Empirical schemes

Nevertheless, the increasing demands for models which can do more in terms of exploring PE surfaces continues to provide an impetus for the inclusion of TM atoms into the more sophisticated semi-empirical schemes. While the general performance of the models (vide infra) is encouraging, it is equally apparent that a significant amount of work remains before such methods enjoy the same level of accuracy for TM systems as they currently achieve for molecules comprising lighter, non-TM atoms. 

Whatever terms are included in the so-called Force Field (FF), optimal values for the various terms describing each energy interaction must be derived a priori. Usually, this is done empirically by fitting computed and experimental data although the results of ab initio calculations can be used to parameterise the FF. Naturally, the computed quantities depend critically on the FF parameters but the pay off is a method which is orders of magnitude faster than even semi-empirical methods. It is not surprising, therefore, that MM finds widespread application in biochemistry where whole protein molecules comprising thousands of atoms can be modelled. 

---

The principal semi-empirical schemes usually involve one of two approaches. The first uses an effective one-electron Hamiltonian, where the Hamiltonian matrix elements are given empirical or semi-empirical values to try to correlate the results of calculations with experiment, but no specified and clear mathematical derivation of the explicit form of this one-electron Hamiltonian is available beyond that given above. The extended Hiickel calculations are of this type. 

Various empirical schemes have also been proposed as predictive tools with respect to the outcome of radical addition reactions.9193 Two-parameter schemes, including the Q-e scheme (Section 7.3.4.1), Patterns of Reactivity (Section 7.3.4.2)... 

This database supersedes those in Cox and Pilcher [54], Pedley 77 [45], and Pedley 86 [38]. An empirical scheme, developed by the author, to estimate enthalpies of formation of organic compounds in gas and condensed phases, is also described. 

For the nitrogen hyperfine tensors, there is no satisfactory empirical scheme for estimating the various contributions, so that Table II compares the total observed tensor to the DSW result. The tensors are given in their principal axis system, with perpendicular to the plane of the heme and along the Cu-N bond. The small values (0.1 - 0.2 MHz) found for A O in the nonrelativistic limit are not a consequence of orbital motion (which must vanish in this limit) but are the result of inaccuracies in the decomposition of the total tensor into its components, as described above. 

The interactions between the components that make up a photopolymer are extremely important in arriving at a working formulation. Here we show that inclusion of pyrrolidone derivatives like NVP or NMP in acrylate systems enhances the ambient cure of a film. From the reactivity parameters of some simple systems we have derived an empirical scheme for the formulation of fully and/or partially reactive systems based on the molar equivalent ratios of the acrylate to pyrrolidone components. The data support the presence of a synergistic effect between NVP and the acrylate components. 

The success of earlier empirical schemes in predicting the sign and magnitude of the enthalpy of formation can in part be attributed to making predictions for AB compounds. This can hide asymmetrical effects which are more apparent when other stoichiometries are considered. The empirical and semi-empirical schemes do... 

Table 6-8 shows values of the various parameters needed to calculate monomer reactivity ratios from Eqs. 6-60 and 6-62 [Jenkins and Jenkins, 1999]. The monomers in Table 6-8 are lined up in order of their u values. The Patterns of Reactivity scheme, like the Q e. scheme, is an empirical scheme. Monomer reactivity ratios calculated by the patterns of reactivity scheme are generally closer to experimental values than those calculated by the Q e scheme, which supports the rationale of assigning different polarity values to a monomer and the radical derived from the monomer. 

We see that the simple rectangular d band model reproduces the behaviour found by experiment and predicted by Miedema s semi-empirical scheme. However, we must stress that the model does not give credence to any theory that bases the heat of formation of transition-metal alloys on ionic Madelung contributions that arise from electronegativity differences between the constituent atoms because in the metallic state the atoms are perfectly screened and, hence, locally charge neutral. Instead, the model supports... 

Given the somewhat ad hoc nature of most specific schemes for evaluating the non-electrostatic components of the solvation free energy, a reliance on a simpler, if somewhat more empirical, scheme has become widely accepted within available continuum models. In essence, the more empirical approach assumes that the free energy associated with the non-electrostatic solvation of any atom will be characteristic for that atom (or group) and proportional to its solvent-exposed surface area. Thus, the molecular Geos may be computed simply as... 

