Non-rigorous methods

A recent comparison (33.) of quantities calculated from molecular electronic wave functions for pyrrole and pyrazole indicated that ARCANA values compared more favorably with large-scale ab initio values than those calculated by other non-rigorous methods in the case of orbital energies of occupied molecular orbitals, gross atomic populations of heteroatoms, and total overlap populations including negative overlap populations between nonbonded atoms. 

Allow for the collection of the greatest amount of data with the least expenditure of time and money, using rapid, non-rigorous methods of analysis to make quick, preliminary assessments of types and levels of pollutants. 

Lyapunov had uttered these words in connection with Poincare s investigation where he had applied some non-rigorous methods to a problem on the stability of the equilibrium shapes of rotating fluids, and which had led him to a wrong conclusion in favor of pear shapes. A more rigorous analysis conducted by Lyapunov had revealed that the pear shape is unstable. Lyapunov s proof was later confirmed by Cartan. 

A more complete and much more rigorous description of bonding in complexes would be provided by a quantum mechanical treatment. Such a treatment is especially needed in the case of departures from the ionic model and increasing contribution of covalent bonding (ion pairs, soft donors and acceptors). However only a few studies have been reported. They are mainly concerned with cation hydration and use either semi-empirical 19—21) or non-empirical methods 22—24). A non-empirical treatment of cation NH3 systems has also been performed recently (25). However the present state of the computations is still far from providing a complete description of the system including the medium. The latter may be taken into account by a Bom-type "solvaton (27,26). Heats of hydration may then be calculated (27). A discussion of this aspect of the problem is deferred to a later date, awaiting especially a more complete analysis of non-empirical calculations. In the course of the discussion of... 

Attempts have been made to calculate the mentioned forces more quantitatively. W. Simon and coworkers195) have been successful in representing their, and other, data by semi-empirical model, so that a reasonable quantitative account of the various reported phenomena can be presented in terms of molecular structure data. To date, little work has been done to apply more rigorous, non-empirical methods of quantum chemistry to problems other than solvation67,157,261,269). 

Before beginning our discussion of wave function-based electronic structure theory, we note that an alternative, rigorous approach to electronic structure is provided by DFT (this volume, chapter by Ayers and Yang). DFT is based on the premise that all information about the electronic system can be extracted from the electron density, rather than from the electronic wave function. The attraction of DFT is that the electron density is a much simpler entity than the wave function, depending on just three spatial coordinates rather than on the An spatial and spin coordinates of n electrons. However, a difficulty of DFT is that no accurate, non-empirical method has yet been devised to extract the necessary information from the electron density. Current DFT calculations are therefore, to a large extent, based on semiempirical functionals [12], in which a set of parameters is fitted to experimental data. Nevertheless, the fitted parameters are universal in the sense that they are not atom-dependent or molecule-dependent. Also, the accuracy achieved in this manner is often high, surpassed only by the most elaborate wave function methods [13]. 

Whilst this argument mirrors simple textbook derivations for nonrelativistic problems, for example [73, 32], such a non-rigorous treatment is not enough for relativistic calculations. We have already seen in Section 1 that early attempts to solve Dirac s equation by matrix methods encountered unexpected... 

At the beginning of this section we presented a non-rigorous version of the Rayleigh-Ritz method for Dirac Hamiltonians. It is now time to examine the conditions under which this can be justified rigorously as a foundation for practical calculations with the DHF(B) equations. Our presentation here is based on [76,86,87] which give much more detail. 

Thus, the non-negativity of solution is not an established constraint in the theoretical foundation of linear methods. On the other hand, the empirically formulated non-linear methods [Eqs. (55-56)] effectively secure positive and stable solutions. Such a weakness of the rigorous linear methods indicates a possible inadequacy in criteria employed for formulating the optimum solutions. In Section 6 we discuss possible revisions in assumptions employed for accounting for random noise in inversions. For example, it will be shown that by using log-normal noise assumptions the non-negativity constraints can be imposed into inversion in a fashion consistent with the presented approach inasmuch as one considers the solution as a noise optimization procedure. 

It is legitimate to ask whether, in the case of a calibration plot whose curvature is not too severe, we could take the spline idea to its simplest conclusion, and plot the curve as a series of straight lines joining successive points. This method is of course entirely non-rigorous, and would not provide any information on the precision with which x-values can be determined. However, its value as a simple initial data analysis (IDA) method (see Chapter 6) is indicated by applying it to the data in the above example. For y-values of 5, 16 and 27 this method of linear interpolation between... 

As mentioned earlier (see p. 122) the previously postulated dioxetane intermediate in firefly bioluminescence has been challenged as no 180 is in-corporated in the carbon dioxide released during oxidation of firefly luciferin with 18C>2. In view of the crucial significance of the 180. experiments De Luca and Dempsey 202> rigorously examined the reliability of their tracer method. They conclude from their experiments that all available evidence is in favour of a linear, not a cyclic peroxide intermediate — in contrast to Cypridina bioluminescence where at least part of the reaction proceeds via a cyclic peroxide (dioxetane) as concluded from the incorporation of 180 into the carbon dioxide evolved 202,203). However, the dioxetane intermediate is not absolutely excluded as there is the possibility of a non-chemiluminescent hydrolytic cleavage of the four-membered ring 204>. 

Sample-standard comparison is more applicable in MC-ICP-MS, in which instrument mass fractionation is fundamentally a steady state phenomenon (Marechal et al. 1999). This method has been used successfully for some non-traditional stable isotopes, particularly involving Fe, in which analyses of samples are bracketed by standards to cope with systematic instrumental drift (e.g., Zhu et al. 2002 Beard et al. 2003). However, other methods have been used for Mo stable isotope work published to date because of concerns about non-systematic changes in instrument mass fractionation, particularly arising from differences in matrices, between samples and standards. Such concerns are more acute for Mo than for Fe and many other elements because Mo is a trace constituent of most samples, increasing the challenge of rigorous, high-yield sample purification. 

---