Energy, van der Waals

We first note errors in total energy means that are not greater than 0.5% for all LN versions tested. Individual energy components show errors that are generally less than 1%, with the exception of the van der Waals energy that can reach 4% for large k2. Of course, this discussion of relative errors reflects practical rather than mathematical considerations, since constants can be added to individual terms without affecting the dynamics. The relative errors... 

Figure 2-117. Dependence of the van der Waals energy on the distance between two non-con-nected atom nuclei. With decreasing atoiTiic distance the energy between the two atoms becomes attraction, going through a minimum at the van der Waals distance. Then, upon a further decrease in the distance, a rapid increase in repulsion energy is observed.

A powerful and general technique to enhance sampling is the use of umbrella potentials, discussed in Section IV. In the context of alchemical free energy simulations, for example, umbrella potentials have been used both to bias the system toward an experimentally determined conformation [26] and to promote conformational transitions by reducing dihedral and van der Waals energy terms involving atoms near a mutation site [67]. 

At best, van der Waals interactions are weak and individually contribute 0.4 to 4.0 kj/mol of stabilization energy. ITowever, the sum of many such interactions within a macromolecule or between macromolecules can be substantial. For example, model studies of heats of sublimation show that each methylene group in a crystalline hydrocarbon accounts for 8 k[, and each C—IT group in a benzene crystal contributes 7 k[ of van der Waals energy per mole. Calculations indicate that the attractive van der Waals energy between the enzyme lysozyme and a sugar substrate that it binds is about 60 k[/mol. 

For each atom type there are two parameters to be determined, the van der Waals radius and the atom softness, Rq and, It should be noted that since the van der Waals energy is calculated between pairs of atoms, but parameterized against experimental data, the derived parameters represent an effective pair potential, which at least partly includes many-body contributions. 

The Self-Consistent Reaction Field (SCRF) model considers the solvent as a uniform polarizable medium with a dielectric constant of s, with the solute M placed in a suitable shaped hole in the medium. Creation of a cavity in the medium costs energy, i.e. this is a destabilization, while dispersion interactions between the solvent and solute add a stabilization (this is roughly the van der Waals energy between solvent and solute). The electric charge distribution of M will furthermore polarize the medium (induce charge moments), which in turn acts back on the molecule, thereby producing an electrostatic stabilization. The solvation (free) energy may thus be written as... 

Table 4. Van der Waals energy for cyclodextrin-phenol systems (—Em(n kJ/mol)...

Fig. 7. Plots of log k2/k vs. Dc 0 determined by the calculation of van der Waals energies for cyclodextrin complexes with Bland p-substituted phenyl acetates...

A Gst is the difference in free energy due to steric constants in reactant and transition state, k is the rate constant of the nonsterically constrained reaction. The contribution of the steric component to the transition-state energy cannot be deduced accurately from DFT calculations because van der Waals energies are poorly computed. Force field methods have to be used to properly account for such interactions. 

At low subplateau temperatures T < co, which we are primarily interested in here, the expression above reduces to the following van der Waals energy ... 

The DLVO theory, with the addition of hydration forces, may be used as a first approximation to explain the preceding experimental results. The potential energy of interaction between spherical particles and a plane surface may be plotted as a function of particle-surface separation distance. The total potential energy, Vt, includes contributions from Van der Waals energy of interaction, the Born repulsion, the electrostatic potential, and the hydration force potential. [Israelachvili (13)]. 

AtHiKlifiATiON KNIiRCY - van der Waal energy - Lattice binding energy TA ... 

Fig. 5 Changes in the van der Waals energy during crystallization by quenching from 600 K to each crystallization temperature Tc (A) at 300 K, (A) at 200 K, ( ) at 100 K, and (o) at 20 K. The energies are expressed in Kelvin/atom...

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