Force field models, empirical simple

A potentially much more adaptable technique is force-field vibrational modeling. In this method, the effective force constants related to distortions of a molecule (such as bond stretching) are used to estimate unknown vibrahonal frequencies. The great advantage of this approach is that it can be applied to any material, provided a suitable set of force constants is known. For small molecules and complexes, approximate force constants can often be determined using known (if incomplete) vibrational specha. These empirical force-field models, in effect, represent a more sophisticated way of exhapolating known frequencies than the rule-based method. A simple type of empirical molecular force field, the modified Urey-Bradley force field (MUBFF), is introduced below. 

Computer programs for empirical force-field calculations that use other concepts have been tested, and some of these will be discussed elsewhere in this book. Among these approaches are one based on a pure central force-field model, used for simple organic compounds [208], an equipotential surface force-field model, used for carbonyl cluster complexes [99,103], and a program that includes ligand field terms, developed for transition metal complexes in the LFMM model (ligand field molecular mechanics) [21, 105, 108]. 

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]. 

The approach presented above is referred to as the empirical valence bond (EVB) method (Ref. 6). This approach exploits the simple physical picture of the VB model which allows for a convenient representation of the diagonal matrix elements by classical force fields and convenient incorporation of realistic solvent models in the solute Hamiltonian. A key point about the EVB method is its unique calibration using well-defined experimental information. That is, after evaluating the free-energy surface with the initial parameter a , we can use conveniently the fact that the free energy of the proton transfer reaction is given by... 

Infrared spectra are straightforward to predict theoretically, demanding development of a force field (FF) to determine frequencies and dipole derivatives for intensities. These parameters were initially obtained using empirically fitted force constants and simple models for transition dipoles (Krimm and Bandekar, 1986 Torii and Tasumi, 1996). 

Molecular mechanics is the modeling of the molecule not as electrons obeying quantum mechanics, but as a number of atoms connected together by flexible bonds, which move according to Newtonian mechanics and empirical force fields. For instance, in a simple diatomic molecule like H2, the length of the bond between the two atoms is assumed to have an energy equal to... 

By claiming that MD simulations can be used to test the approximations in the theoretical models, we assume that the MD simulations themselves do not introduce any new approximations. Naturally, this is not true because there are approximations, specific to the MD simulations. Among the more severe of these are the empirical force fields, which in their simple form cannot be expected to be a perfect description of the interactions. However, the performance of the force field can be examined by comparison to many experimental techniques. Thus, the errors from the force fields can be evaluated accurately and it is not a fatal problem for the use in relation to NMR related issues. 

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