Empirical mean-field potentials

A description of nuclear matter as an ideal mixture of protons and neutrons, possibly in (5 equilibrium with electrons and neutrinos, is not sufficient to give a realistic description of dense matter. The account of the interaction between the nucleons can be performed in different ways. For instance we have effective nucleon-nucleon interactions, which reproduce empirical two-nucleon data, e.g. the PARIS and the BONN potential. On the other hand we have effective interactions like the Skyrme interaction, which are able to reproduce nuclear data within the mean-field approximation. The most advanced description is given by the Walecka model, which is based on a relativistic Lagrangian and models the nucleon-nucleon interactions by coupling to effective meson fields. Within the relativistic mean-field approximation, quasi-particles are introduced, which can be parameterized by a self-energy shift and an effective mass. 

The polarization model is extended to account for the ion-ion and ion-surface interactions, not included in the mean field electrical potential. The role of the disorder on the dipole correlation length A, is modeled through an empirical relation, and it is shown that the polarization model reduces to the traditional Poisson Boltzmann formalism (modified to account for additional interactions) when X, becomes sufficiently small. 

It is instructive to compare the performance of empirically derived force fields with those potentials derived from quantum mechanical calculations. Sierka and Sauer <> compared results from Jackson and Catlow s empirical shell model potential with Schroder and Sauer s Hartree-Fock based and their own density functional based shell model potentials. The mean deviation between computed and observed unit cell parameters was found to be 0.7%, 1.9%, and 1.4%, respectively, for the three potentials. This means that the empirical shell model potential is twice as accurate as the best quantum chemically derived force field for unit cell predictions. However, the calculated vibrational spectra of silicalite are in good agreement with experiment for both of the quantum chemically derived potentials,whereas agreement is not as satisfactory for the empirical force field. ... 

The most general force field of a molecule would include anharmonic as well as harmonic terms. However, with the limited experimental information generally available for refining an empirical force field for complex molecules, the harmonic approximation is the only feasible one at present. This means that, for the isolated molecule, we need to know the force constants, Fy, in the quadratic term of the Taylor series expansion of the potential energy, V ... 

Ab-initio MD (AIMD) can usually target systems of few hundreds atoms and time scales of few tens of picoseconds [26-31], In the case of glasses, this means that it is possible to perform an AIMD melt-and-quench simulation of a system around 100-200 atoms, using quenching rates around 20-100 K/ps [31-34]. The small system size in AIMD prevents to extract a full description of medium-range order features such as the network connectivity (NQ of a glass. However, the AIMD models provide the most accurate description of local structure, such as the coordination of key ions, unbiased from any potential bias introduced by an empirical force field. This could be particularly important in order to target the local stmcture... 

This is an empirical equation of the mean-field type, based on the assumption that should be proportional to the local density of monomers. The magnitude of the excluded volume interactions is described by the volume-like parameter Ve, with typical values in the order of 0.01-1 nm. The factor kT is explicitly included, not only for dimensional reasons, but also in order to stress that excluded volume energies, like hard core interactions in general, are of entropic nature (entropic forces are always proportional to T, as is exemplified by the pressure exerted by an ideal gas, or the restoring force in an ideal rubber, to be discussed in a later chapter). If the local potential experienced by a monomer is given by Eq. (2.78), then forces arise for all non-uniform density distributions. For the coil under discussion, forces in radial direction result since everywhere, with the exception of the center at x = 0, we have dcm/d x < 0. The obvious consequence is an expansion of the chain. 

A very drastic simplification to the above-mentioned procedure to obtain the potential energy hypersurface Ui(R) is to consider the nuclei as point masses that evolve under the Newton mechanics within a conservative potential field created by the electrons. Under this classical approach the electrons do not explicitly appear and the only requirement is to have an expression for the force field. This is the theoretical basis of the MM methods. There are many different empirical force fields. They differ in the way the analytical function of the potential energy is defined and what kind of experimental data are used to fit the different parameters. They can be used for evaluating energies for systems of virtually any size so that supramolecular systems can be customarily obtained. Of course MM methods have also some severe limitations the total energy has only a relative meaning as it cannot be compared with other systems that have different number of atoms or a different structure, and the electronic effects are not considered in the MM scheme. 

Thus one must rely on macroscopic theories and empirical adjustments for the determination of potentials of mean force. Such empirical adjustments use free energy data as solubilities, partition coefficients, virial coefficients, phase diagrams, etc., while the frictional terms are derived from diffusion coefficients and macroscopic theories for hydrodynamic interactions. In this whole field of enquiry progress is slow and much work (and thought ) will be needed in the future. 

This Chapter has outlined several different approaches to the computational determination of solution properties. Two of these address solute-solvent interactions directly, either treating the effects of individual solvent molecules upon the solute explicitly or by means of a reaction field due to a continuum model of the solvent. The other procedures establish correlations between properties of interest and certain features of the solute and/or solvent molecules. There are empirical elements in all of these methods, even the seemingly more rigorous ones, such as the parameters in the molecular dynamics/Monte Carlo intermolecular potentials, Eqs. (16) and (17), or in the continuum model s Gcavitation and Gvdw, Eqs. (40) and (41), etc. 

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