Empirical fit

We have alluded to the comrection between the molecular PES and the spectroscopic Hamiltonian. These are two very different representations of the molecular Hamiltonian, yet both are supposed to describe the same molecular dynamics. Furthemrore, the PES often is obtained via ab initio quairtum mechanical calculations while the spectroscopic Hamiltonian is most often obtained by an empirical fit to an experimental spectrum. Is there a direct link between these two seemingly very different ways of apprehending the molecular Hamiltonian and dynamics And if so, how consistent are these two distinct ways of viewing the molecule ... 

Eurthemiore, the actual Hamiltonians obtained very closely match tliose obtained via the empirical fitting of spectra This consistency lends great confidence that both approaches are complementary, mutually consistent ways of apprehendmg real infomiation on molecules and their internal dynamics. 

In this project, we shall predict the wavelength of the absorption maxima of the same four polyenes using the calculated difference (in units of eV), between the LUMO and HOMO of these four molecules (Fig. 8-6). Bear in mind that this is not an ab initio calculation of wavelengths of maximum absorption, because empirically fitted parameters, Yio exist within the program or are... 

Other techniques that work well on small computers are based on the molecules topology or indices from graph theory. These fields of mathematics classify and quantify systems of interconnected points, which correspond well to atoms and bonds between them. Indices can be defined to quantify whether the system is linear or has many cyclic groups or cross links. Properties can be empirically fitted to these indices. Topological and group theory indices are also combined with group additivity techniques or used as QSPR descriptors. 

Plume rise observations based on single-stack operation were regressed into the above expression and empirically fitted to the following expression, which incorporates atmospheric stability classes into the coefficients ... 

Tlie suffices i and J refer to individual atoms and S and Sj to the species of the atoms involved. The summation over j extends over those neighbors of the atom i for which ry, the separation of atoms i and J, is within the cutoff radii of these potentials. The second term in Equation (la) is the attractive many-body term and both V and are empirically fitted pair potentials. A Justification for the square root form of the many-body function is provided in the framework of a second moment approximation of the density of states to the tight-binding theory incorporating local charge conservation in this framework the potentials represent squares of the hopping integrals (Ackland, et al. 1988). 

As outlined in Section III.A, knowledge of the molecular wavefunction implies knowledge of the electron distribution. By setting a threshold value for this function, the molecular boundaries can be established, and the path is open to a definition of molecular shape. A quicker, but quite effective, approach to this entity is taken by assuming that each atom in a molecule contributes an electron sphere, and that the overall shape of a molecular object results from interpenetration of these spheres. The necessary radii can be obtained by working backwards from the results of MO calculations, or from some kind of empirical fitting ... 

Infrared spectra are straightforward to predict theoretically, demanding development of a force field (FF) to determine frequencies and dipole derivatives for intensities. These parameters were initially obtained using empirically fitted force constants and simple models for transition dipoles (Krimm and Bandekar, 1986 Torii and Tasumi, 1996). 

In the last few years, there has been great progress in the more a-priori method of evolutionary population synthesis , pioneered by Tinsley (1968), using the same stellar libraries as in the empirical fitting, and calculating the spectral evolution of single stellar populations (SSP) as a function of age and metallicity. One problem... 

A simple way to bypass the vagaries of empirical fits is to define an alternative, purely theoretical electronegativity scale based on a proposed functional relationship between /ah and jca, analogous, e.g., to Eq. (3.57) or Eq. (3.58). A particularly simple and attractive one-parameter exponential relationship is... 

Coulson concluded that the most important contribution to H-bonding is ionic resonance (5.29a). However, generations of empirical modelers have found it convenient to employ simple pairwise-additive Coulombic formulas with empirically fitted point charges to model H-bonds, and such empirical models have tended to encourage uncritical belief in the adequacy of a classical electrostatic picture of H-bonding. 

The MQH correlation for the layer temperature rise has found the empirical fit to data ... 

Various attempts have been made to increase the valid range of the Debye- Huckel equation to regions of high ionic strength by the use of empirically fitted parameters. Several of these equations are listed in table I. 

In view of the difficulties discussed in Section 2 it seems that many of the more important equilibria of relevance to power station operation will not be directly measurable. It is certain, therefore, that great emphasis will have to be placed on methods of estimating high temperature data. It also seems clear that, if these are to be checked up to 350 C, a variety of experimental techniques may well prove necessary to sort out usable thermodynamic data from experiments which, on their own, cannot give them. Alternatively, if estimation procedures can be developed which are substantially free from empirical fitting parameters, they may not require extensive checking. 

In transforming the independent variables alone, it is assumed that the dependent variable already has all the properties desired of it. For example, if the /s are normally and independently distributed with constant variance, at least approximately, then any transformations such as described in Section VI,B,1 would be unnecessary. Under such assumptions, Box and Tidwell (B17) have shown how to transform the independent variables to reduce a fitted linear function to its simplest form. For example, a function that has been empirically fitted by... 

Adsorption from liquids is less well understood than adsorption from gases. In principle the equations derived for gases ought to be applicable to liquid systems, except when capillary condensation is occurring. In practice, some offer an empirical fit of the equilibrium data. One of the most popular adsorption isotherm equations used for liquids was proposed by Freundlich 21-1 in 1926. Arising from a study of the adsorption of organic compounds from aqueous solutions on to charcoal, it was shown that the data could be correlated by an equation of the form ... 

Selected values of (1000 A)1/2 for quartz = mineral lsO, and pyrite = mineral 34S equilibria are found in Table 9.2. In other cases, especially at lower temperature, the temperature dependence of IE s on mineral formation may be more complicated and require empirical fits of the form, in a = Ci/T + C2/T2 or in a = Co + Ci/T + C2/T2, the C s are empirical least squares fitting parameters. 

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