The Parameterization

Where do the parameters come from Fundamentally, they result from a fitting procedure, in which many types of experimental data are used. Structural information is crucial. There is a large database of experimentally determined structures for organic molecules, and a good force field should be able to reproduce them. So, parameters are adjusted to properly reproduce experimental structures. 

The value of a force field is directly proportional to the quality of its parameterization, and that in turn depends completely on the quantity and C]uality of experimental structural and energetic data that are available. Thus, good force fields for hydrocarbons exist because there is a wealth of experimental data on such systems. Another issue is that the factors that determine structure and energetics in hydrocarbons are fairly simple, in part becau.se the electrostatic and hydrogen bonding terms are not very relevant. As structures become more complex, with more and more polar groups, parameterization becomes more difficult. 

A recent boon to force field development has been the success of modern, ab initio quantum mechanical methods in predicting the properties of molecules (see Chapter 14 for a thorough description of these methods). These computational methods can now provide reliable data on small prototype systems for which experimental data are unavailable, and then force fields can be developed based on the quantum mechanical calculations. This is a valuable approach, but it is limited in that many interesting systems are too large to be treated by the quantum mechanical methods. 

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It is noteworthy that it is not obligatory to use a torsional potential within a PEF. Depending on the parameterization, it is also possible to represent the torsional barrier by non-bonding interactions between the atoms separated by three bonds. In fact, torsional potentials and non-bonding 1,4-interactions are in a close relationship. This is one reason why force fields like AMBER downscale the 1,4-non-bonded Coulomb and van der Waals interactions. 

Molecular dipole moments are often used as descriptors in QPSR models. They are calculated reliably by most quantum mechanical techniques, not least because they are part of the parameterization data for semi-empirical MO techniques. Higher multipole moments are especially easily available from semi-empirical calculations using the natural atomic orbital-point charge (NAO-PC) technique [40], but can also be calculated rehably using ab-initio or DFT methods. They have been used for some QSPR models. 

Output File 8-1. Parameters for the STO-2G Basis Set. The parameterized STO-2G basis function is... 

Semiempirical methods are parameterized to reproduce various results. Most often, geometry and energy (usually the heat of formation) are used. Some researchers have extended this by including dipole moments, heats of reaction, and ionization potentials in the parameterization set. A few methods have been parameterized to reproduce a specific property, such as electronic spectra or NMR chemical shifts. Semiempirical calculations can be used to compute properties other than those in the parameterization set. 

Many semiempirical methods compute energies as heats of formation. The researcher should not add zero-point corrections to these energies because the thermodynamic corrections are implicit in the parameterization. 

It is occasionally desirable to add new parameters to a molecular mechanics force field. This might mean adding an element that is not in the parameterization set or correctly describing a particular atom in a specihc class of molecules. 

The first step in creating a force field is to decide which energy terms will be used. This determines, to some extent, the ability of the force field to predict various types of chemistry. This also determines how dilficult the parameterization will be. For example, more information is needed to parameterize an-harmonic bond-stretching terms than to parameterize harmonic terms. 

The validation of the prediction equation is its performance in predicting properties of molecules that were not included in the parameterization set. Equations that do well on the parameterization set may perform poorly for other molecules for several different reasons. One mistake is using a limited selection of molecules in the parameterization set. For example, an equation parameterized with organic molecules may perform very poorly when predicting the properties of inorganic molecules. Another mistake is having nearly as many fitted parameters as molecules in the test set, thus fitting to anomalies in the data rather than physical trends. 

PM3/TM is an extension of the PM3 method to transition metals. Unlike the parameterization of PM3 for organics, PM3/TM has been parameterized only to reproduce geometries. This does, of course, require a reasonable description of energies, but the other criteria used for PM3 parameterization, such as dipole moments, are not included in the PM3/TM parameterization. PM3/TM tends to exhibit a dichotomy. It will compute reasonable geometries for some compounds and completely unreasonable geometries for other compounds. It seems to favor one coordination number or hybridization for some metals. 

Use a constant dielectric of 1.0 with TIP3P water molecules in a periodic box. Because of the parameterization of TIP3P molecules, using a distance-dependent dielectric or a value other than 1.0 gives unnatural results. 

If the heat of fonnation parameters are derived on the basis of fitting to a large variety of compounds, a specific set of parameters is obtained. A slightly different set of parameters may be obtained if only certain strainless molecules are included in the parameterization. Typically molecules like straight chain alkanes and cyclohexane are defined as strainless. Using these strainless heat of formation parameters, a strain energy may be calculated as illustrated in Figure 2.14. 

The parameterization process may be done sequentially or in a combined fashion. In the sequential method a certain class of compound, such as hydrocarbons, is parameterized first. These parameters are held fixed, and a new class of compound, for example alcohols and ethers, is then parameterized. Tins method is in line with the basic assumption of force fields parameters are transferable. The advantage is that only a fairly small number of parameters are fitted at a time. The ErrF is therefore a relatively low-dimensional function, and one can be reasonably certain that a good minimum has been found (although it may not be the global minimum). The disadvantage is that the final set of parameters necessarily provides a poorer fit (as defined from the value of the ErrF) than if all the parameters are fitted simultaneously. 

