Reversibility condition, transition path

Just as in a conventional Monte Carlo simulation, correct sampling of the transition path ensemble is enforced by requiring that the algorithm obeys the detailed balance condition. More specifically, the probability n [ZW( ) - z(n)( )]2 to move from an old path z ° 22) to a new one " (2/ ) in a Monte Carlo step must be exactly balanced by the probability of the reverse move from 22) to z<,J> 22)... 

Because of microscopic reversibility, many of these pathways are merely the reverse of others. The transition states for each related pair of paths are similar (they need not be identical, for the reaction conditions are often slightly different). A figure that illustrates the approximate orbital alignment and the transformation of the orbitals of the reactants into those of the product is given for each pair of paths. 

When conditions for reversibility are not satisfied, that is, when the transition from A to B is not much slower than the internal system relaxation, the system cannot be assumed in equilibrium and in particular its temperature may not be well defined during the process. Still AA = Sb — Sa is well defined as the difference between entropies of two equilibrium states of the system. The second law of thermodynamics states that for a nonreversible path between states A and B... 

If the initial addition (A, Scheme 3) is essentially irreversible, the net stereoselectivity can be controlled by interactions that exist in the transition state for the Michael addition. However, if there is not a rapid intervening process (cyclization or proton transfer), the initial dipolar adducts would be expected to reform starting materials at an appreciable rate (vide supra). Based on the reports described previously, a significant possibility exists that this initial addition is reversible, at least in most cases. If indeed step A is reversible or if the configuration of 3.1 is not stable to reaction conditions, then the net stereoselectivity can be determined by the relative stability of the dias-tereomers of 3.1 or by the relative rates of the diastereomeric transition states for some subsequent reaction (e.g., B-F).+ For example, selectivity could be induced by preferential cyclization (paths D and E) or by selective proton transfer (path B) from one of the components of the initial diastereomeric mixture (3.1). Also, it is possible that selective protonation (path F) of enamine 3.5 could give the observed products. This prospect is less likely as the generation of enamine 3.5 is disfavored by allylic strain considerations. 

When reversible addition and elimination reactions are carried out under similar conditions, they follow the same mechanistic path, but in opposite directions. The principle of microscopic reversibility states that the mechanism of a reversible reaction is the same in the forward and reverse directions. The intermediates and transition structures involved in the addition process are the same as in the elimination reaction. Under these circumstances, mechanistic conclusions about the addition reaction are applicable to the elimination reaction and vice versa. The reversible acid-catalyzed reaction of alkenes with water is a good example. Two intermediates are involved a carbocation and a protonated alcohol. The direction of the reaction is controlled by the conditions, which can be adjusted to favor either side of the equilibrium. Addition is favored in aqueous solution, whereas elimination can be driven forward by distilling the alkene from the reaction solution. The reaction energy diagram is show in Figure 5.1. 

---