Elements Empirical method

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. 

Semi-empirical methods, such as those outlined in Appendix F, use experimental data or the results of ab initio calculations to determine some of the matrix elements or... 

Fj v elements differs from one semi-empirical method to another this topic is covered late... 

In semi-empirical methods, complicated integrals are set equal to parameters that provide the best fit to experimental data, such as enthalpies of formation. Semi-empirical methods are applicable to a wide range of molecules with a virtually limitless number of atoms, and are widely popular. The quality of results is very dependent on using a reasonable set of experimental parameters that have the same values across structures, and so this kind of calculation has been very successful in organic chemistry, where there are just a few different elements and molecular geometries. 

Two philosophies have emerged in connection with the various semi-empirical methods (for a review, see Klopman and Evans, 1976). In both cases certain matrix elements are assumed to be negligible, others are computed, and others are chosen according to some criteria. According to one philosophy, the chosen parameters should lead to agreement with exact Hartree-Fock theory. Then, if desired, correlation can be added in some form. Methods called CNDO and INDO are examples of this. A more recent development is the partial retention of diatomic differential overlap (PRDDO) method (see Estreicher et al., 1989). 

Semi-empirical methods (e.g. CNDO, MNDO or PM3), which had been successfully applied to elements of the second period, failed for the heavier elements. This is probably due to the determination of the parameters for the d-atomic orbitals which turned... 

The Schrodinger equation can also be solved semi-empirically, with much less computational effort than ab initio methods. Prominent semi-empirical methods include MNDO, AMI, and PM3 (Dewar 1977 Dewar etal. 1985 Stewart 1989a Stewart 1989b). The relative computational simplicity of these methods is accompanied, however, by a substantial loss of accuracy (Scott and Radom 1996), which has limited their use in geochemical simulations. Historically, semi-empirical calculations have also been limited by the elements that could be modeled, excluding many transition elements, for example. Semi-empirical calculations have been used to predict Si, S, and Cl isotopic fractionations in molecules (Hanschmaim 1984), and these results are in qualitative agreement with other theoretical approaches and experimental results. 

As presented, the Roothaan SCF process is carried out in a fully ab initio manner in that all one- and two-electron integrals are computed in terms of the specified basis set no experimental data or other input is employed. As described in Appendix F, it is possible to introduce approximations to the coulomb and exchange integrals entering into the Fock matrix elements that permit many of the requisite F, v elements to be evaluated in terms of experimental data or in terms of a small set of fundamental orbital-level coulomb interaction integrals that can be computed in an ab initio manner. This approach forms the basis of so-called semi-empirical methods. Appendix F provides the reader with a brief introduction to such approaches to the electronic structure problem and deals in some detail with the well known Htickel and CNDO- level approximations. 

Semi-empirical methods, such as those outlined in Appendix F, use experimental data or the results of ab initio calculations to determine some of the matrix elements or integrals needed to carry out their procedures. Totally empirical methods attempt to describe the internal electronic energy of a system as a function of geometrical degrees of freedom (e.g., bond lengths and angles) in terms of analytical force fields whose parameters have been determined to fit known experimental data on some class of compounds. Examples of such parameterized force fields were presented in Section III. A of Chapter 16. 

The so-called method of orthogonal operators, mentioned in the Introduction, looks fairly promising in semi-empirical calculations [20,34,134]. Its main advantage is that the addition of new parameters practically does not change the former ones. As a rule this approach allows one to reduce the root-mean-deviation of calculated values from the measured ones by an order of magnitude or so in comparison with the conventional semi-empirical method [135]. Unfortunately it requires the calculation of complex matrix elements of many-electron operators. 

Molecular mechanics is an empirical method based on simple elements of theory that every user can and should understand. With modem software the user is able to control the calculations in terms of the energy minimization routine, the potential energy functions and the force field parameters used. A significant advantage of molecular mechanics calculations is that they are relatively rapid and therefore that large series of calculations may be performed. 

Empirical Methods. The empirical methods use calibration standards to derive sensitivity factors that can be used to determine the unknown concentration of given elements in similar matrices [3. The sensitivity factors are derived from calibration curves that plot measured secondary ion intensities versus the known concentration of standards. Three types of sensitivity factors have been used the absolute sensitivity factor, the relative sensitivity factor, and the indexed relative elemental sensitivity factor. 

Calibration Standards. The empirical methods use calibration standards which are typically from glasses or iron alloys that are chemically doped with elements of known concentrations. The National Bureau of Standards (NBS) supplies several characterized standards of this type. 

Leta and Morrison [82, 83] have described a new empirical method for guantitative SIMS analysis. They use the method of solid-state addition in which they implant their specimens with a known concentration of the element of interest. Since the depth profile of the implanted species has a characteristic Gaussian shape, it is easily distinguished from the element originally present in the specimen. Therefore, the known concentration of the implanted element is used as an internal standard to determine the concentration of the unknown. 

The discovery of new chemical elements - the transactinides or superheavy elements - stimulated the work on theoretical predictions of their chemical properties. Our intention is to present empirical methods [20-35], which can be used to predict chemical properties and which are relevant to gas phase chemical studies of transactinides. 

Up to now, only empirical methods can predict the volatility of compounds. These predictions represent an important part in the multiple step process of the chemical characterization of transactinide elements. The main steps in this process are ... 

A Antisymmetrizing operator A Vector potential P First hyperpolarizability P Resonance parameter in semi-empirical theory B Magnetic field (magnetic induction) X, /r, A, cr Basis functions (atomic orbitals), ab initio or semi-empirical methods rraiipp inrliiflinp basis fiinrHon 7] An infinitesimal scalar rj Absolute hardness h Planck s constant H hjl K h Core or other effective one-electron operator hap Matrix element of a one-electron operator in AO basis Matrix element of a one-electron operator in semi-empirical theory... 

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