Force field models, empirical function

Figure 5. Normal modes for vibration of tetrahedral [Cr04] (chromate). There are four distinct vibrational frequencies, including one doubly-degenerate vibration (E symmetry) and two triply-degenerate vibrations (F2 symmetry), for a total of nine vibrational modes. Arrows show the characteristic motions of each atom during vibration, and the length of each arrow is proportional to the magnitude of atomic motion. Only F2 modes involve motion of the central chromium atom, and as a result their vibrational frequencies are affected by Cr-isotope substitution. The normal modes shown here were calculated with an ab initio quantum mechanical model, using hybrid Hartree-Fock/Density Functional Theory (B3LYP) and the 6-31G(d) basis set—other ab initio and empirical force-field models give very similar results.

One important point that we should bear in mind as we undertake a deeper analysis of molecular mechanics is that force fields are empirical-, there is no correct form for a force field. Of course, if one functional form is shown to perform better than another it is likely that form will be favoured. Most of the force fields in common use do have a very similar fqrm, and it is tempting to assume that this must therefore be the optimal functional form Certainly such models tend to conform to a useful picture of the interactions present in a system, but it should always be borne in mind that there may be better forms, particularly when developing a force field for new classes of molecule. The functional forms employed in molecular mechanics force fields are often a compromise between accuracy and computational efficiency the most accurate functional form may often be unsatisfactory for efficient computation. As the performance of computers increases so it becomes pcKsible to incorporate more sophisticated models. An additional consideration is that in order to use techniques such as energy minimisation and molecular dynamics, it is usually desirable to be able to calculate the first and second derivatives of the energy with respect to the atomic coordinates. 

The Universal Force Field, UFF, is one of the so-called whole periodic table force fields. It was developed by A. Rappe, W Goddard III, and others. It is a set of simple functional forms and parameters used to model the structure, movement, and interaction of molecules containing any combination of elements in the periodic table. The parameters are defined empirically or by combining atomic parameters based on certain rules. Force constants and geometry parameters depend on hybridization considerations rather than individual values for every combination of atoms in a bond, angle, or dihedral. The equilibrium bond lengths were derived from a combination of atomic radii. The parameters [22, 23], including metal ions [24], were published in several papers. 

Empirical energy functions can fulfill the demands required by computational studies of biochemical and biophysical systems. The mathematical equations in empirical energy functions include relatively simple terms to describe the physical interactions that dictate the structure and dynamic properties of biological molecules. In addition, empirical force fields use atomistic models, in which atoms are the smallest particles in the system rather than the electrons and nuclei used in quantum mechanics. These two simplifications allow for the computational speed required to perform the required number of energy calculations on biomolecules in their environments to be attained, and, more important, via the use of properly optimized parameters in the mathematical models the required chemical accuracy can be achieved. The use of empirical energy functions was initially applied to small organic molecules, where it was referred to as molecular mechanics [4], and more recently to biological systems [2,3]. 

A classical description of M can for example be a standard force field with (partial) atomic charges, while a quantum description involves calculation of the electronic wave function. The latter may be either a semi-empirical model, such as AMI or PM3, or any of the ab initio methods, i.e. HF, MCSCF, CISD, MP2 etc. Although the electrostatic potential can be derived directly from the electronic wave function, it is usually fitted to a set of atomic charges or multipoles, as discussed in Section 9.2, which then are used in the actual solvent model. 

In the ONIOM(QM MM) scheme as described in Section 2.2, the protein is divided into two subsystems. The QM region (or model system ) contains the active-site selection and is treated by quantum mechanics (here most commonly the density functional B3LYP [31-34]). The MM region (referred to as the real system ) is treated with an empirical force field (here most commonly Amber 96 [35]). The real system contains the surrounding protein (or selected parts of it) and some solvent molecules. To analyze the effects of the protein on the catalytic reactions, we have in general compared the results from ONIOM QM MM models with active-site QM-only calculations. Such comparisons make it possible to isolate catalytic effects originating from e.g. the metal center itself from effects of the surrounding protein matrix. 

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