Saddle point of reaction

At R > 400 pm the orientation of the reactants looses its importance and the energy level of the educts is calculated (ethene + nonclassical ethyl cation). For smaller values of R and a the potential energy increases rapidly. At R = 278 pm and a = 68° one finds a saddle point of the potential energy surface lying on the central barrier, which can be connected with the activated complex of the reaction (21). This connection can be derived from a vibration analysis which has already been discussed in part 2.3.3. With the assistance of the above, the movement of atoms during so-called imaginary vibrations can be calculated. It has been attempted in Fig. 14 to clarify the movement of the atoms during this vibration (the size of the components of the movement vector... 

The transition-state energy is defined as the saddle point of the energy of the system when plotted as a function of the reaction coordinates illustrated in Figure 1.1. 

Botschwina, P, 1998, The Saddle Point of the Nucleophilic Substitution Reaction Cl- + CH3C1 Results of Large-Scale Coupled Cluster Calculations , Theor. Chem. Acc., 99, 426. 

If only the solvation of the gas-phase stationary points are studied, we are working within the frame of the Conventional Transition State Theory, whose problems when used along with the solvent equilibrium hypothesis have already been explained above. Thus, the set of Monte Carlo solvent configurations generated around the gas-phase transition state structure does not probably contain the real saddle point of the whole system, this way not being a correct representation of the conventional transition state of the chemical reaction in solution. However, in spite of that this elemental treatment... 

Let us take a simple example, namely a generic Sn2 reaction mechanism and construct the state functions for the active precursor and successor complexes. To accomplish this task, it is useful to introduce a coordinate set where an interconversion coordinate (%-) can again be defined. This is sketched in Figure 2. The reactant and product channels are labelled as Hc(i) and Hc(j), and the chemical interconversion step can usually be related to a stationary Hamiltonian Hc(ij) whose characterization, at the adiabatic level, corresponds to a saddle point of index one [89, 175]. The stationarity required for the interconversion Hamiltonian Hc(ij) defines a point (geometry) on the configurational space. We assume that the quantum states of the active precursor and successor complexes that have non zero transition matrix elements, if they exist, will be found in the neighborhood of this point. 

The relationship between the geometry of the saddle point of index one (SPi-1) and the accessibility to the quantum transition states cannot be proved, but it can be postulated [43,172], To some extent, invariance of the geometry associated with the SPi-1 would entail an invariance of the quantum states responsible for the interconversion. Thus, if a chemical process follows the same mechanism in different solvents, the invariance of the geometry of the SPi-1 to solvent effects would ensure the mechanistic invariance. This idea has been proposed by us based on computational evidence during the study of some enzyme catalyzed reactions [94, 96, 97, 100-102, 173, 174, 181-184],... 

Strong interactions are observed between the reacting solute and the medium in charge transfer reactions in polar solvents in such a case, the solvent effects cannot be reduced to a simple modification of the adiabatic potential controlling the reactions, since the solvent nuclear motions may become decisive in the vicinity of the saddle point of the free energy surface (FES) controlling the reaction. Also, an explicit treatment of the medium coordinates may be required to evaluate the rate constant preexponential factor. 

A theoretical study at a HF/3-21G level of stationary structures in view of modeling the kinetic and thermodynamic controls by solvent effects was carried out by Andres and coworkers [294], The reaction mechanism for the addition of azide anion to methyl 2,3-dideaoxy-2,3-epimino-oeL-eiythrofuranoside, methyl 2,3-anhydro-a-L-ciythrofuranoside and methyl 2,3-anhydro-P-L-eiythrofuranoside were investigated. The reaction mechanism presents alternative pathways (with two saddle points of index 1) which act in a kinetically competitive way. The results indicate that the inclusion of solvent effects changes the order of stability of products and saddle points. From the structural point of view, the solvent affects the energy of the saddles but not their geometric parameters. Other stationary points geometries are also stable. 

The three rate coefficients indicated in eqn. (1) are those for the formation of the psuedo species (AB) from separated A and B, fed, the backward rate coefficient for this step, k d, and the first-order rate coefficient, fea, for the reaction of (AB) to form products. Depending on the nature of the interaction between A and B once they come close to each other, (AB) might be described by several terms. If the psuedo species (AB) is at the saddle point of the potential energy surface between A + B, it would be described as an activated complex. When A and B are in contact, but little or no interaction occurs between them, then the term contact or collision pair is appropriate. Usually, the exact details of the interaction energy between A and B are not known, but will probably be between... 

Transition state theory (Chapter 2, section A) was derived for chemical bonds that obey quantum theory. An equation analogous to that for transition state theory may be derived even more simply for protein folding because classical low energy interactions are involved and we can use the Boltzmann equation to calculate the fraction of molecules in the transition state i.e., = exp(— AG -D/RT), where A G D is the mean difference in energy between the conformations at the saddle point of the reaction and the ground state. Then, if v is a characteristic vibration frequency along the reaction coordinate at the saddle point, and k is a transmission coefficient, then... 

We can now see the difference between the GE and TRIM methods. In the GE algorithm we first minimize the transverse modes and then maximize the reaction mode. In the TRIM method we minimize and maximize simultaneously by introducing an auxiliary function Eq. (6.21). The underlying idea of the GE method are lines connecting stationary points. The idea behind the TRIM method is an auxiliary function (the image function) whose minima coincide with the saddle points of the original function. 

When lithium vapor was reacted with SiCU in a Knudsen cell, and the reaction product was treated with excess methyl chloride, Me4Si was obtained in 5-10% yield. This result was interpreted in terms of the formation of SiLi4117. Interestingly, tetrahedral SiLi4 is a saddle point of 3rd order at the 3-21G level of theory118. The minimum structure of... 

The methods of constructing different reaction paths are described in numerous papers and reviews (see, for example, Truhlar and Garrett [1984, 1987], Garrett et al. [1988], Ischtwan and Collins [1988], and references therein). In the IRC method proposed by Fukui [1970], the steepest descent path from the saddle point of a multidimensional PES V(X) to the reactant and product valleys is found by numerically solving the equation... 

The entropy of activation AS may be regarded as the saddle point of energy over which reactant molecules must pass as activated complexes (Frost and Pearson, 1961 Laidler, 1965). The AS conveys whether a particular reaction proceeds more quickly or slowly than another individual reaction. Negative AS values would depict a system that could ascertain a more ordered molecular arrangement in a shorter period of... 

Fig. 5.1.4 Potential energy surface for the reaction in Eq. (5.54) as a function of the distance y c.AB and tab for a fixed angle between the two distance vectors. The marks the position of the saddle point of the potential energy surface and S the surface spanning the reaction channel.

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