The Force-Field Method

Finally, we come to the force-field method, which is also known as the Westheimer method or the molecular mechanics method. An excellent review of this method was published earlier (Williams et al., 

1968) covering the literature through January 1968. This chapter will concentrate on more recent results. 

This method differs fundamentally from the other calculational methods previously mentioned. The Born-Oppenheimer approximation says that one can separate the nuclear from the electronic motions in a molecule, and the previously discussed quantum mechanical methods have to do with the electronic system, after the nuclear positions have been established (or assumed). To determine structures by such methods, one must repeat the calculation for a number of different nuclear positions, and locate the energy minimum in some way. Unless the structure is known at the outset, one therefore requires not just a single calculation, but many calculations, in order to determine the actual structure. 

The force-field method involves the other part of the Born-Oppenheimer approximation, that is the positioning of the nuclei. The electronic system is not considered explicitly, but its effects are of course taken into account indirectly. This method is often referred to as a classical approach, not because the equations and parameters are derived from classical mechanics, but rather because it is assumed that a set of equations exist which are of the form of the classical equations of motion. The problem from this point of view is one of establishing just which equations are necessary, and determining the numerical values for the constants which appear in the equations. In general there is no limit as to what functions may be chosen or what parameters arc to be used, except that the force-field must duplicate the experimental data. 

The force-field method which will be discussed here presents a useful method for determining the structure and energy of a molecule. Both structure and energy seem to be of primary importance. There are many other properties of a molecule that can be determined, after the structure and/or energy are known, or they can be determined simultaneously with the structure and energy. These include the Raman and infrared spectra, and the other thermodynamic functions. Since these quantities are of limited use to organic chemists, we will concentrate here on just structure and energy (gas phase) at 25°C. 

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The various terms in this formula have the meaning of the potential function (force field) V(r1,r2,0) the vibrational, 7 rotational, fx, Ty, Tz and rotational-vibrational, Tvr kinetic energy terms. The latter are differential operators acting in the space of wave functions )/(/ ,r2,0 a,P,y). The potential function V(rur2,Q) is either calculated ab initio or parametrized in a suitable fashion. A commonly used parametrization is that provided by the force-field method... 

The affinity of interactions can be calculated in a number of ways. One example is the force field method to calculate the free energy of binding for the Ugand-protein system before and after the docking, as given by the equation... 

An additional advantage of the force field method 32) is its power to predict the energy levels of conformations which are not populated and even complete rotational potentials of bonds. Again, statisfying agreement with results from dynamic nmr-measurements for a series of crowed hydrocarbons was found 36). Knowledge of the shape of rotational potentials proved to be helpful for the interpretation of entropy effects in these series of compounds and in their thermolysis reactions. 

This conformational situation is also responsible for entropy effects. It has been shown 161 that the entropy differences between two diastereomers in this series is mainly dependent on the shapes of the rotational profiles about the central bonds. The observed differences in AS (D,L-meso) for the thermolyses can be reproduced semiquantitatively by differences in ground state entropy (see Table 2) which were calculated by the force field method 53). 

Meso- and D,L-2,3-dimethoxy-2,3-diphenylsuccinonitrile 39 were obtained in a 1 1 ratio by a preparative dimerization procedure. Their configurations were assigned by crystal structure analysis 74). According to nmr both diastereomers have a configuration with the phenyl groups in the anti position. From a conformational analysis by the force field method it was concluded that these conformations remain the preferred ones also for the isolated molecules meso-39 was calculated to be 1.2 kcal mol-1... 

A model calculation for Type A molecules using the force-field method CHARMm (Chemistry in Harvard Molecular Modeling) showed a linear arrangement of the poly (ethylene glycol) chains for the compound 31a (Figure 6.5). 

Summarizing, the purpose of this review is not to give a more or less complete coverage of the literature, but rather to illuminate the present scope and limitations of the force-field method and to draw guidelines for the future. We prefer to concentrate our attention mainly on theoretical results, on open questions, and on selected applications taken from recent work of leading experts in the area. A larger body of older but still relevant literature on the FF method and related topics has been covered in an extensive review by Williams et al. 

N.L. Allinger and J.T. Sprague, Conformational analysis. LXXXIV. A study of the structures and energiesof some aUcenes and cycloalcanes by the force field method. J. Am. Chem. Soc., 94 (1972) 5734. 

