Static Approach

In this approach properties of potential energy surfaces are investigated from the point of view of all possible monomolecular transformations of the given reactants. A plausible suggestion concerning the mechanism of the reaction under study is usually made on the basis of reaction barriers or activation energies. Moreover, in some studies, partition functions are evaluated and rate constants are obtained within the framework of the absolute rate theory. 

Results of nonempirical quantum chemical energy and geometry calculations are available in the literature (117,118) for hundreds of molecules. 

Recently, a technique (129) for localization of a point on the crossing seam of two energy surfaces has been suggested, followed by a method of going downhill to the minima on both surfaces, which is equivalent to IRC for single PES reactions. 

To predict a reasonable energy and structure of a transition state, an extensive basis set must be used and the correlation energy must be included. 

Chemical intuition, which is very useful in the location of minima, usually fails in the search for saddle points. The symmetry of the transition-state structure was discussed and it was shown that many transition states exhibit low symmetry (130-132). Some empirical rules have been developed for approximate localization of the transition state. The rule referred to in the literature as the Bell-Evans- Polanyi-Leffler-Hammond rule is well known the more exoergic the process, the more the transition-state geometry resembles that structure which belongs to a higher energy minimum on the given potential energy surface. This idea has been supported by calculations on SN2-activated complexes (133). 

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From 1933 85>, several theoretical approaches to the problem of the chemical reactivity of planar conjugated molecules began to appear, mainly by the Huckel molecular orbital theory. These were roughly divided into two groups 36>. The one was called the "static approach 35,37-40)j and the other, the "localization approach 41,42). in 1952, another method which was referred to as the "frontier-electron method was proposed 43> and was conventionally grouped 44> together with other related methods 45 48> as the "delocalization approach". 

There are two basic approaches to foundation design equivalent static and dynamic. The equivalent static approach is almost always selected because of its simplicity. However, sometimes an overly conservative design could result. The dynamic approach involves a very complex analysis, although it should result in a more realistic design. 

Calculations on dynamics of solvation shells are still in their infancy. However, very recent papers on this subject, show that in most examples we cannot expect a realistic picture of solvent shells from a purely static approach. Most probably, molecular dynamics calculations and Monte Carlo methods will produce a variety of interesting data and will improve our knowledge on solvation of ions substantially. 

For compounds that are usually not available in larger amounts or are expensive, e.g., many transition metal complexes, the static approach was applied as well (Fig. 14). Here, the cell is loaded with a known amount of the solute and SCCO2 of known density [124]. This technique is convenient, because it allows for in situ analysis, and the solvatation equilibrium is obtained easily. 

Static Approach. There appears to be only one author, namely Passler (1974a,b, 1975a,b, 1976a,b, 1977a,b, 1978, 1980a,b, 1981), who has extensively worked with the static approximation, Eq. (31), in application to semiconductors, with a very recent additional such treatment by Morante et al. (1982). Other recent work, on rare earth ions in ionic crystals, is, for instance, given by Pukhov and Sakun (1979). 

Two dynamic alternatives to the static approach have been used in HO calibration and measurement. In the CSTR (continuously stirred tank reactor) approach, air containing the tracer or tracers flows into the reactor to balance the bulk flow out to the HO measuring devices, and the contents are stirred by a fan or other means. The HO chemical tracer is measured in the inlet flow to obtain [T]() and in the outlet flow to obtain [T], Mass balance requires... 

The kinetic method produced results concordant with those derived from the direct static approach (13), and leads to a value of 331 liter/mole for Kn as compared with KD= 240 30 liter/mole obtained by extrapolation of K d to M0= 0. In view of the steepness of the extrapolation, the kinetic value seems to be more reliable. 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

The focus of previous sections was on cases where spectroscopic parameters in condensed phases could be computed by an essentially static approach the PCM was able to effectively reproduce the influence of the solvent on the EPR parameters in some instances, the explicit introduction of some first-shell solvent molecules [33] also proved necessary. 

IR, Raman and related phenomena) to describe with a static approach the salient aspects of phenomena, which are essentially of a dynamical nature [1], This regime was later shown to be essential for a correct description of the photophysical phenomena. It introduces in the QM formalism aspects that are not present in the standard formulation, particularly, that the excited states activated by the excitation process are not orthogonal to the fundamental one (a similar effect is present in the emission process). The orthogonality among states is a basic tenet of the standard formulation, and the selection rules are based on this property. The description obtained with this model is more realistic than the standard one, when the chromophore is immersed into a responsive medium. Discrete solvent simulation methods could hardly describe these effects. 

In the preceding, we considered the use of electron densities of excited states in predicting photochemical reactivity. This is the static, starting state technique of the type considered in the introduction. Another static approach makes use of excited state bond orders to predict reactivity. The basic assumption is that where the excited state has a high bond order between two orbitals at two centers of the molecule, there will be a tendency for these two centers to bond. Where the bond order is low, or especially where negative, the two centers will tend to repel. Two such centers certainly will not tend to form a bond. If already attached, the bond between the centers will tend to break. 

