Parameterization

4 The Independence of Steepest Descent Paths from Parameterization and Coordinate System 

We assume a solution curve x=x(t) of the gradient system Eg.(25) in Sect.3.3 

the differential equation system for a steepest descent path in the curve length parameterization of the Cartesian coordinates is 

Remark that we treat one and the same curve line of steepest descent from x°, namely x x(t) or x=x(s) in the configuration space r of the Cartesian coordinates x. The difference is the parameter interval [0,t ] or [0,s ] to describe this piece of the curve. 

If we have a PES of purely harmonic character (in the neighborhood of a minimizer as starting point) 

The simulated struaures, hence the calculated properties, critically depend on the potential energy function and the parameters used. Thus, parameterization and force field development is an integral part of molecular modeling research. The reliability of the force field is determined by its ability to reproduce experimentally measured quantities from the simulations. It is also desirable that the force field be useful for a wide range of applications. 

The quality of a force field can be established only through extensive test simulations. A large number of force fields that are widely used for biomolecu-lar simulations are available in the literature. These have been used to simulate a variety of systems. For example, the OPLS parameter set developed by Jorgensen et al. ° has been shown to be satisfactory for the simulation of a large number of neat liquid hydrocarbons and polar molecules. Similarly, the AMBER force field nd the CHARMM force field have been used for a variety of protein and nucleic acid simulations. 

Parameterization of force fields for lipids is at an early stage of development in comparison to what has been done for proteins and nucleic acids. Most simulations on lipid systems using force fields have employed parameter sets and parameterization procedures that have proven satisfactory for related systems (i.e., alkane simulations, etc.). This appears to be a reasonable approach, but there are clear cases warranting improvement in the model (e.g., obtaining accurate order parameter profiles). 

Typically, force field parameters are determined by fitting calculated results to experimental data. This may sound straightforward, but there are many problems involved in parameter development, especially with respect to conformational energies. Ideally, gas phase enthalpy data should be used. Such data are scarce, however, and very often it is necessary to use free energy data in solution for the parameterization (and validation) of a force field. In some cases, moreover, there are large variations in experimental data, and it is not an easy task to select data from which to parameterize the force field. 

To overcome this limitation, force fields such as CFF91 and MMFF93 are parameterized using molecular properties determined by ab initio quantum chemical methods. This interesting development requires high level calculations, including electron correlation, to be useful.23 

The UFF force field represents still another type of force field in that it is neither parameterized in the classical way nor based on ab initio quantum chemical calculations. Instead, the parameters in UFF are evaluated from general rules based on the elements, their hybridization and connectivity. The reason for this choice of parameterization scheme is that the goal of the UFF force field is to extend the calculations to the entire periodic table, making the large number of parameters required for such a task untenable by traditional prescriptions. 

In this context it should be noted that the combination of potential functions and the parameters strongly depend on each other. Thus, parameters from one force field cannot be safely transferred to another force field. 

Other important,differences in the set of potential energy functions include the form of the van der Waals function, the number of terms in the torsional energy function, and the way electrostatic interactions and conjugated systems are handled. 

The reliability of a molecular mechanics calculation is dependent on the potential energy equations and the numerical values of the parameters that are incorporated into those equations. In general, parameters are not transferable from one force field to another because of the different forms of equations that have been used and because of parameter correlation within a force field. That is, when one is carrying out the parameterization, if one makes some kind of error, or arbitrary decision, regarding one parameter, other parameters in the 

The first question the user may ask is why they do not put all of the parameters in the program The following example shows the difficulty in putting all of them in. There are currently in MM3 68 different atom types. (Because bond lengths, etc., depend not only on the atom itself, but also on whether it is single, double, or triple bonded, there needs to be a multitude of possible atom types for each element. These types cover only a tiny fraction of 

In general, there is an art and a science to molecular mechanics parameterization. On one extreme, least-squares methods can be used to optimize the parameters to best fit the available data set, and reviews on this topic are avail-able. Alternatively, parameters can be determined on a trial-and-error basis. The situation in either case is far from straightforward because the data usually available come from a variety of sources, are measured by different kinds of experiments in different units, and have relative importances that need subjective assessment. Therefore, straightforward applications of least-squares methods are not expected to give optimum results. 

