Parameterization accuracy/generality

Methods such as G3 and CBS-QB3 do reach the goal of chemical accuracy (generally defined as 1 kcal/mol) on average, but worst-case errors for problematic molecules may exceed this criterion by almost an order of magnitude. In addition, almost all of these approaches involve some level of parameterization and/or empirical correction against experimental data. While this is by and large possible (albeit not without pitfalls) in the kcal/mol accuracy range for first-and second-row compounds, experimental data of sub-kcal/mol accuracy are thin on the... 

To improve the accuracy of implicit-solvent potential energy surfaces, non-electrostatic interactions must be included, although such interactions have received only a brief mention here. The smooth, linear-scaling PCM technology that is discussed here is immediately ready for use in MM/PBSA applications [36, 38], as a replacement for finite-difference electrostatics. Other formulas for the non-electrostatic interactions [47] can also be used in PCM calculations, possibly after some re-parameterization. In general these non-electrostatic interaction formulas depend in some way on the cavity surface area, which is smooth and easily calculable by means of the PCM algorithms discussed herein. 

The parameters in the original parameterization are adjusted in order to reproduce the correct results. These results are generally molecular geometries and energy differences. They may be obtained from various types of experimental results or ah initio calculations. The sources of these correct results can also be a source of error. Ah initio results are only correct to some degree of accuracy. Likewise, crystal structures are influenced by crystal-packing forces. 

During last decades the DFT based methods have received a wide circulation in calculations on TMCs electronic structure [34,85-88]. It is, first of all, due to widespread use of extended basis sets, allowing to improve the quality of the calculated electronic density, and, second, due to development of successful (so called - hybrid) parameterizations for the exchange-correlation functionals vide infra for discussion). It is generally believed, that the DFT-based methods give in case of TMCs more reliable results, than the HER non-empirical methods and that their accuracy is comparable to that which can be achieved after taking into account perturbation theory corrections to the HER at the MP2 or some limited Cl level [88-90]. 

The G2 and G3 methods go beyond extrapolation to include small and entirely general empirical corrections associated with the total numbers of paired and unpaired electrons. When sufficient experimental data are available to permit more constrained parameterizations, such empirical corrections can be associated with more specific properties, e.g., with individual bonds. Such bond-specific corrections are employed by the BAG method described in Section 7.7.3. Note that this approach is different from those above insofar as the fundamentally modified quantity is not Feiec, but rather A/7. That is, the goal of the method is to predict improved heats of formation, not to compute more accurate electronic energies, per se. Irikura (2002) has expanded upon this idea by proposing correction schemes that depend not only on types of bonds, but also on their lengths and their electron densities at their midpoints. Such detailed correction schemes can offer very high accuracy, but require extensive sets of high quality experimental data for their formulation. 

A more widely used approach for organic molecules is based on second-order perturbation theory. Here the dipolar contribution to the field induced charge displacement is calculated by inclusion of the optical field as a perturbation to the Hamiltonian. Since the time dependence of the field is included here, dispersion effects can be accounted for. In this approach the effect of the external field is to mix excited state character into the ground state leading to charge displacement and polarization. The accuracy of this method depends on the parameterization of the Hamiltonian in the semi-empirical case, the extent to which contributions from various excited states are incorporated into the calculation, and the accuracy with which those excited states are described. This in turn depends on the nature of the basis set and the extent to which configuration interaction is employed. This method is generally referred to as the sum over states (SOS) method. 

The fact that, in the case of solution spectra, the spectroscopic data are obtained from the solvated species while the corresponding molecular mechanics structure usually represents the naked species is somewhat unsatisfactory. This is one reason for some observed, albeit small, differences between structural parameters in solution and in the solid state. Nevertheless, the accuracy is often surprisingly high, and this might be explained by the fact that the parameterization of the force field is generally based on crystal structural data, and the calculated structures therefore repre-... 

Having described the variety of ways in which the core-valence interaction may be parameterized it is clear that we should examine how they perform in actual calculations. Generally the advances in the complexity of the parameterization have produced commensurate improvements in the accuracy of the results. However, by introducing a large number of parameters the simplicity of the core-valence concept is lost and, in practice, the fitting of the parameters themselves can be expensive in terms of computer time, although they only need to be obtained once for each atom. 

In general, the accuracy of semiempirical methods, particularly in energetics, falls short of that of current routine ab initio methods (this may not have been the case when AMI was developed, in 1985 [125]). Parameters may not be available for the elements in the molecules one is interested in, and obtaining new parameters is something rarely done by people not actively engaged in developing new methods. Semiempirical errors are less systematic than ab initio, and thus harder to correct for. Clark has soberly warned that All parameterized techniques can interpolate and none can extrapolate consistently and well , thus we can expect on occasion a catastrophic failure but semiempirical methods will do what they are designed to do [11]. 

Structural uncertainties in models can be dealt with in a variety of ways, including (1) parameterization of a general model that can be reduced to alternative functional forms (e.g. Morgan Henrion, 1990), (2) enumeration of alternative models in a probability tree (e.g. Evans et al., 1994), (3) assessment of critical assumptions within a model, (4) assessment of the pedigree of a model and (5) assessment of model quality. The first two of these are quantitative methods, whereas the others are qualitative. In practice, a typical approach is to compare estimates made separately with two or more models. However, the models may not be independent of each other with respect to their basis in theory or data. Thus, an apparently favourable comparison of models may indicate only consistency, rather than accuracy. 

The computational methods developed to deal with the various (explicitly correlated) resonance-theoretic models turn out to be most powerful for the more highly simplified models, of the preceding section 2. It is emphasized that these more highly simplified schemes need not necessarily entail significant approximation or loss of accuracy when properly parameterized. Moreover, these more simplified schemes often allow general conclusions for whole sequences or sets of molecules. For the higher-level models the manner of solution turns out to... 

To achieve just a rudimentary assignment of the value of one parameter, at least 3-4 independent data should be available. To parameterize MM2 for all molecules described by the 71 atom types would thus require of the order of 10 independent experimental between different molecules. A compromise between accuracy and generality must thus be made. In MM2(91) for example the actual number of parameters compared to the theoretical estimated possible (based on the 30 effective atom types above) is shown in TnWp 9 9... 

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