Semi-empirical method of model potential

In conclusion of this chapter let us discuss briefly the use of model (effective) potentials. They represent one more variety of simplified theoretical or semi-empirical calculations. Let us notice at once that these methods have a rather narrow domain of applicability. They are most frequently 

Certain semi-empirical methods are aimed at choosing the effective potential, allowing one to determine as accurately as possible values of oscillator strengths of electronic transitions others, more complex, are also directed at achieving the best correspondence between calculated energy spectra and experimental ones. 

Model potential methods and their utilization in atomic structure calculations are reviewed in [139], main attention being paid to analytic effective model potentials in the Coulomb and non-Coulomb approximations, to effective model potentials based on the Thomas-Fermi statistical model of the atom, as well as employing a self-consistent field core potential. Relativistic effects in model potential calculations are discussed there, too. Paper [140] has examples of numerous model potential calculations of various atomic spectroscopic properties. 

The main advantage of the effective potential method consists in the relative simplicity of the calculations, conditioned by the comparatively small number of semi-empirical parameters, as well as the analytical form of the potential and wave functions such methods usually ensure fairly high accuracy of the calculated values of the energy levels and oscillator strengths. However, these methods, as a rule, can be successfully applied only for one- and two-valent atoms and ions. Therefore, the semi-empirical approach of least squares fitting is much more universal and powerful than model potential methods it combines naturally and easily the accounting for relativistic and correlation effects. 

The energy levels and eigenfunctions, obtained in one or other semi-empirical approach, may be successfully used further on to find fairly accurate values of the oscillator strengths, electron transition probabilities, lifetimes of excited states, etc., of atoms and ions [18, 141-144]. 

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In the most elementary models of orbital structure, the quantities that explicitly define the potential V are not computed from first principles as they are in so-called ab initio methods (see Section 6). Rather, either experimental data or results of ab initio calculations are used to determine the parameters in terms of which V is expressed. The resulting empirical or semi-empirical methods discussed below differ in the sophistication used to include electron-electron interactions as well as in the manner experimental data or ab initio computational results are used to specify V. 

The construction of exchange correlation potentials and energies becomes a task for which not much guidance can be obtained from fundamental theory. The form of dependence on the electron density is generally not known and can only to a limited extent be obtained from theoretical considerations. The best one can do is to assume some functional dependence on the density with parameters to satisfy some consistency criteria and to fit calculated results to some model systems for which applications of proper quantum mechanical theory can be used as comparisons. At best this results in some form of ad-hoc semi-empirical method, which may be used with success for simulations of molecular ground state properties, but is certainly not universal. 

Fig. 17. (a) Orientation of molecular dipole moments in 3-methyl-4-nitropyridine. Y-oxide fih molecular dipole moment from direct integration methods fi2, from multipolar model and Hj from semi-empirical calculation, (b) Electrostatic potential around the molecule in the plane of ring atoms. Contours at 0.2kcal/mol (reproduced with permission from Hamazaoui et al. [79]). 

By restricting this article to FP calculations, we exclude the large body of ab initio computational work based on the cluster as a surface model. Other reviews compare both proaches [2]. We likewise do not consider semi-empirical methodologies or pair-potential methods. A very recent comprehensive look at FP calculations in theory and in practice is that of Lindan et al. [6]. Equally, Ref. [7] is a review aimed somewhat at the geological community, discussing atomistic and ab initio methods for modelling minerals. 

The authors also use semi-empirical methods to estimate the equilibrium constants for mononuclear An(lV) hydroxide complexes. Two methods are suggested Model A assumes a linear correlation between the formation constants log,o (ML ) and the ion potential Z/riAn u, while Model B is based on an electrostatic approach described in detail in [2000NEC/KIM]. The model assumes a relationship between the consecutive equilibrium constants of the form ... 

The combination of the discrete variable with the finite element method allows not only to compute atomic data for the hydrogen atom. Atomic data for alkali-metal atoms and alkaU-like ions can be obtained by a suitable phenomenological potential, which mimics the multielectron core. The basic idea of model potentials is to represent the influence between the non-hydrogenic multielectron core and the valence electron by a semi-empirical extension to the Coulomb term, which results in an analytical potential function. The influence of the non-hydrogenic core on the outer electron is represented by an exponential extension to the Coulomb term [18] ... 

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