The Molecular-Mechanics Method

Some people use the term strain energy to designate Fin (17.83) but other people use strain energy to denote a different quantity (see Burkert and AUinger, pp. 184-189). 

The explicit expressions used for each of the terms in (17.83) define what is called a molecular-mechanics force field, since the derivatives of the potential-energy function determine the forces on the atoms. A force field contains analytical formulas for the terms in (17.83) and values for all the parameters that occur in these formulas. The MM method is sometimes called the empirical-force-field method. Empirical force fields are used not only for single-molecule molecular-mechanics calculations of energy differences, geometries, and vibrational frequencies, but also for molecular-dynamics simulations of liquids and solutions, where Newton s second law is integrated to follow the motions of atoms with time in systems containing thousands of atoms. 

An MM force field assigns each atom in a molecule to one of a number of possible atom types, depending on the atom s atomic number and molecular environment. For example, some commonly used atom types in force fields for organic compounds are sp (saturated) carbon, (doubly bonded) carbon, sp (triply bonded) carbon, carbonyl carbon, aromatic carbon, and so on, H bonded to C, H bonded to O, H bonded to N, and so on. Different force fields contain somewhat different numbers and kinds of atom types, based on the decisions made by their constructors. A force field for organic compounds typically contains 50 to 75 atom types. 

In ab initio and semiempirical molecular electronic single-point or geometry-optimization calculations, one inputs the atomic numbers of the atoms and a set of coordinates (Cartesian or internal) for each atom, and no specification is made as to which 

FIGURE 17,8 The heavily shaded atoms are 1,2 atoms, 1,3 atoms, and 1,4 atoms. 

The molecular-mechanics (MM) method is quite different fi om the semiempirical methods of the last section. Molecular mechanics is not a quantum-mechanical method, since it does not deal with an electronic Hamiltonian or wave function or an electron density. Instead, the method uses a model of a molecule as composed of atoms held together by bonds. Using such parameters as bond-stretching and bond-bending 

In ab initio and semiempirical molecular electronic single-point or geometry-optimization calculations, one inputs the atomic numbers of the atoms and a set of coordinates (Cartesian or internal) for each atom, and no specification is made as to which atoms are bonded to which atoms (Section 15.16). In a molecular-mechanics calculation, one must specify not only the initial atomic coordinates, but also which atoms are bonded to each atom, so that the V expression can be properly constructed. This 

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The semi-empirical methods of HyperChem are quantum mechanical methods that can describe the breaking and formation of chemical bonds, as well as provide information about the distribution of electrons in the system. HyperChem s molecular mechanics techniques, on the other hand, do not explicitly treat the electrons, but instead describe the energetics only as interactions among the nuclei. Since these approximations result in substantial computational savings, the molecular mechanics methods can be applied to much larger systems than the quantum mechanical methods. There are many molecular properties, however, which are not accurately described by these methods. For instance, molecular bonds are neither formed nor broken during HyperChem s molecular mechanics computations the set of fixed bonds is provided as input to the computation. 

An example of the application of molecular mechanics in the investigation of chemical reactions is a study of the correlation between steric strain in a molecule and the ease of rupture of carbon-carbon bonds. For a series of hexasubstituted ethanes, it was found that there is a good correlation between the strain calculated by the molecular mechanics method and the rate of thermolysis. Some of the data are shown in Table 3.3. 

The molecular mechanics method has been applied to the calculation of conformational properties of the thiane, dithiane and trithiane oxide systems which are... 

The final step in the molecular-mechanics calculation of molecular conformation involves the minimization of the energy Approximations are involved whose importance is not always clear. Usually, all first derivatives with respect to the various internal coordinates are set equal to zero - although these coordinates are often not independent (see Section 10.6). Furthermore, the final conformation obtained depends on the assumed initial structure. Therefore, (he method must be applied with care and a certain amount of chemical intuition. In spite of these uncertainties the molecular mechanics method has been employed with considerable success, particularly in the conformational analysis of branched alkanes. For molecules containing hetero-atoms, it can be applied, but with somewhat less confidence. 

The molecular mechanics method is usually limited to the determination of molecular geometry and thermodynamic quantities. However, it is sometimes employed to estimate vibrational frequencies - at least in those cases in which 7r electrons are not involved in the determination of the molecular geometry. It should be emphasized that this method, as well as those presented in Chapter 12, are applicable only to isolated molecules, as intermolecular forces are not included in the model. 

