Potential energy function, for

Jones D T and J M Thornton 1996. Potential Energy Functions for Threading. Current Opinion in Structural Biology 6 210-216. 

In the two-dimensional theory of solids, the potential energy functional for the shallow shell with the mid-surface is as follows ... 

A. Potential Energy Functions for the Treatment of Biological Molecules... 

Jones, D.T., Thornton, J. Potential energy functions for threading. Curr. Opin. Struct. Biol. 6 210-216, 1996. 

The conclusion that each inner-core spheron in a stable core should ligate its neighbors about itself in a way corresponding to local stability is a reasonable consequence of the self-generating character of the potential energy function for nucleons in nuclei (mutual interdependence of structure and potential energy function) and the short range of internucleonic forces. 

Similarly, we often try to interpret changes in the free energy in terms of contributions to the potential energy function. For instance, one might want to know whether AA is primarily driven by electrostatic or van der Waals interactions. Alternatively, one might be interested in finding out what are the contributions to AA arising from... 

Table 16-4 shows the IUPAC classification of pores by size. Micropores are small enough that a molecule is attracted to both of the opposing walls forming the pore. The potential energy functions for these walls superimpose to create a deep well, and strong adsorption results. Hysteresis is generally not observed. (However, water vapor adsorbed in the micropores of activated carbon shows a large hysteresis loop, and the desorption branch is sometimes used with the Kelvin equation to determine the pore size distribution.) Capillary condensation occurs in mesopores and a hysteresis loop is typically found. Macropores form important paths for molecules to diffuse into a par-... 

Consider a one-dimensional system in which the potential energy functions for the oxidized and reduced states are ... 

At the same time that Heisenberg was formulating his approach to the helium system, Born and Oppenheimer indicated how to formulate a quantum mechanical description of molecules that justified approximations already in use in treatment of band spectra. The theory was worked out while Oppenheimer was resident in Gottingen and constituted his doctoral dissertation. Born and Oppenheimer justified why molecules could be regarded as essentially fixed particles insofar as the electronic motion was concerned, and they derived the "potential" energy function for the nuclear motion. This approximation was to become the "clamped-nucleus" approximation among quantum chemists in decades to come.36... 

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). 

One of the important electrochemical interfaces is that between water and liquid mercury. The potential energy functions for modeling liquid metals are, in general, more complex than those suitable for modeling sohds or simple molecular liquids, because the electronic structure of the metal plays an important role in the determination of its structure." However, based on the X-ray structure of liquid mercury, which shows a similarity with the solid a-mercury structure, Heinzinger and co-workers presented a water/Hg potential that is similar in form to the water/Pt potential described earlier. This potential was based on quantum mechanical calculations of the adsorption of a water molecule on a cluster of mercury atoms. ... 

In order to study the behavior of ions at the water/metal interface using the molecular dynamics method, the potential energy functions for the interaction between the ions and the water and between the ions and the metal surface must be specified. 

In principle, it is a simple matter to include solvent water molecules directly in MD simulations, since appropriate intermolecular potential energy functions for water are available (1Z 37,38) one would just surround the solute molecules with a sufficient number of water molecules to approximate a bulk solution. Unfortunately, a "sufficient number of water molecules might be enormous, since many of the effects of aqueous solvation are long range or are due to entropic contributions arising from "structuring of the solvent, which may be cooperative in nature. 

PEF95SAC optimized potential energy function for alcohols and carbohydrates. J. Carbohydr. Chem. 16, 751-772. 

Schlenkrich, M., Brickmann, J., MacKerell Jr., A. D., and Karplus, M. (1996) An empirical potential energy function for phospholipids criteria for parameter optimization and application. In Biological membranes a molecular perspective from computational and experiment, Merz Jr., K. M. and Roux, B. (eds.), Birkhauser, Boston, 31-81. 

Varshni, Y. P. (1957). Comparative study of potential energy functions for diatomic molecules. Rev. Mod. Phys. 29, 664-682. 

Let us consider for the moment the potential energy function in an abstract form. A useful potential energy function for a bond between atoms A and B should have an analytic form. Moreover, it should be continuously differentiable. Finally, assuming the dissociation energy for the bond to be positive, we will define the minimum of the function to have a potential energy of zero we will call the bond length at the minimum r. We can determine the value... 

Having discussed the form of analytical potentials for three special categories of surfaces 1 will now describe a more general method for obtaining analytical potential energy functions for polyatomic molecules from spectroscopic data (755). The method is probably simpler to follow if a specific example is taken. 

The potential energy function, for those vibrations in which the two H atoms lie always along the O 0 lines, is a function of four variables r% r4. But wo may consider, to simplify matters, the symmetrical... 

From the above assumptions the potential energy function for the... 

Figure 4.20 Variation of potential energy function for ground and excited states of biphenyl (single bond) and ethylene (double bond).

Computer simulations are regularly performed of the energetics of protein structure and their interactions with ligands using potential energy functions for the forces described so far. The energy functions are of necessity simplifications, but they are calibrated on experimental data. A minimalist model uses equation 11.3.15... 

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