Energy-minimization

Lattice dynamics simulations provide a powerful extension of energy minimization methods by evaluating the dynamical matrix that relates forces and atomic displacements for a crystd. Originally developed by Bom and Huang (1954), this method incorporates a statistical mechanics approach to determine the vibrational modes and thermodynamic 

Molecular simulations include the application of FF to model the lowest potential energy of a conformation (energy minimization, EMin), the dynantic properties of macromolec-ular structure (molecular dynamics, MoID) and search for the optimum conformation of a macromolecule (conformation search). In all these techniques, the positions of atoms are perturbed in small increments, and it is difficult to sample all of the possible arrangements of atoms in conformational space. The successful simulation is the one that reproduces the experimentally observed properties of the molecule. It is essential to incorporate as much empirical information as possible into the initial model for simulation. While the overall system must be accurately defined, it is important to understand the limitations of FF since a choice of FF may affect the outcome of the simulation results. 

The most widely used methods fall into two general categories  

steepest descent and related methods such as conjugate gradient, which uses first derivatives, and 

Newton-Raphson procedure, which additionally uses second derivatives. 

The Newton-Raphson methods of EMin (Berkert and Allinger, 1982) use the curvature of the strain energy surface to locate minima. The computations are considerably more complex than the first-derivative methods, but they use the available information more fuUy and therefore converge more quickly. These methods involve setting up a system of simultaneous equations of size (3N - 6) (3N - 6) and solving for the atomic positions that are the solution of the system. Large matrices must be inverted as part of this approach. 

The search for an advantageous condition of a system is a common problem in science and many algorithms are available to optimize objective functions. In this chapter we will discuss the methods commonly used in molecular mechanics. For a more detailed discussion we refer to specialist texts on computational chemistry193-961. 

Prior to minimization, little information is available about the high-dimensional energy surface (3N- 6 dimensions with N atoms). In simple words, the program cannot see the landscape . Ideally, the minimization process should adapt to the shape of the surface and the distance from the minimum. Also, the type of energy minimization procedure used should depend on whether a specific local minimum, or any minimum, is sought. Most programs offer a choice of different optimization methods and the step size may often be chosen interactively. 

The first derivatives of a potential energy function define the gradient of the potential and the second derivatives describe the curvature of the energy surface (Fig. 3.4). In most molecular mechanics programs the potential functions used are relatively simple and the derivatives are usually determined analytically. The second derivatives of harmonic oscillators correspond to the force constants. Thus, methods using the whole set of second derivatives result in some direct information on vibrational frequencies. 

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The effective moment of inertia / and the friction coefficient / could easily be estimated. The force constant k associated with the relative motion of the lobes was determined from an empirical energy function. To do so, the molecule was opened in a step-wise fashion by manipulating the hinge region and each resulting structure was energy minimized. Then, the interaction energy between the two domains was measured, and plotted versus 0. 

M. Levitt and Shneior Lifson. Refinement of protein conformation using a macromolecular energy minimization procedure. J. Mol. Biol., 46 269-279, 1969. 

Bernhard R. Brooks, Robert E. Bruccoleri, Barry D. Olafson, David J. States, S. Swaminathan, and Martin Karplus. CHARMM A program for macro-molecular energy, minimization, and dynamics calculations. J. Comp. Chem., 4(2) 187-217, 1983. 

Quenched dynamics is a combination of high temperature molecular dynamics and energy minimization. This process determines the energy distribution of con formational families produced during molecular dynamics trajectories. To provide a better estimate of conformations, you should combine quenched dynamics with simulated annealing. 

Gibson K D and H A Scheraga 1987. Revised Algorithms for the Build-up Procedure for Predicting lAotein Conformations by Energy Minimization, journal of Computational Chemistry 8 826-834. 

If a molecule is strained, atoms may not be ver y close to the minimum of their individual potential energy wells when the best compromise geometry is reached. In such a case, the geometric criterion does not provide an exit from the loop. Programs are usually written so that they can automatically switch from a geometric minimization criterion to an energy minimization procedure. 

For sueh a funetion, the CI part of the energy minimization is absent (the elassie papers in whieh the SCF equations for elosed- and open-shell systems are treated are C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951) 32, 179 (I960)) and the density matriees simplify greatly beeause only one spin-orbital oeeupaney is operative. In this ease, the orbital optimization eonditions reduee to ... 

Model optimization is a further refinement of the secondary and tertiary structure. At a minimum, a molecular mechanics energy minimization is done. Often, molecular dynamics or simulated annealing are used. These are frequently chosen to search the region of conformational space relatively close to the starting structure. For marginal cases, this step is very important and larger simulations should be run. 

Sometimes, the system of interest is not the inhnite crystal, but an anomaly in the crystal, such as an extra atom adsorbed in the crystal. In this case, the inhnite symmetry of the crystal is not rigorously correct. The most widely used means for modeling defects is the Mott-Littleton defect method. It is a means for performing an energy minimization in a localized region of the lattice. The method incorporates a continuum description of the polarization for the remainder of the crystal. 

Chem3D uses a MM2 force field that has been extended to cover the full periodic table with the exception of the /block elements. Unknown parameters will be estimated by the program and a message generated to inform the user of this. MM2 can be used for both energy minimization and molecular dynamics calculations. The user can add custom atom types or alter the parameters used... 

Brooks, B.R. Bruccoleri, R.E. Olafson, B.D. States, D.J. Swaminathan, S. Karpins, M. CHARMM A program for macromolecular energy, minimization, and dynamics calculations J. Comput. Chem. 4 187-217, 1983. 

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... 

A range of plasticizer molecule models and a model for PVC have been generated and energy minimized to observe their most stable conformations. Such models highlight the free volume iacrease caused by the mobiHty of the plasticizer alkyl chains. More detailed models have also been produced to concentrate on the polar region of the plasticizer and its possible mode of interaction with the polymer. These show the expected repulsion between areas on the polymer and plasticizer of like charge as weU as attraction between the negative portions of the plasticizer and positive portions of the PVC. 

Reactor types modeled A, stoichiometric conversion B, equiUbrium/free-energy minimization, continuous stirred tank, and plug flow C, reactive distillation. Some vendors have special models for special reactions also, private company simulators usually have reactors of specific interest to their company. 

When the kinetics are unknown, still-useful information can be obtained by finding equilibrium compositions at fixed temperature or adiabatically, or at some specified approach to the adiabatic temperature, say within 25°C (45°F) of it. Such calculations require only an input of the components of the feed and produc ts and their thermodynamic properties, not their stoichiometric relations, and are based on Gibbs energy minimization. Computer programs appear, for instance, in Smith and Missen Chemical Reaction Equilibrium Analysis Theory and Algorithms, Wiley, 1982), but the problem often is laborious enough to warrant use of one of the several available commercial services and their data banks. Several simpler cases with specified stoichiometries are solved by Walas Phase Equilibiia in Chemical Engineering, Butterworths, 1985). 

This is a 4 2 reaction, and is thus pressure dependent. However, it is necessary to compute the equilibrium partial pressure of some alternative gaseous species, such as SiCls, and other hydrocarbons such as C2H2 and for this a Gibbs energy minimization calculation should be made. 

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