The application of the molecular mechanics method is carried out in three steps, namely,... 

The molecular mechanics method, often likened to a ball and spring model of the molecule, represents the total energy of a system of molecules with a set of simple analytical functions representing different interactions between bonded and non-bonded atoms, as shown schematically in Figure 1. 

These results indicate that the random walk procedure is an efficient tool to improve the performance of the molecular mechanics methods and to provide a better description of oligosaccharide conformations. While Figures 3 and 4 illustrate the effects of changes in the pendant group orientations, in normal use the entire structure would be optimized after the random walk procedure had detennined low-energy positions for the pendant groups. 

The computational methods for the structure of a molecule are divided into the ab initio, the semi-empirical, and the molecular mechanics methods. 

The molecular mechanics method is extremely parameter dependent. A force field equation that has been empirically parameterized for calculating peptides must be used for peptides it cannot be applied to nucleic acids without being re-parameterized for that particular class of molecules. Thankfully, most small organic molecules, with molecular weights less than 800, share similar properties. Therefore, a force field that has been parameterized for one class of drug molecules can usually be transferred to another class of drug molecules. In medicinal chemistry and quantum pharmacology, a number of force fields currently enjoy widespread use. The MM2/MM3/MMX force fields are currently widely used for small molecules, while AMBER and CHARMM are used for macromolecules such as peptides and nucleic acids. 

The conformational properties of trimer molecules modeling PVDB (S 100) and PVDF are analyzed by the molecular mechanics method of Boyd and Kesner [J. Chem. Phys. 1980, 72, 21791, which takes into account both steric and electrostatic energy. Total conformational energies are used to calculate a set of intramolecular interaction energies that, by means of the RIS model, allowed estimation of the characteristic ratios and dipole moment ratios of PVDB and PVDF under unperturbed conditions. 

In this part of the book we give practical advice on how to apply molecular mechanics to problems involving metal complexes and describe how to interpret and use the results. The practice of molecular mechanics is outlined in two chapters Chapter 15 covers the development of a force field and Chapter 16 minimization of the strain energy to produce a structure. In both of these aspects there is the potential to make serious errors, and we have attempted to highlight as many of the pitfalls as possible. In Chapter 17 we discuss the interpretation and application of the results obtained using the molecular mechanics method. 

In the molecular mechanics methods, the energy of the molecule or assemblage of molecules is broken down into a sum of energy terms which are minimized separately, i.e.,... 

DoSen-Midovid L, Jeremid D, Allinger NL (1983) Treatment of electrostatic effects within the molecular mechanics method. J Am Chem Soc 105 1716-1722... 

N.L. Allinger and J.T. Sprague, Calculation of the structures of hydrocarbons containing delocalized electronic systems by the molecular mechanics method, J. Am. Chem. Soc., 95 (1973) 3893-3907. 

Allinger, N.L. and Sprague, J.T. (1973) Calculations of the Structures of Hydrocarbons Containing Delocalized Electronic Systems by the Molecular Mechanics Method, J. Am. Chem. Soc. 95, 3893-3907. 

As this method is the most commonly encountered, and is often the basis for geometries used in higher-level calculations, it is worth describing the interactions in detail. The molecular mechanics method assumes that the total energy of the system may be broken down into the following components ... 

The consistent force field (CFF) method developed by Ermer and Lifson (62,63) and the molecular mechanics method described by Allinger (MM 1 and MM2) (64) have been used widely for computational investigations of strained hydrocarbons. Other force fields have been developed and their merits have been critically evaluated (65,66). 

Nevertheless, as with any theory or technique there are certain limitations with this method that should be understood. The users of the molecular mechanics method must be careful about extending these methodologies beyond... 

To more fully appreciate the molecular mechanics method, it is important to examine the underlying mathematical treatment. One of the fundamental theorems of molecular mechanics is that the total energy of a molecule can be divided into various readily identifiable parts. The total energy, E,ot.,i, of a molecule is thus divided into several parts, one of which is attributed to bond... 

L. Dosen-Micovic, D. Jeremic, and N. L. Allinger, /. Am. Chem. Soc., 105, 1716 (1983). Treatment of Electrostatic Effects within the Molecular Mechanics Method. 

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