Potential functions minimization

It is beyond the scope of this short review to list every available molecular mechanics program. Only a selected few programs are mentioned here, without descriptive details of the potential functions, minimization algorithms, or comparative evaluations. Both the CHARMM and AMBER force fields use harmonic potential functions to calculate protein structures. They were developed in the laboratories of Karplus and Kollman, respectively, and work remarkably well. The CFF and force fields use more complex potential functions. Both force fields were developed in commercial settings and based extensively or exclusively on results obtained from quantum mechanics. Unlike the other molecular mechanics methods, the OPLS force field was parameterized by Jorgensen to simulate solution phase phenomena. 

Jorgensen, W.L. Tirado-Rives, J. The OPLS potential functions for proteins. Energy minimizations for crystals of cyclic peptides and crambin. J. Am. Chem. Soc. 110 1657-1666, 1988... 

Order 2 minimization algorithms, which use the second derivative (curvamre) as well as the first derivative (slope) of the potential function, exhibit in many cases improved rate of convergence. For a molecule of N atoms these methods require calculating the 3N X 3N Hessian matrix of second derivatives (for the coordinate set at step k)... 

The OPLS Potential Function for Proteins. Energy Minimization for Crystals of Cyclic Peptides and Crambin William L. Jorgensen and Julian Tirado-Rives Journal of the American Chemical Society 110 (1988) 1657... 

Potential functions induced-dipole terms, 84-85 minimization, 113-116 nonbonded interactions, 84-85 Potential of mean force, 43, 144 Potential surfaces, 1,6-11, 85, 87-88, 85 for amide hydrolysis, 176-181,178,179, 217-220, 218... 

Minimization of this quantity gives a set of new coefficients and the improved instanton trajecotry. The second and third terms in the above equation require the gradient and Hessian of the potential function V(q)- For a given approximate instanton path, we choose Nr values of the parameter zn =i 2 and determine the corresponding set of Nr reference configurations qo(2n) -The values of the potential, first and second derivatives of the potential at any intermediate z, can be obtained easily by piecewise smooth cubic interpolation procedure. 

Optimization problems in process design are usually concerned with maximizing or minimizing an objective function. The objective function might typically be to maximize economic potential or minimize cost. For example, consider the recovery of heat from a hot waste stream. A heat exchanger could be installed to recover the waste heat. The heat recovery is illustrated in Figure 3.1a as a plot of temperature versus enthalpy. There is heat available in the hot stream to be recovered to preheat the cold stream. But how much heat should be recovered Expressions can be written for the recovered heat as ... 

It is finally assumed that with all force constants and potential functions correctly specified in terms of the electronic configuration of the molecule, the nuclear arrangement that minimizes the steric strain corresponds to the observed structure of the isolated (gas phase) molecule. In practice however, the adjustable parameters, in virtually all cases, are chosen to reproduce molecular structures observed by solid-state diffraction methods. The parameters are therefore conditioned by the crystal environment and the minimized structure corresponds to neither gas phase nor isolated molecule [109],... 

A new feature in MM3 is the full Newton-Raphson minimization algorithm. This allows for the location and verification of transition states and for the calculation of vibrational spectra. Indeed, many of the new potential functions in MM3 were included to provide a better description of the potential energy surface which is required for an accurate calculation of vibrational spectra. 

One can be easily convinced that this prescription is correct, if one compares the variation of the functionals minimized in Eqs. (87) and (64) the last minimization being equivalent to that in Eq. (28), solved via Eq. (33). Because the effective external potential (89) depends functionally on n r), an iterative method, leading to self-consistency, must be employed. 

Next, when we compare the variation of the functional minimized in Eq. (108) with that in Eq. (101), we conclude that the exact GS problem can be solved by algorithms of the OP method, if an OP effective external potential... 

The Gordon-Kim interaction functions may be compared with empirical potential functions derived by energy- or net-force minimization methods using known crystal structures. The O—O Gordon-Kim potentials are more repulsive, as illustrated in Fig. 9.2. Spackman points out that the empirical potentials likely contain a significant attractive component because of the inadequate allowance for electrostatic interactions in their derivation. This attractive component is included in the electrostatic interaction in the density functional model. 

Considering additional functionalities in an aromatic ring allows for conclusions with implications for coal chemistry. Coal is a vital fossil fuel about 50% of the United States is dependent on coal for electric power generation, and its use accounts for 90% of Ohio s electrical power. Current clean-coal engineering efforts are underway to maximize coal s energy potential while minimizing harmful environmental emissions (i.e., Hg, SO, NO, and C02). ... 

Potential functions that have been used include the Morse type or the Lennard-Jones type. The potential function is generally minimized by the greatest number of nearest-neighbor bonds. 

Jorgensen WL, J Tirado-Rives (1988) The OPLS Potential Functions for Proteins - Energy Minimizations for Crystals of Cyclic-Peptides and Crambin. J. Am. Chem. Soc. 110 (6) 1657-1666... 

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