Minimum perturbation method

To overcome the limitations of the database search methods, conformational search methods were developed [95,96,109]. There are many such methods, exploiting different protein representations, objective function tenns, and optimization or enumeration algorithms. The search algorithms include the minimum perturbation method [97], molecular dynamics simulations [92,110,111], genetic algorithms [112], Monte Carlo and simulated annealing [113,114], multiple copy simultaneous search [115-117], self-consistent field optimization [118], and an enumeration based on the graph theory [119]. 

The design of fluorescent sensors is of major importance because of the high demand in analytical chemistry, clinical biochemistry, medicine, the environment, etc. Numerous chemical and biochemical analytes can be detected by fluorescence methods cations (H+, Li+, Na+, K+, Ca2+, Mg2+, Zn2+, Pb2+, Al3+, Cd2+, etc.), anions (halide ions, citrates, carboxylates, phosphates, ATP, etc.), neutral molecules (sugars, e.g. glucose, etc.) and gases (O2, CO2, NO, etc.). There is already a wide choice of fluorescent molecular sensors for particular applications and many of them are commercially available. However, there is still a need for sensors with improved selectivity and minimum perturbation of the microenvironment to be probed. Moreover, there is the potential for progress in the development of fluorescent sensors for biochemical analytes (amino acids, coenzymes, carbohydrates, nucleosides, nucleotides, etc.). 

Ion [392] and the related ions [397]-[399] have been studied theoretically by MINDO/3 and perturbation methods (Jorgensen, 1977). No energy minimum was obtained for the trishomocyclopropenium ion [398]. The energy proximity of the ions in (255) suggests that complete carbon scrambling should be facile for isolated [392] in the gas phase. The energies have... 

To solve this equation the one electron approximation is used. The differences in the treatments of the various authors lie in the differences in their assumptions about orthogonality in the wave functions used and in the use of either a variational or perturbation method to find the minimum energy. 

This corresponds to describing the function/in an M-dimensional space of the basis functions x- For a fixed basis set size M, only the components of/that lie within this space can be described, and/is therefore approximated. As the size of the basis set M is increased, the approximation becomes better since more and more components of / can be described. If the basis set has the property of being complete, the function / can be described to any desired accuracy, provided that a sufficient number of functions are included. The expansion coefficients C are often determined either by variational or perturbational methods. For the expansion of the molecular orbitals in a Flartree-Fock wave function, for example, the coefficients are determined by requiring the total energy to be a minimum. 

The method provides a means of studying biochemical and biophysical characteristics of many lipids, proteins, and amino acids. Many investigations have been devoted to the synthesis of radicals having a suitable reactive functional group to couple as selectively as possible and with minimum perturbation of the biological system. The latter requirement... 

Here, a and b are constants and p, q, and s are the chemical potential, hardness, and local softness, respectively. Based on their proof, they proposed the following rule for hard-hard interaction cases, the site of the minimum Fukui function is preferred then again, the site with the maximum Fukui function is preferred in the soft-soft interaction cases. They have further shown that the global hardness indicates the chemical reactivity of a molecule as a whole, whereas the Fukui function determines chemical selectivity. From the above discussions, the energy-density perturbation method with the properly chosen perturbation variables, within the framework of DFT, indeed provides a formal proof for the HSAB principle. There have also been many numerical studies that have supported the HSAB principle. ... 

The kind of information obtainable through the use of the isotopic perturbation method is manifold and the problems which can be investigated cover a broad range. Fundamental questions connected to the interplay between molecular dynamics and structure like the distinction between time-averaged symmetry of molecules with a multiple minimum energy... 

Historically, the basis of gas permeation module design was first proposed by Weller and Steiner in 1950. Nowadays, modem computation techniques provide numerical solutions to the problems thanks to dedicated routines. Orthogonal collocation methods or perturbation methods " are reported to be particularly attractive when a minimum resolution time and computational efforts are required. Several of these routines have been implemented in commercial process simulation software, where advantage can be taken of thermodynamics or unit operation design packages in order to simulate hybrid or multi-stage operations with gas separation membranes. Nevertheless, much effort has been devoted to... 

Many groups are now trying to fit frequency shift curves in order to understand the imaging mechanism, calculate the minimum tip-sample separation and obtain some chemical sensitivity (quantitative infonuation on the tip-sample interaction). The most conunon methods appear to be perturbation theory for considering the lever dynamics [103], and quantum mechanical simulations to characterize the tip-surface interactions [104]. Results indicate that the... 

The systematic, or grid, search is only possible for small molecules. For larger systems tlrere are methods which can be used for perturbing a geometry from a local minimum to another minimum. Some commonly used methods for conformational sampling are ... 

In stochastical methods the random kick is typically somewhat larger, and a standard minimization is carried out starting at the perturbed geometry. This may or may not produce a new minimum. A new perturbed geometry is then generated and minimized etc. There are several variations on how this is done. 

Deuterium Substitution. The a and P secondary isotope effects affect the rate in various ways (p. 298). The measurement of a secondary isotope effects provides a means of distinguishing between SnI and Sn2 mechanisms, since for Sn2 reactions the values range from 0.95 to 1.06 per a D, while for S l reactions the values are higher. This method is especially good because it provides the minimum of perturbation of the system under study changing from a H to a D hardly affects the reaction, while other probes, such as changing a substituent or the polarity of the solvent, may have a much more complex effect. 

Steepest descent can terminate at any type of stationary point, that is, at any point where the elements of the gradient of /(x) are zero. Thus you must ascertain if the presumed minimum is indeed a local minimum (i.e., a solution) or a saddle point. If it is a saddle point, it is necessary to employ a nongradient method to move away from the point, after which the minimization may continue as before. The stationary point may be tested by examining the Hessian matrix of the objective function as described in Chapter 4. If the Hessian matrix is not positive-definite, the stationary point is a saddle point. Perturbation from the stationary point followed by optimization should lead to a local minimum x. ... 

The exchange contribution in an ab initio perturbation theory is the only repulsive term ) around the energy minimum in most of the stable complexes and consequently we would expect no net repulsion between two closed shell molecules in semiempirical calculations. On the other hand it is known from actual calculations that intermolecular interactions are described more or less correctly by the CNDO/2 procedure. Indeed, strong repulsion is obtained between closed shell molecules. Evidently there must be another approximation which compensates for the neglect of exchange energy. With regard to the simplifications of the CNDO/2 method we find that this is in fact the case. The approximation shown in Eq. (17) is responsible for the repulsive term. 

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