Validation of a force field is typically done by showing how accurately it reproduces reference data, which may or may not have been used in the actual parameterization. Since different force fields employ different sets of reference data, it is difficult to compare their accuracy directly. Indeed there is no single best force field, each has its advantages and disadvantages. They perform best for the type of compounds used in the parameterization, but may give questionable results for other systems. Table 2.6 gives some typical accuracies for AH( that can be obtained with the MM2 force field. 

To compensate for these approximations, the remaining integrals are made into parameters, and their values are assigned on the basis of calculations or experimental data. Exactly how many integrals are neglected, and how the parameterization is done, defines the various semi-empirical methods. 

The MNDO, AMI and PM3 methods are parameterizations of the NDDO model, where the parameterization is in terms of atomic variables, i.e. referring only to the nature of a single atom. MNDO, AMI and PM3 are derived from the same basic approximations (NDDO), and differ only in the way the core-core repulsion is treated, and how the parameters are assigned. Each method considers only the valence s- and p-functions, which are taken as Slater type orbitals with corresponding exponents, (s and... 

The Huckel methods perform the parameterization on the Fock matrix elements (eqs. (3.50) and (3.51)), and not at the integral level as do NDDO/INDO/CNDO. This means that Huckel methods are non-iterative, they only require a single diagonalization of the Fock (Huckel) matrix. The Extended Huckel Theory (EHT) or Method (EHM), developed primarily by Hoffmann again only considers the valence electrons. It makes use of Koopmans theorem (eq. (3.46)) and assigns the diagonal elements in the F... 

The parameterization of MNDO/AM1/PM3 is performed by adjusting the constants involved in the different methods so that the results of HF calculations fit experimental data as closely as possible. This is in a sense wrong. We know that the HF method cannot give the correct result, even in the limit of an infinite basis set and without approximations. The HF results lack electron correlation, as will be discussed in Chapter 4, but the experimental data of course include such effects. This may be viewed as an advantage, the electron correlation effects are implicitly taken into account in the parameterization, and we need not perform complicated calculations to improve deficiencies in fhe HF procedure. However, it becomes problematic when the HF wave function cannot describe the system even qualitatively correctly, as for example with biradicals and excited states. Additional flexibility can be introduced in the trial wave function by adding more Slater determinants, for example by means of a Cl procedure (see Chapter 4 for details). But electron cori elation is then taken into account twice, once in the parameterization at the HF level, and once explicitly by the Cl calculation. 

The Parameterized Configuration Interaction (PCI-X) method simply takes the correlation energy and scales it by a constant factor X (typical value 1.2), i.e. it is assumed that the given combination of method and basis set recovers a constant fraction of the correlation energy. 

For solvent models where the cavity/dispersion interaction is parameterized by fitting to experimental solvation energies, the use of a few explicit solvent molecules for the first solvation sphere is not recommended, as the parameterization represents a best fit to experimental data without any explicit solvent present. 

The last published report of the IPCC acknowledges that the single largest uncertainty in determining the climate sensitivity to either natural or anthropogenic changes are clouds and their effects on radiation and their role in the hydrological cycle. .. At the present time, weaknesses in the parameterization of cloud formation and dissipation are probably the main impediment to improvements in the simulation of cloud effects on climate (IPCC, 1995, p. 346). 

This problem can be cast in linear programming form in which the coefficients are functions of time. In fact, many linear programming problems occurring in applications may be cast in this parametric form. For example, in the petroleum industry it has been found useful to parameterize the outputs as functions of time. In Leontieff models, this dependence of the coefficients on time is an essential part of the problem. Of special interest is the general case where the inputs, the outputs, and the costs all vary with time. When the variation of the coefficients with time is known, it is then desirable to obtain the solution as a function of time, avoiding repetitions for specific values. Here, we give by means of an example, a method of evaluating the extreme value of the parameterized problem based on the simplex process. We show how to set up a correspondence between intervals of parameter values and solutions. In that case the solution, which is a function of time, would apply to the values of the parameter in an interval. For each value in an interval, the solution vector and the extreme value may be evaluated as functions of the parameter. 

Oxidant Formation. The role of HO. in controlling the time-scale and severity of tropospheric oxidant pollution may be seen from the parameterization of O Brien and co-workers (75,76). The simplest possible mechanism for oxidant (Le. ozone, PAN, H2O2, etc.) formation consists simply of the reaction of an individual NNlHCj with HO. to convert the NMHCj to a generic product(s) PRODj, followed by removal of the product by HO. (PROD photolysis may be important, but is ignored here)... 

Zhang, G. ]. and McFarlane, N. A. (1995). Sensitivity of climate simulations to the parameterization of cumulus convection in the Canadian Climate... 

If the inverse in Eq. (2.8) does not exist then the metric is singular, in which case the parameterization of the manifold of states is redundant. That is, the parameters are not independent, or splitting of the manifold occurs, as in potential curve crossing in quantum molecular dynamics. In both cases, the causes of the singularity must be studied and revisions made to the coordinate charts on the manifold (i.e. the way the operators are parameterized) in order to proceed with calculations. 

We have considered the larger AI4-AI6 clusters using both ab initio calculations and the parameterized model (9). The results for AI4 and AI5, summarized in Table IV, show that the parameterized model and ab initio calculations agree well on the relative energetics if both the two- and three-body interactions are included. For Ale it is difficult to treat all the structures at the TZ2P-CPF level, but for the structures considered, there is reasonable agreement between the ab initio and model results. 

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