Lennard-Jones 12-6 variant

But the methods have not really changed. The Verlet algorithm to solve Newton s equations, introduced by Verlet in 1967 [7], and it s variants are still the most popular algorithms today, possibly because they are time-reversible and symplectic, but surely because they are simple. The force field description was then, and still is, a combination of Lennard-Jones and Coulombic terms, with (mostly) harmonic bonds and periodic dihedrals. Modern extensions have added many more parameters but only modestly more reliability. The now almost universal use of constraints for bonds (and sometimes bond angles) was already introduced in 1977 [8]. That polarisability would be necessary was realized then [9], but it is still not routinely implemented today. Long-range interactions are still troublesome, but the methods that now become popular date back to Ewald in 1921 [10] and Hockney and Eastwood in 1981 [11]. 

Detonation pressure may be computed theoretically or measured exptly. Both approaches are beset with formidable obstacles. Theoretical computations depend strongly on the choice of the equation of state (EOS) for the detonation products. Many forms of the EOS have been proposed (see Vol 4, D269—98). So.far none has proved to be unequivocally acceptable. Probably the EOS most commonly, used for pressure calcns are the polytropic EOS (Vol 4, D290-91) and the BKW EOS (Vol 4, D272-74 Ref 1). A modern variant of the Lennard Jones-Devonshire EOS, called JCZ-3, is now gaining some popularity (Refs -11. 14). Since there is uncertainty about the correct form of the detonation product EOS there is obviously uncertainty in the pressures computed via the various types of EOS ... 

Fig. 10.12. Vapor-liquid phase behavior for the Lennard-Jones fluid. Solid triangles and hollow squares indicate the results of the particle addition/deletion and volume scaling variants of the flat-histogram simulation using the Wang-Landau algorithm. Crosses are from a histogram reweighting study based on grand-canonical measurements at seven state points. The solid line is from Lotfi, et al. [76], Reprinted figure with permission from [75]. 2002 by the American Physical Society...

The three variants of this model are described in Fig. 5.9. The ligands are simple Lennard-Jones particles, at the center of which a point dipole of strength d is embedded. The orientation of the ligand on the site is always the same, say upward, as in Fig. 5.9. The adsorbent macromolecule consists of three subunits denoted by a, b, and c, each of which has one binding site to which we also refer as a, b, and c. Near each site the macromolecule has a dipole of strength D, which can be oriented either upward or downward. The orientation of the dipole D is determined... 

Auerbach et al. (101) used a variant of the TST model of diffusion to characterize the motion of benzene in NaY zeolite. The computational efficiency of this method, as already discussed for the diffusion of Xe in NaY zeolite (72), means that long-time-scale motions such as intercage jumps can be investigated. Auerbach et al. used a zeolite-hydrocarbon potential energy surface that they recently developed themselves. A Si/Al ratio of 3.0 was assumed and the potential parameters were fitted to reproduce crystallographic and thermodynamic data for the benzene-NaY zeolite system. The functional form of the potential was similar to all others, including a Lennard-Jones function to describe the short-range interactions and a Coulombic repulsion term calculated by Ewald summation. 

Except for oxygen-balanced expls, the computation of detonation products depends strongly on the choice of the equation of state (EOS) for these products. In the US the BKW EOS (see Vol 4, D272-R) has been favored and most of the computed product compns below will be based on it. Some of these will be compared with the relatively few calcns based on a Lennard-Jones-Devonshire (UD) EOS (see Vol 4, D287-L) CJ state product compns calcd via the BKW EOS are compared with compns computed with LJD types of EOS in Tables 2—4. For PETN (Table 2) an early variant of the LJD EOS (Ref 1) shows no solid C in the products and somewhat more CO than the BKW computation. Note that for PETN both EOS give product compn that show relatively little variation with p q, the initial density of the expl. This is not the case for RDX and TNT (Tables 3 4) where a change in Pq results in substantial changes in product compn. 

This algorithm has been applied to calculate the thermal conductivity of a variant of the Gay-Beme fluid where the Lennard-Jones core has been replaced by a purely repulsive 1/r core [20]. Two systems were studied, one consisting of prolate ellipsoids with a length to width ratio of 3 1 and another one consisting of oblate ellipsoids with a length to width ratio of 1 3. The potential parameters are given in Appendix II. They both form nematic phases at high densities. 

In conclusion, the Cahn-Hilliard theory is a modernization of van der Waals, confirming cmd extending the latter. With these theories and their many variants and extensions the framework is basically available for computing interfacial tensions from molecular interactions. Carrying out the computations is no easy matter, especially if the molecules are not spherical and if their interactions contain contributions other than those of the Lennard-Jones type. Then the quality of the results is determined by the quality of the choice of the parameters, analytical approximations, truncations, etc. A promising alternative is to invoke computer solutions, which will be treated in the next section. 

The bond between atoms is described by the curve that traces the way the energy changes as the atomic coordinates change. The potential of the interatomic attraction is the energy as a function of the normal vibration coordinate strain, which is usually taken to be the distance between the bound atoms (Figure 2.1). Many variants exist. A Morse curve is representative for the potential between covalently bonded atoms. If the bond is between ions the interatomic potential is well described by a Lennard-Jones potential with a Coulomb term. 

At the Hartree-Fock level (Hartree 1928 Fock 1930), the energies for closed-shell systems are evaluated using the restricted Hartree-Fock (RHF) method (Hall and Lennard-Jones 1951 Roothaan 1951). For the open-shell molecules, there are several methods that are available in most programs the unrestricted Hartree-Fock (UHF) method (Pople and Nesbet 1954), several variants of the restricted open-shell Hartree-Fock (ROHF) method (Hsu et al. 1976 McWeeny and Diercksen 1968), and the generalized valence bond (GVB) method (Bobrowicz and Schaefer 1977). 

This bond potential is obviously asymmetric. Another frequently used variant is an asymmetric form of the bond potential that consists of the combination of the repulsive part of the Lennard-Jones potential and the attractive part of the FENE potential [50,54] ... 

The simulations were performed by employing a variant of the Wang—Landau algorithm, in which the energy E space and the k. space were sampled simultaneously [184]. The Lennard-Jones parameters for the nonbonded pairs of monomers were e = 1 and a = 2 . The FENE parameters were set to tto = 1-2 and K = 2, whereas the LJ parameters of the bonded monomers were estimated in such a way that the equilibrium bond length was unity, i.e., Fb(rjj+i = 1) = min. 

PEP300 uses Buckingham exp-6 functions for non-bonded Interactions. A variant of it, PEP301, which uses Lennard-Jones 12-6... 

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