Empirical parameter

Car and Parrinello [202] proposed a teclmique for efficiently solving the Scluodinger equation which has had an enonuous impact on materials simulation (for reviews, see [203. 204. 205. 206]). The technique is an ab initio one, i.e., free of empirical parameters, and is based on the use of a quantum mechanical orthononual... 

Parameters for elements (basis liinctions in ah miiw methods usually derived from experimental data and empirical parameters in semi-empirical methods nsually obtained from empirical data or ah initu> calcii la lion s) are in depen den t of th e ch em -leal environment, [n contrast, parameters used in molecular mechanics methods often depend on the chem ical en viron-ment. 

Solvents exert their influence on organic reactions through a complicated mixture of all possible types of noncovalent interactions. Chemists have tried to unravel this entanglement and, ideally, want to assess the relative importance of all interactions separately. In a typical approach, a property of a reaction (e.g. its rate or selectivity) is measured in a laige number of different solvents. All these solvents have unique characteristics, quantified by their physical properties (i.e. refractive index, dielectric constant) or empirical parameters (e.g. ET(30)-value, AN). Linear correlations between a reaction property and one or more of these solvent properties (Linear Free Energy Relationships - LFER) reveal which noncovalent interactions are of major importance. The major drawback of this approach lies in the fact that the solvent parameters are often not independent. Alternatively, theoretical models and computer simulations can provide valuable information. Both methods have been applied successfully in studies of the solvent effects on Diels-Alder reactions. 

Any orbital-based scheme can be used for crystal-structure calculations. The trend is toward more accurate methods. Some APW and Green s function methods use empirical parameters, thus edging them toward a semiempirical classification. In order of preference, the commonly used methods are ... 

HyperChem quantum mechanical calculations are ab initio and semi-empirical. Ab initio calculations use parameters (contracted basis functions) associated with shells, such as an s shell, sp shell, etc., or atomic numbers (atoms). Semi-empirical calculations use parameters associated with specific atomic numbers. The concept of atom types is not used in the conventional quantum mechanics methods. Semi-empirical quantum mechanics methods use a rigorous quantum mechanical formulation combined with the use of empirical parameters obtained from comparison with experiment. If parameters are available for the atoms of a given molecule, the ab initio and semi-empirical calculations have an a priori aspect when compared with a molecular mechanics calculation, letting... 

HyperChem includes semi-empirical parameters for all the atoms with atomic numbers less than or equal to 54 and can perform calculations on any molecule composed of only those atoms. The few restrictions or exceptions to this rule will be pointed out as appropriate. Nevertheless, the probability of having appropriate parameters for a random calculation are much higher for semi-empirical calculations than for molecular mechanics calculations. 

We shall presently examine the physical significance of the shift factors, since they quantitatively embody the time-temperature equivalence principle. For the present, however, we shall regard these as purely empirical parameters. The following Ust enumerates some pertinent properties of a ... 

Size-dependent Crystal Growth. A number of empirical expressions correlate the apparent effect of crystal size on growth rate (30). The most commonly used correlation uses three empirical parameters to correlate growth rate with crystal size ... 

Two empirical parameters are evident in equation 7, the heat of vaporization and the integration constant, I. Experimental data indicate that the linear relationship suggested by Clausius-Clapeyron may not be followed over a large temperature range (4) therefore additional adjustable parameters have been added to equation 7 to improve its correlating abiUty. The most prominent of these is the Antoine equation ... 

In addition to qualitative identification of the elements present, XRF can be used to determine quantitative elemental compositions and layer thicknesses of thin films. In quantitative analysis the observed intensities must be corrected for various factors, including the spectral intensity distribution of the incident X rays, fluorescent yields, matrix enhancements and absorptions, etc. Two general methods used for making these corrections are the empirical parameters method and the fimdamen-tal parameters methods. 

The empirical parameters method uses simple mathematical approximation equations, whose coefficients (empirical parameters) are predetermined from the experimental intensities and known compositions and thicknesses of thin-film standards. A large number of standards are needed for the predetermination of the empirical parameters before actual analysis of an unknown is possible. Because of the difficulty in obtaining properly calibrated thin-film standards with either the same composition or thickness as the unknown, the use of the empirical parameters method for the routine XRF analysis of thin films is very limited. 

Z varies from -0.23 to - 0.64 over the heat treatment temperature range 2673 to 3273 K, and the term (e +Z) in Eq. (5) therefore varies from 0 to -0.4 over the same temperature range. Equation (6) was then substituted back into eq. (5) which was used to refit the data to determine the relationship between X and the heat treatment conditions (time and temperature, Eqs. 7 and 8). The empirical parameter X in Eq. (5) was found to be given by... 

Davis, R.H. and Gecol, H., 1994. Hindered settling function with no empirical parameters for disperse suspensions. American Institution of Chemical Engineers Journal, 40, 570-575. 

This is taken to be the atomic valence state ionization energy, invariably written 0)i and treated as an empirical parameter to be determined by fitting an experimental result. 

The CNDO method has been modified by substitution of semiempirical Coulomb integrals similar to those used in the Pariser-Parr-Pople method, and by the introduction of a new empirical parameter to differentiate resonance integrals between a orbitals and tt orbitals. The CNDO method with this change in parameterization is extended to the calculation of electronic spectra and applied to the isoelectronic compounds benzene, pyridine, pyri-dazine, pyrimidine and pyrazine. The results obtained were refined by a limited Cl calculation, and compared with the best available experimental data. It was found that the agreement was quite satisfactory for both the n TT and n tt singlet transitions. The relative energies of the tt and the lone pair orbitals in pyridine and the diazines are compared and an explanation proposed for the observed orders. Also, the nature of the lone pairs in these compounds is discussed. 

Regarding current ab initio calculations it is probably fair to say that they are not really ab initio in every respect since they incorporate many empirical parameters. For example, a standard HF/6-31G calculation would generally be called "ab initio", but all the exponents and contraction coefficients in the basis set are selected by fitting to experimental data. Some say that this feature is one of the main reasons for the success of the Pople basis sets. Because they have been fit to real data these basis sets, not surprisingly, are good at reproducing real data. This is said to occur because the basis set incorporates systematical errors that to a large extent cancel the systematical errors in the Hartree-Fock approach. These features are of course not limited to the Pople sets. Any basis set with fixed exponent and/or contraction coefficients have at some point been adjusted to fit some data. Clearly it becomes rather difficult to demarcate sharply between so-called ab initio and semi-empirical methods.4... 

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