Variable transition state

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

The fact that the ratios of rates were much greater in chlorination than in nitration, prompted Dewar to suggest that the actual transition state was intermediate between the Wheland model and the isolated molecule model. He accommodated this variation in the relative rates within his discussion by treating yS as a variable whose value depended on the nature of the reaction. With the notation that y ) is the... 

The optimal control is calculated using equation (9.55) and the values of the state variables are found using the plant state transition equation (9.56)... 

We will discuss shortly the most important structure-reactivity features of the E2, El, and Elcb mechanisms. The variable transition state theoiy allows discussion of reactions proceeding through transition states of intermediate character in terms of the limiting mechanistic types. The most important structural features to be considered in such a discussion are (1) the nature of the leaving group, (2) the nature of the base, (3) electronic and steric effects of substituents in the reactant molecule, and (4) solvent effects. 

Fig. 6.3. Variable transition state theoiy of elimination reactions. J. F. Bunnett, Angew. Chem. Int. Ed. Engl. 1, 225 (1962) J. F. Bunnett, Surv. Prog. Chem. 5, 53 (1969) W. H. Saunders, Jr., and A. F. Cockerill, Mechanisms of Elimination Reactions, Wiley, New York, 1973, pp. 48—55 D. J. McLennan, Tetrahedron 31, 2999 (1975) W. H. Saunders, Jr., Acc. Chem. Res. 9, 19 (1976).

There is another useiiil way of depicting the ideas embodied in the variable transition state theory of elimination reactions. This is to construct a three-dimensional potential energy diagram. Suppose that we consider the case of an ethyl halide. The two stepwise reaction paths both require the formation of high-energy intermediates. The El mechanism requires formation of a carbocation whereas the Elcb mechanism proceeds via a caibanion intermediate. 

Fig. 6.5. Representation of changes in transition-state character in the variable transition state E2 elimination reaetion, showing displacement of transition-state location as a result of substituent effects (a) substituent Z stabilizes catfaanion character of Elcb-like transition state (b) substituent R stabilizes carbocation character of El-like transitions state.

Three-dimensional potential energy diagrams of the type discussed in connection with the variable E2 transition state theory for elimination reactions can be used to consider structural effects on the reactivity of carbonyl compounds and the tetrahedral intermediates involved in carbonyl-group reactions. Many of these reactions involve the formation or breaking of two separate bonds. This is the case in the first stage of acetal hydrolysis, which involves both a proton transfer and breaking of a C—O bond. The overall reaction might take place in several ways. There are two mechanistic extremes ... 

The essential goal is to locate the transition state on the RIP diagram. This involves speeifying two eoordinates, so two quantitative progress variables are required. One approaeh, fairly widely applied, ean be illustrated with the study by Hill et al. of the general aeid-eatalyzed addition of substituted anilines to diey-anamide. The overall reaetion is... 

Dipolar aprotic solvents have similar effects on the transition state any significant differences arise from variable effects on the reactants. 

The first SN2 reaction variable to look at is the structure of the substrate. Because the S, j2 transition state involves partial bond formation between the incoming nucleophile and the alkyl halide carbon atom, it seems reasonable that a hindered, bulky substrate should prevent easy approach of the nucleophile, making bond formation difficult. In other words, the transition state for reaction of a sterically hindered alkvl halide, whose carbon atom is "shielded" from approach of the incoming nucleophile, is higher in energy... 

This is a reaction in which neutral molecules react to give a dipolar or ionic transition state, and some rate acceleration from the added neutral salt is to be expected53, since the added salt will increase the polarity or effective dielectric constant of the medium. Some of the rate increases due to added neutral salts are attributable to this cause, but it is doubtful that they are all thus explained. The set of data for constant initial chloride and initial salt concentrations and variable initial amine concentrations affords some insight into this aspect of the problem. 

An alternate possibility is that one of the component ions of the salt is actually involved in the rate-determining transition state and can catalyze the reaction as does the second molecule of amine. This possibility is supported by the fact that at constant initial concentrations of the substrate and the amine and variable salt concentrations, the experimental data also give a linear plot of k versus the initial concentration of the salt. 

Terms in the denominator represent the competing reactions of an intermediate. One of the two steps reverses the reaction by which the intermediate was formed. Imagine letting each of the denominator terms, in turn, become much larger than the others, either in one s mind or in practice by adjusting the concentration variables. In the limit where one term dominates, there is a change in rate control from one step to another. In each of these limits, the composition of the transition state for the step that is then rate-controlling can be deduced from the application of Rule 1. 

The experimental side of the subject explores the effects of certain variables on the rate constant, especially temperature and pressure. Their variations provide values of the activation parameters. They are the previously mentioned energy of activation, entropy of activation, and so forth. The chemical interpretations that can be realized from the values of the activation parameters will be explored in general terms, but readers must consult the original literature for information about those chemical systems that particularly interest them. On the theoretical side, there is the tremendously powerful transition state theory (TST). We shall consider its origins and some of its implications. 

After the discovery of the remarkable acceleration of some Diels Alder reactions performed in water, a number of polar non-aqueous solvents and their salty solutions were investigated as reaction medium. This revolutionized the concept that the Diels-Alder reaction is quite insensitive to the effect of the medium and emphasized that a careful choice of the solvent is crucial for the success of the reaction. The polarity of the reaction medium is an important variable which also provides some insights into the mechanism of the reaction. If the reaction rate increases by using a polar medium, this means that the transition state probably has polar character, while the absence of a solvent effect is generally related to an uncharged transition state. 

We must now connect these various predicates to be able to derive state variables from other state variables. The process of transitioning from one... 

---