Two-body potential

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. 

In his early survey of computer experiments in materials science , Beeler (1970), in the book chapter already cited, divides such experiments into four categories. One is the Monte Carlo approach. The second is the dynamic approach (today usually named molecular dynamics), in which a finite system of N particles (usually atoms) is treated by setting up 3A equations of motion which are coupled through an assumed two-body potential, and the set of 3A differential equations is then solved numerically on a computer to give the space trajectories and velocities of all particles as function of successive time steps. The third is what Beeler called the variational approach, used to establish equilibrium configurations of atoms in (for instance) a crystal dislocation and also to establish what happens to the atoms when the defect moves each atom is moved in turn, one at a time, in a self-consistent iterative process, until the total energy of the system is minimised. The fourth category of computer experiment is what Beeler called a pattern development... 

In the case of micellar solutions, studied in this work, the monomers interact via two-body potentials. The non-bonded particles interact via the repulsive part of a Lennard-Jones potential... 

For better comparison, these values have been calculated using the two-body potential calibrated with the experimental AI2 data (42,46) (Dg 1.55 eV and rg... 

The experimental second and third virial coefficients for steam are however widely available. But these experimental quantities should be used with more care than has been usual in the past. The prevailing notion asserts that a good two-body potential should yield the second virial in full agreement with the exper-... 

The remaining four energy contributions depend on the reference density only. This is an important observation, which allows to combine these contribution into the so called repulsive energy term Erep pu, which is treated in a simplified way by approximating it by a sum of two-body potentials [44], Erep[po] = Hap Uap Rap)... 

ViJ interaction energy between atoms i and j (two-body potential)... 

After this computer experiment, a great number of papers followed. Some of them attempted to simulate with the ab-initio data the properties of the ion in solution at room temperature [76,77], others [78] attempted to determine, via Monte Carlo simulations, the free energy, enthalpy and entropy for the reaction (24). The discrepancy between experimental and simulated data was rationalized in terms of the inadequacy of a two-body potential to represent correctly the n-body system. In addition, the radial distribution function for the Li+(H20)6 cluster showed [78] only one maximum, pointing out that the six water molecules are in the first hydration shell of the ion. The Monte Carlo simulation [77] for the system Li+(H20)2oo predicted five water molecules in the first hydration shell. A subsequent MD simulation [79] of a system composed of one Li+ ion and 343 water molecules at T=298 K, with periodic boundary conditions, yielded... 

One assumes that a mixture A + B with mole fractions xA, xB behaves like a pure component with the average two-body potential ... 

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. 

If two-body potentials and the three-body contribution of Li+(H20)2 are taken into account the optimum coordination number for a static Li+(H20)n complex turns out to be 4. For the most stable conformation of Li+(H20)6 they found that two water molecules are bound in a second, outer hydration layer. 

Although experimental evidence seems to support these results, the fact that the optimum coordination number obtained depended to a large extent on the use of three body contributions warns against uncritical belief in the convergency of the expansion [Eq. (49)] and stresses the preliminary nature of the conclusions. When two-body potentials were considered exclusively, Kollman and Kuntz 219>... 

There are many empirical and semi-empirical pair potentials which describe quite satisfactorily the properties of liquids and solids, see chapter 5 in book The parameters in these potentials are not real parameters of a true two-body interaction, their values depend upon properties of a medium. So these effective two-body potentials include nonadditive interactions through their parameters. The latter can not be directly related to the definite physical... 

The ultimate space-based approach is to explore systematically every possible spatial arrangement of the atoms in the formula unit and to determine which has the lowest energy. The energy may be calculated using quantum mechanics, but it is more usual in complex solids to use two-body potentials where the... 

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... 

To compile quantitatively reliable information, we need a source of experimental measurements. One way to determine the nature of inter-molecular forces between biopolymer molecules in a solvent medium is to measure the so-called osmotic second virial coefficient A2. Expressed in molar (biopolymer) terms, the quantity A2 can be related to the two-body potential of mean force W(r) by the following equation (Vrij, 1976 de Kruif, 1999 Prausnitz, 2003 de Kruif and Tuinier, 2005) ... 

We see that there are two possible limits for H2 + 0- We take the one of lower energy as appropriate and this will depend on the H—H separation. The D—separation of the oxygen atom is 1.958 eV and calculations by Kolos and Wolniewicz 136) show that the singlet-triplet separation of H2 is equal to this at 1.669 A. Thus the two-body potential that we use in (53) will be the singlet state potential for... 

One source of information on intermolecular potentials is gas phase virial coefficient and viscosity data. The usual procedure is to postulate some two-body potential involving 2 or 3 parameters and then to determine these parameters by fitting the experimental data. Unfortunately, this data for carbon monoxide and nitrogen can be adequately represented by spherically symmetric potentials such as the Lennard-Jones (6-12) potential.48 That is, this data is not very sensitive to the orientational-dependent forces between two carbon monoxide or nitrogen molecules. These forces actually exist, however, and are responsible for the behavior of the correlation functions and - In the gas phase, where orientational forces are relatively unimportant, these functions resemble those in Figure 6. On the other hand, in the liquid these functions behave quite differently and resemble those in Figures 7 and 8. 

For a full discussion we must include momentum dependent interactions. For instance, starting from a two-body potential of finite range we can expand the Fourier transform (q2) as... 

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 accurate parameterization of the effective core potential has shown that the reduction of the pseudopotential to the form of a one-particle operator is adequate. The scaling of the two-body potentials by the use of an operator65... 

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