecules involved). Nevertheless, we can demonstrate with this simplified model the two major experimental findings regarding the proton-proton correlation in these series of molecules, as shown in Table 4.6. 

Figure 4.31. The total rotational potential, in kcal/mol, as a function of the dihedral angle

Fig, 19. Potential of mean force for a liquid of Lennard-Jones spheres with near close packing and reduced temperature near the boiling point versus the reduced separation, r/a, ----. a is the dimension of the Lennard-Jones potential. The effects of an activa-... 

Typical forms of the radial distribution function are shown in Fig. 38 for a liquid of hard core and of Lennard—Jones spheres (using the Percus— Yevick approximation) [447, 449] and Fig. 44 for carbon tetrachloride [452a]. Significant departures from unity are evident over considerable distances. The successive maxima and minima in g(r) correspond to essentially contact packing, but with small-scale orientational variation and to significant voids or large-scale orientational variation in the liquid structure, respectively. Such factors influence the relative location of reactants within a solvent and make the incorporation of the potential of mean force a necessity. 

The effect of density on the velocity autocorrelation function was studied by Verlet and Levesque [519]. In Fig. 52, two velocity autocorrelation functions are shown which correspond to two densities of the Lennard—Jones spheres used in the numerical study (see also Gubbins 1520]. Rdsibois and De Leener [490] have made the following observations on these results. 

In more complex molecular systems, increased coupling between the translational motion and both rotational and vibrational modes occurs. It is difficult to separate these effects completely. Nevertheless, the velocity autocorrelation functions of the Lennard—Jones spheres [519] (Fig. 52) and the numerical simulation of the carbon tetrachloride (Fig. 39) are quite similar [452a]. 

More realistic model systems based on restricted interaction site (RISM) models have also been tested [7], The interaction sites are usually Lennard-Jones spheres decorated with charges, dipoles and quadrupoles. Simulation of these models sometimes yield results that agree with experimental measurements. However, it is very time consuming to study these systems so that only very small systems have been studied so far, but it is reasonable to assume that larger systems will be simulated in the near future as the computers grow faster. 

A similar conclusion has been reached by Ciccotti et al. s-iao jj, their studies of the model ion association reaction. Their system consisted of two equally massive ions, modeled as Lennard-Jones spheres with a positive or negative charge, in a solvent of dipolar molecules. Each solvent molecule was modeled as a Lennard-Jones sphere with a dipole moment of either 2.4 or 3.0 D and with a mass equal to that of the ionic mass. As with the simulations of Karim and McCammon, Ciccotti et al. started the dynamics at the transition state, as determined from the free energy calculations, and ran 104-144 trajectories to determine the transmission coefficient. The values of the transmission coefficient they found were 0.18 in the 2.4 D solvent and 0.16 in the 3.0 D solvent (which are surprisingly, and perhaps coincidentally, close to the results of Karim and McCammon e). Ciccotti et al. also calculated the frequency-dependent friction that the solvent exerted on the reaction coordinate in order to compare the simulation results with Grote-Hynes theory for the rates. The comparison with Grote-Hynes theory was quite close, although within the outer reaches of the calculated uncertainties in the molecular dynamics transmission coefficients. 

The molecular theory considers a dipolar liquid where the two constituents are Lennard-Jones spheres each with an embedded dipole moment at the center. The Lennard-Jones parameters (sizes, interaction strength parameters) and also values of the dipole moments are different for the two species. The theoiy properly includes the differing inter- and intramolecular correlations that are present in a binary mixture. As a result, the theory can explain several important aspects of the nonideality of equilibrium solvation energy (broadly known as preferential solvation) observed in experiments. The non-ideality of solvation is found to depend on both the molecular sizes and the magnitude of the dipole moments of the solvent... 

GEMD has been used to calculate phase diagrams for atomic as well as molecular systems, such as Lennard-Jones spheres or ra-hexane, as shown in Figs. 1 and 2. 

Kotelyanskii, M.J. Hentschke, R. Gibbs-Ensemble Molecular Dynamics Liquid-Gas Equilibria for Lennard-Jones Spheres and n-Hexane. Molecular Simulations 1996, 17, 95-112. 

The calibration procedure is quite straightforward for the alkaline earth metals and these ions can be reasonably well modelled by simple charged Lennard-Jones spheres. That is, the non-bonded parameters can be adjusted so that the solvation energy and the first peak of the radial distribution function (REF) in water coincide with experimental values. For transition metals, however, the situation becomes more complicated and we return to this issue below. 

An empirical model for transition metals. In addition to the alkaline earth metal ions, which lack d-orbital valence electrons, it is important to try to extend the applicability of our model to also include transition metals. Unfortunately, the hydration energies of the transition metal ions cannot be well modelled by a simple Lennard-Jones sphere with a charge in the centre in order to reproduce the observed hydration energy, the ion radius must be... 

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