One of the deficiencies of the force fields in use today is that they commonly use the fixed charge approximation, whereas in reality the atomic charges vary in response to changes of both the molecular conformation and the environment. Obviously, a scheme with geometry-dependent charges would extend capabilities of the force field methods. 

With this information in hand, it seemed reasonable to attempt to use force field methods to model the transition states of more complex, chiral systems. To that end, transition state.s for the delivery of hydrogen atom from stannanes 69 71 derived from cholic acid to the 2.2,.3-trimethy 1-3-pentyl radical 72 (which was chosen as the prototypical prochiral alkyl radical) were modeled in a similar manner to that published for intramolecular free-radical addition reactions (Beckwith-Schicsscr model) and that for intramolecular homolytic substitution at selenium [32]. The array of reacting centers in each transition state 73 75 was fixed at the geometry of the transition state determined by ah initio (MP2/DZP) molecular orbital calculations for the attack of methyl radical at trimethyltin hydride (viz. rsn-n = 1 Si A rc-H = i -69 A 6 sn-H-C = 180°) [33]. The remainder of each structure 73-75 was optimized using molecular mechanics (MM2) in the usual way. In all, three transition state conformations were considered for each mode of attack (re or ) in structures 73-75 (Scheme 14). In general, the force field method described overestimates experimentally determined enantioseleclivities (Scheme 15), and the development of a flexible model is now being considered [33]. 

The force-field method is a method which, like the other calcu-lational procedures mentioned above, has certain shortcomings. It is an empirical method, and is based on a large volume of experimental data. These data must exist for a given class of compounds before the method can be developed and applied to any particular compound in that class. 

The main strength of the force-field method lies in its speed and ease of application. To determine the structure and energy of an average molecule in which one might be interested, and which does... 

Figure 3. Bond lengths and bond angles in some representative bicyclic and related hydrocarbons calculated by the force-field method (Wertz and Allinger, 1974).

A few non-conjugated polyenes have been studied by the force-field method. It was calculated that 1,4-cyclohcxadiene is planar (D2 ), although one electron diffraction study (Oberhammer and Bauer, 1969) on the molecule indicated a boat form (C2 ). An independent electron diffraction work indeed gave a planar structure (Dallingaand Toneman, 1967). The problem with the former electron diffraction study seems to have been a misinterpretation of the observed 3,6-distance, a problem related to what is sometimes referred to as shrinkage (see Bartell and Kohl, 1963 and references therein). This effect occurs because the atomic nuclei are undergoing vibrational motion. Thus, one mode of vibration of 1,4-cyclo-hexadiene is as shown in eqn (11). 

Cyclo-octatetraene presents an interesting calculational problem. The molecule exists in a tub conformation, the structure of which is well established experimentally (Traetteberg, 1966). Two barriers to inversion have also been determined. The molecule can invert (or reach a planar conformation) with or without a bond shift, so that the double bonds become single, and the single bonds become double. The barriers to both processes are known. The simple mechanical inversion can be treated adequately by force-field methods, and that barrier has been calculated to be 15-1 kcal mole-1 (Allinger et al., 1973) [experimental (Anet, 1962 Anet et al., 1964) 13 7 kcal mole-1 for AG+]. Because of quantum mechanical complications, the force-field method is not suitable in its present form for calculation of the barrier to bond shifting. 

We continue to believe that the force-field method offers a rapid, convenient and reliable method for the determination of molecular structures and energies. While there are limitations to the method, as there are with each of the experimental methods, the usefulness of this technique now seems generally appreciated. We can forsee only a continuing expansion of the development and applications of force-field calculations in many areas of chemistry. 

Fig. 4.17 Unit cells of PAFs, a PAF-301, b PAF-302, c PAF-303, and d PAF-304, derived from topology design and geometry optimization with the force field method. Here, gray and pink spheres represent carbon and hydrogen atoms, respectively, while the blue polyhedron represents the tetrahedrally bonded carbon atoms. In addition, the yellow sphere denotes the pores in 3D PAFs. Reproduced from Ref. [103] with permission from the American Chemical Society...

Kao J, Allinger NL (1977) Conformational Analysis — 122 - Heats of Formation of Conjugated Hydrocarbons by the Force Field Method. J Am Chem Soc 99 975... 

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