The quality of a particular parameter is directly dependent on the quality and nature of the experimental or theoretical data available. There is also a dependency on the level of accuracy that is desired. For some purposes, generalized approximate parameters based on known trends may be developed. However, such generalized parameters can lead to serious problems if an exact value is essential for understanding some property that is being studied. 

The bond angle bending constants are typically such that to move an atom a given distance by bending requires an order of magnitude less energy than to 

For many years, chemists have been using theory to determine molecular properties. One property of particular interest is the dipole moment. The dipole moment is a measure of the overall electronic charge separation 

The next section in this chapter provides a brief comparison of the dipole moment (magnitude and direction) for a set of simple alcohols. Experimental gas phase dipole moments45 are compared to ab initio and as molecular mechanics computed values. It is important to note that the direction of the vector dipole used by chemists is defined differently in classical physics. In the former definition, the vector points from the positive to the negative direction, while the latter has the orientation reversed. 

Since the parameters used in molecular mechanics contain all of the electronic interaction information to cause a molecule to behave in the way that it does, proper parameters are important for accurate results. MM3(2000), with the included calculation for induced dipole interactions, should model more accurately the polarization of bonds in molecules. Since the polarization of a molecular bond does not abruptly stop at the end of the bond, induced polarization models the pull of electrons throughout the molecule. This changes the calculation of the molecular dipole moment, by including more polarization within the molecule and allowing the effects of polarization to take place in multiple bonds. This should increase the accuracy with which MM3(2000) can reproduce the structures and energies of large molecules where polarization plays a role in structural conformation. 

In this next section the dipole magnitude and directionality from MM3(2000) is compared to results obtained by MM3(96), Hartree-Fock and Mdller-Plesset minimized structures (calculated using GAUSSIAN94),77 as well as experimental dipole moment measurements.78 For the molecular mechanics geometry optimizations, full matrix energy minimizations were carried out, and ground state structures were verified by the vibrational 

These two measurements give a clear picture of the orientation of the dipole in 3D space and how the dipole moment of a molecule moves when different computational methods are used. 

In general, we know bond lengths to within an uncertainty of 0.00.5 A — 0.5 pm. Bond angles are reliably known only to one or twx) degrees, and there arc many instances of more serious angle enxirs. Tn addition to experimental uncertainties and inaccuracies due to the model (lack of coincidence between model and molecule), some models present special problems unique to their geometry. For example, some force fields calculate the ammonia molecule. Nlln to be planar when there is abundant ex p er i m en ta I evidence th at N H is a 11 i g o n a I pyramid. 

The five semi-empirical methods in HyperChem differ in many technical details. Treatment of electron-electron interactions is one major distinguishing feature. Another important distinguishing feature is the approach used to parameterize the methods. Based on the methods used for obtaining parameters, the NDO methods fall into two classes  

MINDO/3, MNDO, and AMI were developed by the Dewar group at the University of Texas at Austin. This group chose many parameters, such as heats of formation and geometries of sample molecules, to reproduce experimental quantities. The Dewar methods yield results that are closer to experiment than the UNDO and INDO methods. 

developed by James J.P. Stewart, is a reparameterization of AMI, which is based on the neglect of diatomic differential overlap (NDDO) approximation. NDDO retains all one-center differential overlap terms when Coulomb and exchange integrals are computed. PM3 differs from AMI only in the values of the parameters. The parameters for PM3 were derived by comparing a much larger number and wider variety of experimental versus computed molecular properties. Typically, non-bonded interactions are less repulsive in PM3 than in AMI. PM3 is primarily used for organic molecules, but is also parameterized for many main group elements. 

ZINDO/1 is based on a modified version of the intermediate neglect of differential overlap (INDO), which was developed by Michael Zerner of the Quantum Theory Project at the University of Florida. Zerner s original INDO/1 used the Slater orbital exponents with a distance dependence for the first row transition metals only. However, in HyperChem constant orbital exponents are used for all the available elements, as recommended by Anderson, Edwards, and Zerner, Inorg. Chem. 28, 2728-2732,1986. 

ZINDO/S is an INDO method parameterized to reproduce UV visible spectroscopic transitions when used with the singly excited Cl method. It was developed in the research group of Michael Zerner of the Quantum Theory Project at the University of Florida. 

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Suppose that x [Q)) is the adiabatic eigenstate of the Hamiltonian H[q]Q), dependent on internal variables q (the electronic coordinates in molecular contexts), and parameterized by external coordinates Q (the nuclear coordinates). Since x Q)) must satisfy... 