A series of empirical schemes targeting the assembly of ternary complexes on beads have been attempted with varying degrees of success (reviewed in Simons et al, 2005). In Fig. 1, the molecular assemblies that are the subject of this chapter are based on the tethering of G protein heterotrimers to beads by an epitope tag (either hexahistidine or FLAGt ) fused to the N-terminus of the y subunit, and projected away from its binding partners in three-dimensional reconstructions. 

The authors of this work proposed a semi-empirical scheme for the calculation of 13C chemical shift tensors based on the bond polarization theory (5). This method can reproduce 13C chemical shift tensors with deviations from experiment comparable to the errors of the ab initio methods. One major advantage is that the calculations can be performed for large molecular systems with hundreds of atoms even on a PC computer. In contrast to the ab initio method a set of empirical parameters is needed for the calculations. In the case of the bond polarization theory these parameters can be estimated directly from experimental chemical shifts solving a set of linear equations. 

Given that this revised calibration scheme together with the correction of the weighted mean calculation are nearly identical (< 150 m difference) to the previous model results all of the comparisons of the modern isotopic compositions of precipitation from other regions are effectively unchanged and hence these comparisons demonstrates that the model yields quite reasonable fits without adjustment (see Rowley et al. 2001 and Rowley and Garzione 2007). However it should always be made clear that the empirical scheme to model precipitation from condensate does not represent the microphysics of water droplet formation, coalescence,... 

The methods available for computing enthalpies of formation fall into two general groups those based on purely empirical schemes and those founded on quantum chemistry. The quantum chemical methods can be further divided into three types ab initio molecular orbital theory, density functional theory, and semiempirical molecular orbital theory. A summary of the types of method used to calculate enthalpies of formation is given in Table 2, along with some specific examples. This is not meant to be a comprehensive tabulation, but rather a list of some of the more popular approaches in use today. Table 3 names some of the commercially available computer programs having capabilities to calculate thermochemical data. 

The results obtained by Menon et al. (2003) revealed substantial differences in the use of physically substantiated prognostic schemes of how aerosol acts taking into account vertical velocity and empirical schemes based on diagnostic data on the vertical velocity at cloud bottom level. Prognostic schemes are characterized by the stronger variability of results compared with diagnostic ones because of differences in scheme of the interaction between the processes of aerosol activation and precipitation of drizzle when calculating N. 

The advantages of SQMF however can be understood only in comparison with the purely empirical schemes for normal mode calculations. Despite of their wide diversity VFF is always present as a compulsory element, as far as compounds with well pronounced covalent bonding are considered (isolated molecules, molecular crystals, polymers, etc.). In the context of the SQMF technique VFF is also an important model from a conceptual point of view. For this reason in the Introduction we will consider the main ideas which underlie the VFF model. The SQMF method itself will be considered in the following sections. 

For the purpose of theoretical (quantum-mechanical) study of the H02 radical s ability to detach two hydrogen atoms from substrate molecule in one stage, in the framework of non-empirical scheme [28] and on the basis STO-3G the heat effect of the following model reaction was calculated ... 

The fundamental reasons for the difficulties faced by the MM methods when metal (both transition and nontransition) complexes are involved can be understood if one does not consider the MM as a purely empirical scheme (as it is frequently done), but think about them as of some reflection of specific features of molecular electronic structure, formalized by the form of the trial wave function of that class of compounds where such a parameterization might be possible. As shown in Chapter 3, organic compounds for which the MM methods are known to demonstrate significant successes can be described by the QC method, which directly leads to local and transferable two-center bonds. It is shown in Chapter 3 that the derivation of the MM method from the QC description is possible due to a common background of the MM and SLG description, which consists in the physical presence of two-center, two-electron bonds in organic molecules (in strict terms of Section 1.7 - numbers of electrons in each of the geminals weakly fluctuate). 

There is no quantum-mechanical guidance to elucidate the relative stability of different nuclides or define the number of naturally occurring stable elements and their isotopes. Nuclear periodicity in terms of the Elsasser magic numbers remains an empirical scheme. 

