Potential energy function additivity

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... 

The additional penalty function that is added to the empirical potential energy function in restrained dynamics X-ray refinement has the form ... 

A number of more general force fields for the study of small molecules are available that can be extended to biological molecules. These force fields have been designed with the goal of being able to treat a wide variety of molecules, based on the ability to transfer parameters between chemical systems and the use of additional terms (e.g., cross terms) in their potential energy functions. Typically, these force fields have been optimized to... 

Here we present and discuss an example calculation to make some of the concepts discussed above more definite. We treat a model for methane (CH4) solute at infinite dilution in liquid under conventional conditions. This model would be of interest to conceptual issues of hydrophobic effects, and general hydration effects in molecular biosciences [1,9], but the specific calculation here serves only as an illustration of these methods. An important element of this method is that nothing depends restric-tively on the representation of the mechanical potential energy function. In contrast, the problem of methane dissolved in liquid water would typically be treated from the perspective of the van der Waals model of liquids, adopting a reference system characterized by the pairwise-additive repulsive forces between the methane and water molecules, and then correcting for methane-water molecule attractive interactions. In the present circumstance this should be satisfactory in fact. Nevertheless, the question frequently arises whether the attractive interactions substantially affect the statistical problems [60-62], and the present methods avoid such a limitation. 

The assumption of additivity that underlies many empirical intermolecular potentials can be stated more formally as follows. Suppose that A, B, C,... represent chosen molecules in a given spatial configuration. The potential-energy function V(A, B, C,...) (relative to isolated molecules61) will be said to be pairwise additive (and denoted Vpw) if... 

The study of liquids near solid surfaces using microscopic (atomistic-based) descriptions of liquid molecules is relatively new. Given a potential energy function for the interaction between liquid molecules and between the liquid molecules and the solid surface, the integral equation for the liquid density profile and the liquid molecules orientation can be solved approximately, or the molecular dynamics method can be used to calculate these and many other structural and dynamic properties. In applying these methods to water near a metal surface, care must be taken to include additional features that are unique to this system (see later discussion). 

Potential Energy Functions and Parameters. The second point is the importance of non-bonded interactions. The program was developed to optimize also on unit cell dimensions in addition to the usual conformational properties, because this gives the possibility of optimizing on properties that are very sensitive to non-bonded interactions. 

Consider the physical significance of the additional terms in (4.67) as compared to (4.39). The fourth term on the right side of (4.67) represents a shift in the vibrational levels. The constant involves the third and fourth derivatives of V evaluated at Re, and is therefore a consequence of the deviation of the potential energy function from the (quadratic) harmonic-oscillator potential ... 

Before we go on to consider functional forms for all of the components of a molecule s total steric energy, let us consider the limitations of Eq. (2.2) for bond stretching. Like any truncated Taylor expansion, it works best in regions near its reference point, in this case req. Thus, if we are interested primarily in molecular structures where no bond is terribly distorted from its optimal value, we may expect Eq. (2.2) to have reasonable utility. However, as the bond is stretched to longer and longer r, Eq. (2.2) predicts the energy to become infinitely positive, which is certainly not chemically realistic. The practical solution to such inaccuracy is to include additional terms in the Taylor expansion. Inclusion of the cubic term provides a potential energy function of the form... 

A third question which is important for the unsubstituted sugars is whether it is necessary to provide for additional conformational stability resulting from the formation of intramolecular H-bonds in solution. (Intramolecular H-bonds are difficult to incorporate into limiting van der Waals radii because of their vectorial character and uncertainty about their potential energy function.)... 

Fig. 2.7. The bonding potential V(r) and its representation by a harmonic potential energy function and an additional cubic term.

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