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

For this reason, there has been much work on empirical potentials suitable for use on a wide range of systems. These take a sensible functional form with parameters fitted to reproduce available data. Many different potentials, known as molecular mechanics (MM) potentials, have been developed for ground-state organic and biochemical systems [58-60], They have the advantages of simplicity, and are transferable between systems, but do suffer firom inaccuracies and rigidity—no reactions are possible. Schemes have been developed to correct for these deficiencies. The empirical valence bond (EVB) method of Warshel [61,62], and the molecular mechanics-valence bond (MMVB) of Bemardi et al. [63,64] try to extend MM to include excited-state effects and reactions. The MMVB Hamiltonian is parameterized against CASSCF calculations, and is thus particularly suited to photochemistry. 

The choice of parameterization and the design of a discretization method are not independent Some choices of parameters will facilitate symplec-tic/reversible discretization while others may make this task very difficult or render the resulting scheme practically useless because of the computational expense involved. 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

A force field does not consist only of a mathematical eiqjression that describes the energy of a molecule with respect to the atomic coordinates. The second integral part is the parameter set itself. Two different force fields may share the same functional form, but use a completely different parameterization. On the other hand, different functional forms may lead to almost the same results, depending on the parameters. This comparison shows that force fields are empirical there is no "correct form. Because some functional forms give better results than others, most of the implementations within the various available software packages (academic and commercial) are very similar. 

For each pair of interacting atoms (/r is their reduced mass), three parameters are needed D, (depth of the potential energy minimum, k (force constant of the par-tictilar bond), and l(, (reference bond length). The Morse ftinction will correctly allow the bond to dissociate, but has the disadvantage that it is computationally very expensive. Moreover, force fields arc normally not parameterized to handle bond dissociation. To circumvent these disadvantages, the Morse function is replaced by a simple harmonic potential, which describes bond stretching by Hooke s law (Eq. (20)). 

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. 

In contrast to the point charge model, which needs atom-centered charges from an external source (because of the geometry dependence of the charge distribution they cannot be parameterized and are often pre-calculated by quantum mechanics), the relatively few different bond dipoles are parameterized. An elegant way to calculate charges is by the use of so-called bond increments (Eq. (26)), which are defined as the charge contribution of each atom j bound to atom i. 

The first quantum mechanical improvement to MNDO was made by Thiel and Voityuk [19] when they introduced the formalism for adding d-orbitals to the basis set in MNDO/d. This formalism has since been used to add d-orbitals to PM3 to give PM3-tm and to PM3 and AMI to give PM3(d) and AMl(d), respectively (aU three are available commercially but have not been published at the time of writing). Voityuk and Rosch have published parameters for molybdenum for AMl(d) [20] and AMI has been extended to use d-orbitals for Si, P, S and Q. in AMI [21]. Although PM3, for instance, was parameterized with special emphasis on hypervalent compounds but with only an s,p-basis set, methods such as MNDO/d or AMI, that use d-orbitals for the elements Si-Cl are generally more reliable. 

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. 

This quantity is found to be related to the local polarization energy and is complementary to the MEP at the same point in space, making it a potentially very useful descriptor. Reported studies on local ionization potentials have been based on HF ab-initio calculations. However, they could equally well use semi-empirical methods, especially because these are parameterized to give accurate Koopmans theorem ionization potentials. 

A molecular modeling and simulation package with various implemented force field parameterizations. Free of charge for academic use. Available for different platforms. 

Using semi-einpirical methods, which are also based on approximate solutions of the Schrodingcr equation but use parameterized equations, the computation times can be reduced by twu orders of magnitude. HyperChem from Hypercubc,... 

The classical introduction to molecular mechanics calculations. The authors describe common components of force fields, parameterization methods, and molecular mechanics computational methods. Discusses th e application of molecular mechanics to molecules comm on in organic,and biochemistry. Several chapters deal w ith thermodynamic and chemical reaction calculations. 

The accuracy of a molecular mechanics or seim-eni pineal quantum mechanics method depends on the database used to parameterize the method. This is true for the type of molecules and the physical and chemical data in the database. Frequently, these methods give the best results for a limited class of molecules or phen omen a. A disad van tage of these methods is that you m u si have parameters available before running a calculation. Developing param eiers is time-consuming. 

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