Design of molecular materials with specific properties often requires interdisciplinary research involving various experimental and theoretical techniques. Molecular modeling by ab initio methods based on quantum-mechanics is now commonly used in such studies. However, theoretical investigations are still dominated by traditional, static approaches in which the stationary points on the respective potential... 

The main purpose of this chapter is to present the basics of ab initio molecular dynamics, focusing on the practical aspects of the simulations, and in particular, on modeling chemical reactions. Although CP-MD is a general molecular dynamics scheme which potentially can be applied in combination with any electronic structure method, the Car-Parinello MD is usually implemented within the framework of density functional theory with plane-waves as the basis set. Such an approach is conceptually quite distant from the commonly applied static approaches of quantum-chemistry with atom-centered basis sets. Therefore, a main... 

In this section we will present results of ab initio molecular dynamics simulations performed for more complex chemical reactions. Catalytic copolymerization of a-olefins with polar group containing monomers, chosen here as an example, is a complex process involving many elementary reactions. While for many aspects of such a process the standard approach by static quantum chemical calculations performed for the crucial reaction intermediates provides often sufficient information, for some aspects it is necessary to go beyond static computations. In the case of the process presented here, MD was priceless in exploring the potential energy surfaces for a few elementary reactions that were especially difficult for a static approach, due to a large number of alternative transition states and thus, alternative reaction pathways.77... 

It is the purpose of this review to present an outline of both the dynamic and static approaches to theoretical studies of reaction mechanisms. The dynamic approach may be regarded—as far as the physics involved is concerned—as the more sophisticated of the two. However, it has limitations in answering many practical questions which a chemist may ask. The static approach, on the other hand, may seem not to be so sophisticated, but at present it provides answers to everyday chemistry in a variety of fields at different levels of theory. 

The calculation of reaction profiles is one of the main subjects of the static approach (see later) the relevance of reaction profiles in representing the potential energy determining the course of elementary processes is given by the adequacy of expansion Eq. (36) and of neglecting the second-order term. It is obvious that expansion Eq. (36) tends to be more adequate when the kinetic energy content in the evolving polyatomic system is small. 

In order to be able to describe kinetic phenomena in terms of the statistical methods, the knowledge of the critical configuration on the PES is needed as well as the eigenvalues of all the nonelectronic degrees of freedom of the system in the transition state. As shown in the previous sections, information on the critical configurations is the subject of quantum chemical calculations within the static approach. 

For planar reference structures, the bnear term is absent for symmetry reasons and the key role is played by mean square amplitudes, which, however, can be quite large and badly described at the harmonic level. From a quantitative point of view, vibrational averaging changes the equilibrium value of Ah by about 2 G (10%) and that of ac by about 10 G (30%) thus quantitative (and even semiquantitative) agreement with experiment cannot be obtained by static approaches, irrespective of the quality of the electronic model. 

The adjustment of the symmetric function to the energy flows from wood carbonization is an original and dynamic (and no more static) approach. The analysis of the symmetric logistic function demonstrates again the dramatic effect of water. As for mass flows, energy flows are delayed and slowed down for wet wood samples (H37). Water intervenes through the large quantities of heat it requires to he evaporated and eliminated from the solid matrix. 

In the present article, we report a study concerning the reaction mechanism of a prototype reaction using both static and dynamic approaches to explore a DFT potential surface. The static approach is the standard IRC model, while the dynamic one is based on a Carr-Parrinello method performed with localized (Gaussian) orbitals, the so-called atom-centered density matrix propagation (ADMP) model.25 Our aim is to elucidate the differences, and the common aspects, between the two approaches in the analysis of bond breaking/formation. To this end, we have chosen topological quantities as probe molecular descriptors. 

In order to rationalize the differences observed between the static approach and dynamic process, we investigated the intra- and intermolecular charge transfers along the different reaction paths. As a first step, we looked at the AIM atomic populations for the two states of interest, i.e. the transition state and ion-molecule complex. The results are gathered in Table 3. At the transition state, each chlorine atom bears a negative charge of -0.70 e-. This electron excess corresponds to the sharing of the unit... 

To explain the mechanism of migration and separation of compounds on the column, the oldest model, known as Craig s theoretical plate model is a static approach now judged to be obsolete, but which once offered a simple description of the separation of constituents. 

Testing Conditions and Analysis. The fracture behavior was investigated at room temperature at nominal piston velocities, from 10-4 m/s to 10 m/s. For test speeds higher than 10-1 m/s, the damped test procedure described in reference 15 was used. Quasi-static stress conditions therefore prevailed in the specimen, even at high loading rates. This fact allowed the analysis of fracture-mechanics parameters to be performed using a static approach. 

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