Potential parameters empirical fitting

The interatomic potentials define the force field parameters that contribute to the lattice energy of a relaxed or energy minimized structure. The fundamental question is how reliable is a force field The force field used in evaluating a potential function must be consistent and widely applicable to all similar systems. It must be able to predict the crystal properties as measured experimentally. Two main approaches, namely empirical and semi-empirical, are usually employed in the derivation of potential parameters. Empirical derivations involve a least square fitting routine where parameters are chosen such that the results achieve the best correlation with the observed properties. The semi-empirical approach uses an approximate formulation of the quantum mechanical calculations. An example of such an approximation is the electron gas method [57] which treats the electron density at any point as a uniform electron gas. The following is the analytical description of the potential energy function and interatomic potentials we recommend for use in simulation of zeolites and related system. 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

The virial coefficients B(T), C(T), D(T),... are functions of temperature only. Although these coefficients might be treated simply as empirical fitting parameters, they are actually deeply connected to the theory of intermolecular clustering, as developed by J. E. Mayer (Sidebar 13.5). More specifically, the second virial coefficient B(T) is related to the intermolecular potential for pairs of molecules, the third virial coefficient C(T) to that for triples of molecules, and so forth. For example, knowledge of the intermolecular pair potential V(R) (see Sidebar 2.8) allows B T) to be explicitly evaluated by statistical mechanical methods as... 

Fitting of model potentials to supermolecular interactions like in Eq. (3-1) has its disadvantages the calculations have to be repeated many times and a predefined analytical expression of the model potentials in terms of atomic parameters is required. These parameters are not easily transferred to other situations. The internal state of, e.g., system A depends on the presence of another system X, and this will be different near system Y. Furthermore, the chosen analytical form of the potential may give rise to problems and errors. For example, Hartree-Fock (HF) calculations fitted to a power series in 1/r suggest that the 1/r6 terms have to do with dispersion, which is not part of the HF energy. Finally, such empirical potentials are best for describing situations close to those to which the parameters are fitted. Whenever the situation is very different from that, the results will be doubtful. 

Before we leave this topic, it is well to remember that we have confined our attention to the leading contribution in the transformed Hamiltonian in (7.247). If the transformation operator is not small, we may have to include higher-order effects from the second and subsequent pairs of braces in this equation. This corresponds to retention of the terms in y2 in equation (7.241). Furthermore, the commutator [F. in the second brace has the potential to modify the parameters and forms of all the terms in 3( - If a particular choice is made for, V, as say in equations (7.250) or (7.253), parameters determined in the empirical fit will also be modified. This modification must be taken into account if the parameters so obtained are to be used to provide... 

From the scientific point of view, however, all approaches in the sense of the Kd concept (Henry, Freundlich or Langmuir isotherm) are unsatisfactory, since the complex processes on surfaces cannot be described by empirical fitting parameters. Boundary conditions like pH value, redox potential, ionic strength, competition reactions for binding sites are not considered. Thus results from laboratory and field experiments are not transferable to real systems. They are only advisable to provide a suitable prognosis model, if no changes concerning boundary conditions are to be expected and if no parameters for deterministic or mechanistic approach can be determined. 

This is undertaken by two procedures first, empirical methods, in which variable parameters are adjusted, generally via a least squares fitting procedure to observed crystal properties. The latter must include the crystal structure (and the procedure of fitting to the structure has normally been achieved by minimizing the calculated forces acting on the atoms at their observed positions in the unit cell). Elastic constants should, where available, be included and dielectric properties are required to parameterize the shell model constants. Phonon dispersion curves provide valuable information on interatomic forces and force constant models (in which the variable parameters are first and second derivatives of the potential) are commonly fitted to lattice dynamical data. This has been less common in the fitting of parameters in potential models, which are onr present concern as they are required for subsequent use in simulations. However, empirically derived potential models should always be tested against phonon dispersion curves when the latter are available. 

The experimental results are compared with theoretical calculations using strictly empirical as well as Sato potential energy surfaces and a transition state theory formulation of the rate coefficient ratio. No one set of potential parameters could be found to fit all the data but it is not possible to attribute the deviations either to experimental error or the theory. 

The first term represehts the repulsive branch and the second term represents the attractive branch of the interaction potential between two atoms. By performing a non linear least square fit procedure the parameters (Al, ai, Ai A2, a2, A2) of the empirical pair potential are determined. In the fit procedure we have used the binding energy values of Au-dimer calculated at various interatomic distances by RDFT. The estimated points by RDFT and the fitted function are shown in Fig. 1. The potential parameters for the gold dimer interaction are determined as Ai = 1222.86345, A2 = -3.93623329, Ai = 2.94056151, A2 = -1.30223862, ai = 0.806351693, 2 = 0.216139972. In these parameters energy is in eV, and distance is in... 

The atom-atom potential fitted to the ab initio data gives fairly realistic results for the equilibrium structure (unit cell parameters and molecular orientations in the cell), the cohesion energy and the phonon frequencies of the molecular crystal. The latter have been obtained via both a harmonic and a self-consistent phonon lattice dynamics calculation and they were compared with IR and Raman spectra. About some of the empirical hydrocarbon atom-atom potentials , which are fitted to the crystal data, we can say that they correspond reasonably well with the ab initio results (see figs. 6, 7,8), their main defect being an underestimate of the electrostatic multipole-multipole interactions. 

As noted in Chapter 1, there are two principal methods by which potential parameters may be obtained (i) empirical fitting and (ii) direct calculation. Empirical fitting will be described in more detail here, as direct calculation by electronic structure methods will be described in Chapter 8. A comparison of the strengths and weaknesses of each method is given following the description of the methods. 

A molecular dynamics calculation was performed for thorium mononitride ThN(cr) in the temperature range from 300 to 2800 K to evaluate the thermophysical properties, viz. the lattice parameter, linear thermal expansion coefficient, compressibility, heat capacity (C° ), and thermal conductivity. A Morse-type function added to the Busing-Ida type potential was employed as the potential function for interatomic interactions. The interatomic potential parameters were semi-empirically determined by fitting to the experimental variation of the lattice parameter with temperature. 

Completely empirical treatments, in which the potential energy surfaces of singlet and triplet states are represented as a function of potential parameters fitted to the available experimental information (equilibrium geometries, vibrational frequencies), have had a considerable success for molecules for which a localized electronic description is applicable -io). In the case of conjugated molecules, the important delocalization of the tc electrons introduces difficulties in such treatments and the two following approaches appear preferable ... 

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