A convenient way to classify solvents of chromatographic interest in terms of their polarity and the specific chemical interactions is the empirical scheme proposed by Snyder [214,215]. This scheme is based on experimental (gas chromatographic) distribution coefficients for three test solutes ( probes ) on a large number of stationary phases, which were published by Rohrschneider [216]. The probe compounds are ethanol (e), 1,4-dioxane (d) and nitromethane (n). The experimental values for the distribution coefficients undergo several empirical modifications ... 

In the present paper, we have discussed the ab-initio evaluation of the static polarizabilities and magnetizabilities of molecular systems, with emphasis on the principles underlying such calculations. With the recent widespread availability of powerful computers, these second-order molecular properties may nowadays be calculated a priori for large molecular systems, allowing us to explore, for instance, the relationship between the properties and molecular structure. Such calculations complement the experimental work in the area and may help in reassessing and improving on the empirical schemes... 

In addition to the chemistry that may be addressed, ab-initio calculations are needed to provide important guidance on which to develop semi-empirical and empirical schemes that use, in fact, the same formalism. There is an important interplay between all methods, and the information that we gain from ab-initio calculations is not only used to examine the chemistry of the model systems, but also used to help develop semi-empirical methods. The numbers obtained are often used to guide parameter choices where these are not available from experiment. Most importantly, discoveries in the underlying theory that we use are made in semi- empirical, density functional and ab-initio studies, and these improvements are adopted, perhaps in modified form, to affect all theories used in computational chemistry. 

Some cases found empirically by Hall to fall in the zwitterionic category were redesignated as diradicals through these calculational findings, but the overall trends of the empirical scheme were duplicated. 

Current interest in high temperature chemistry and the closely related thermodynamics of the actinides will provide additional stimuli for determining precise thermodynamic data in cryogenic as well as in higher temperature regions. The utopian era in the chemical thermodynamics of the lanthanides is sufficiently far off to occasion extension of shrewdly devised schemes to other classes of compounds. Use of the semi-empirical schemes already discussed—or theoretically based ones— plus the key compound concept may prove as effective here (desipte magnetic and electronic complications) as it has for hydrocarbon thermodynamics. 

The question about these empirical schemes is whether they can be generalized or automated in any way. At the moment it seems that for each specific material, the process needs to done from scratch, involving a lot of human interaction and judgement. Also with the diversity of models and empirical schemes, parameterization is strongly model dependent - the bead size and interaction parameters in MesoDyn, the bond-fluctuation or the fee lattice chain model of the same system will all be different. The situation hence seems confusing. Any emerging efforts aimed at some more coherence in this area would be beneficial. 

Some attempts to employ much less rigorous empirical schemes in the estimation of the electronic distribution for the considered structure and its correlation with spectral parameters such as chemical shifts were suggested by Bangov. The charge densities of each candidate structure may be calculated by a fast empirical scheme based on either full or partial equalization of orbital or atomicelectronegativity and further correlated with the chemical shifts from the spectrum of the query structure. The structure providing the best correlation is considered to be the correct solution for the unknown compound. However, this approach, as subsequently discussed, also yields rather vague results. 

For cyclic jr-electron hydrocarbons two more schemes, introduced by Clar (1952) and Platt (1949), are widely used. Clar s empirical scheme is based on the appearance of the absorption bands and designates the first three bands in the spectrum of phenanthrene as the a, p, and j3 bands, re-... 

As the name implies, semi-empirical calculations simplify the computational problem presented by the Schrodinger equation. Semi-empirical approaches eliminate some of the direct number crunching involved in ab initio calculations and instead replace portions of the calculation with values that may be taken from experimental data or other calculations that are parameterized to agree with empirical data. There is a variety of semi-empirical schemes that differ in the types of parameterizations that are made. Three common semi-empirical methods that are included in Spartan02 are called AMI, PM3, and MNDO. Each has strengths and weaknesses depending on the specific molecular environment one wishes to model. 

In this procedure the repulsion between electrons in a Slater orbital was too big, no allowance being made for the relaxation of the orbital in the atomic configuration. However the over-estimate could be allowed for. When the one-centre repulsion was estimated empirically, and combined with non-empirical values for the others, the role of Cl was less, but the mixture not altogether satisfying. Later new fully empirical schemes were introduced, notably by Pariser and Parr [208]. Inclusion of Cl remains a standard part of molecular electronic calculations. 

---