Forms of Potential Energy Functions

One of the most popular and most extensively used molecular mechanics force fields is MM2, developed by Allinger and co-workers, The latest version of this program is MM2(91). The MM3 method - is an enhancement of MM2 with significant improvements in the functional forms of the potential energy functions. The current version of MM3 is MM3 (94). We will use these force fields as reference points in our principal discussion. 

During the last 10 years several different molecular mechanics force fields have emerged. These are today most often distributed together with a graphics interface. A number of these force fields, such as MM2 and MM3 in the 

The behavior of bond stretching/compression is well described by the Morse function given in Eq. [2]  

In a molecular mechanics force field, the starting point for describing bond stretching and compression is the harmonic approximation. The simplest approach is the use of a quadratic potential energy function as shown in Eq. [3]  

The TRIPOS force fields in the SYBYL and Alchemy programs and the Chem-X, CHARMm, and COSMIC force fields all employ a simple harmonic potential (Eq. [1]) for bond stretching/compression. The CVFF, DREIDING, and UFF force fields support a Morse potential as well as a harmonic potential. The harmonic function is the default in DREIDING and UFF. 

---

The reduced variables are employed as in the theory of the corresponding states therefore a universal form of potential-energy function has to be assumed by all components of the mixture. The validity of this hypothesis is doubtful even in the case of small molecules, and even more so in the case of polymer solutions. 

The molecules are represented as objects exerting forces on their neighbors (see, e.g.. Berry et al., 2000, chapter 10), so that a potential energy of pairwise interaction may be defined. As was mentioned in section 5.1, the simplest form of potential energy function usable with atoms is that between hard spheres. A more realistic function, but one which is still reasonably easy to handle mathematically, is the Lennard-Jones (1924) potential, a common form of which is represented by Equation 5.4 ... 

The parameter sets were modified slightly, as described in detail in the paper. This was done partly to conform to our new forms of potential energy functions, see sections 9 1, 9 2 and 11.6.2, partly to take into account the special problems encountered with torsional angles in spiro compounds containing small rings. The parameter sets were checked on cyclohexane, cyclopentane, cyclobutane and cyclopropane with good results except for the vibrational spectrxim of cyclopropane and the structure of cyclobutane which came out planar as in some of its derivatives. 

The techniques collectively termed molecular mechanics (MM) employ an empirically derived set of equations to describe the energy of a molecule as a function of atomic position (the Born—Oppenheimer surface). The mathematical form is based on classical mechanics. This set of potential energy functions (usually termed the force field) contains adjustable parameters that are optimized to fit calculated values of experimental properties for a known set of molecules. The major assumption is, of course, that these parameters are transferable from one molecule to another. Computational efficiency and facile inclusion of solvent molecules are two of the advantages of the MM methods. 

The emphasis in this section is on the form of the kinetic energy operator. The choice of potential energy functions is considered in Section IV dealing with specific molecules. Emphasis is also placed on treatment of the vibrational data since a review emphasizing treatment of microwave data for ring puckering has appeared recently19). 

The molecular mechanical approach to simulating biomacromolecular structures by the use of potential energy functions has been discussed. These potential energy functions are an enthalpic contribution to the free energy, but the free energy contains entropic contribution. There are two forms of entropy in a biomacromolecular system namely the conformational entropy, which entails the inherent entropy of the biomacromolecular structure and the solvent entropy, which results from interactions between a biomacromolecule and solvent molecules. 

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 Finnis-Sinclair analytic functional form was introduced at about the same time as two other similar forms, the embedded-atom method > and the glue model." ° However, the derivation of the Finnis-Sinclair form from the second-moment approximation is very different from the interpretation of the other empirical forms, which are based on effective medium theory as discussed later. This difference in interpretation is often ignored, and all three methods tend to be put into a single class of potential energy function. In practice, the main difference between the methods lies in the systems to which they have been traditionally applied. In developing the embedded-atom method, for example, Baskes, Daw, and Foiles emphasized close-packed lattices rather than body-centered-cubic lattices. Given that angular interactions are usually ig-... 

Finally, we note that the use of the term force field in the literature is somewhat arbitrary. It is usually applied to both the set of potential energy functions and their corresponding parameters, but it is also common for some authors to refer to just a certain functional form as the force field. 

U, H, A, and G form the short list, but not the whole list, of potential energy functions. Potentials such as <1> are equally valid and obtain by Legendre transformation through ... 

Establishment of CG protocols for proteins having broader applicability than elastic network models requires the definition of potential energy functions containing terms that are based on more general assumptions. The Holy Grail of the present research would be the definition of a universal form for such terms that would guarantee transferability over multiple systems keeping reasonable levels of reliability over several properties of interest. 

This Just shows that the actual analytical form of the non-bonded functions is of no consequence, and that the inclusion of the unit cell of ethane into optimisation has only marginal effect. When the functional form has little or no importance, the simplest can be used, which is the (A,B) form with one-atom parameters. This conclusion is encouraging for later optimisation of potential energy functions which include heteroatoms. 

A particularly important application of molecular dynamics, often in conjunction with the simulated annealing method, is in the refinement of X-ray and NMR data to determine the three-dimensional structures of large biological molecules such as proteins. The aim of such refinement is to determine the conformation (or conformations) that best explain the experimental data. A modified form of molecular dynamics called restrained moleculai dynarrdcs is usually used in which additional terms, called penalty functions, are added tc the potential energy function. These extra terms have the effect of penalising conformations... 

Of the biomolecular force fields, AMBER [21] is considered to be transferable, whereas academic CHARMM [20] is not transferable. Considering the simplistic form of the potential energy functions used in these force fields, the extent of transferability should be considered to be minimal, as has been shown recently [52]. As stated above, the user should perform suitable tests on any novel compounds to ensure that the force field is treating the systems of interest with sufficient accuracy. 

---