Two-body potential models

A third approach, the bond valence model see Bond Valence Method), is related to the two-body potential model but is expressed in terms of the electrostatic field rather than the energy. It assumes that the valence electrons of the atoms can be assigned to individual bonds and that these determine the bond length. In spite of its conceptual and computational simplicity, this model gives good predictions of the bond length and provides useful chemical insights. ... 

A simpler approach that avoids the use of quantum mechanics is the two-body potential model (ionic model) that treats a crystal or molecnle as a collection of atoms that... 

Conformational Adjustments The conformations of protein and ligand in the free state may differ from those in the complex. The conformation in the complex may be different from the most stable conformation in solution, and/or a broader range of conformations may be sampled in solution than in the complex. In the former case, the required adjustment raises the energy, in the latter it lowers the entropy in either case this effect favors the dissociated state (although exceptional instances in which the flexibility increases as a result of complex formation seem possible). With current models based on two-body potentials (but not with force fields based on polarizable atoms, currently under development), separate intra-molecular energies of protein and ligand in the complex are, in fact, definable. However, it is impossible to assign separate entropies to the two parts of the complex. 

Water Potentials. The ST2 (23), MCY (24), and CF (2J5) potentials are computationally tractable and accurate models for two-body water-water interaction potentials. The ST2, MCY and CF models have five, four, and three interaction sites and have four, three and three charge centers, respectively. Neither the ST2 nor the MCY potentials allow OH or HH distances to vary, whereas bond lengths are flexible with the CF model. While both the ST2 and CF potentials are empirical models, the MCY potential is derived from ab initio configuration interaction molecular orbital methods (24) using many geometrical arrangements of water dimers. The MCY+CC+DC water-water potential (28) is a recent modification of the MCY potential which allows four body interactions to be evaluated. In comparison to the two-body potentials described above, the MCY+CC+DC potential requires a supercomputer or array processor in order to be computationally feasible. Therefore, the ST2, MCY and CF potentials are generally more economical to use than the MCY+CC+DC potential. 

Although simple, a model system containing one solvent molecule together with one ion already provides valuable insight into the nature of the ion-solvent interaction. There is also convincing evidence that this two body potential dominates in much more complicated situations like in the liquid state 88,89,162). Molecular data for one to one complexes can be calculated with sufficient accuracy within reasonable time limits. Gas-phase data reported in Chapter III provide a direct basis for comparison of the calculated results. 

The bond valence model may also be used to refine the structure since it is based on the same assumptions as the two-body potential method. The network equations (3.3) and (3.4), can be used to predict the theoretical bond valences as soon as the bond graph is known. From these one can determine the expected bond... 

We finally note that this discussion gains additional importance with respect to the continuous chain limit. In Chap. we have shown that we can construct the continuous chain model only after an additive renormalization. which essentially extracts a one-body part from the two-body potential. If we... 

The lattice energy based on the Born model of a crystal is still frequently used in simulations [14]. Applications include defect formation and migration in ionic solids [44,45],phase transitions [46,47] and, in particular, crystal structure prediction whether in a systematic way [38] or from a SA or GA approach [ 1,48]. For modelling closest-packed ionic structures with interatomic force fields, typically only the total lattice energy (per unit cell) created by the two body potential,... 

On the other hand, empirical or semi-empirical models will be strictly two-body, if only based on data of second virial coefficient of the real gases and of the isolated molecule, or effective two-body potentials, if they keep a two-body form, but also experimental data relevant to the condensed phase are used to construct them. 

As the empirical or semiempirical potentials, those obtained in the supermolecular approach with the PCM, are effective two-body potentials that implicitly include non-additive effects, modeling the solvent molecules as a continuum. 

An attempt to satisfy both needs relies on effective two-body potentials, such as that developed with the polarizable continuum model (PCM) of the solvent, briefly described above (see equations (43)-(48)). 

Finally, there is another model commonly used in simulations - a simple bead-spring model for chain molecules. The bead-spring model is often referred to as a meso-scale model because the beads and springs represent the average properties of much larger molecules. In this model, monomers separated by distance r interact through a two -body potential, often of the truncated LJ form ... 

The construction of practical interatomic potential models for silica has a long history (see Refs. [5, 6] and references therein). Initially, two-body potentials were used for the interaction between ions i and j separated by a distance r ... 

The potential of Eq. (1) with parameters determined in Refs. [10, 11] was thoroughly tested in computer simulations of silica polymorphs. In Ref. [10], the structural parameters and bulk modulus of cc-quartz, a-cristobalite, coesite, and stishovite obtained from molecular dynamics computer simulations were found to be in good agreement with the experimental data. The a to / structural phase transition of quartz at 850 K ha.s also been successfully reproduced [12]. The vibrational properties computed with the same potential for these four polymorphs of crystalline silica only approximately reproduce the experimental data [9]. Even better results were reported in Ref. [5] where parameters of the two-body potential Eq. (1) were taken from Ref. [11]. It was found that the calculated static structures of silica polymorphs are in excellent agreement with experiments. In particular, with the pressure - volume equation of state for a -quartz, cristobalite, and stishovite, the pressure-induced amorphization transformation in a -quartz and the thermally induced a — j3 transformation in cristobalite are well reproduced by the model. However, the calculated vibrational spectra were only in fair agreement with experiments. 

The reliability of other statements concerning the structure of the glassy silica surface depends on the reliability of the interatomic potential model and the adequacy of the simulation procedure. We have already mentioned that if the two-body potential of Eq. (2)... 

We start with the two-body Smoluchowski model (2BSM) the details of the formulation (matrix and starting vector) are discussed in Section II.C. A stochastic system made of two spherical rotators in a diffusive (Smoluchowski) regime has been used recently to interpret typical bifurcation phenomena of supercooled organic liquids [40]. In that work it was shown that the presence of a slow body coupled to the solute causes unusual decay behavior that is strongly dependent on the rank of the interaction potential. 

Attempts to represent the three-body interactions for water in terms of an analytic function fitted to ab initio results date back to the work of dementi and Corongiu [191] and Niesar et al. [67]. These authors used about 200 three-body energies computed at the Hartree-Fock level and fitted them to parametrize a simple polarization model in which induced dipoles were generated on each molecule by the electrostatic field of other molecules. Thus, the induction effects were distorted in order to describe the exchange effects. The three-body potentials obtained in this way and their many-body polarization extensions have been used in simulations of liquid water. We know now that the two-body potentials used in that work were insufficiently accurate for a meaningful evaluation of the role of three-body effects. 

Two different forms for the three-body interactions have been used together with the above two-body potential. The original form models the interaction between three atoms i,j, and k according the the following formula ... 

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