Least-motion principle

FLUORESCENCE ENERGY OF ACTIVATION ARRHENIUS EQUATION LEAST MOTION, PRINCIPLE OF MARCUS EQUATION MARCUS RATE THEORY TRANSITION-STATE THEORY Om... 

LEAST-CONDUCTIVE STEP LEAST MOTION, PRINCIPLE OF Least squares fit... 

This intuitive parallel can be best demonstrated by the example of electrocye-lic reactions for which the values of the similarity indices for conrotatory and disrotatory reactions systematically differ in such a way that a higher index or, in other words, a lower electron reorganisation is observed for reactions which are allowed by the Woodward-Hoffmann rules. In contrast to electrocyclic reactions for which the parallel between the Woodward-Hoffmann rules and the least motion principle is entirely straightforward, the situation is more complex for cycloadditions and sigmatropic reactions where the values of similarity indices for alternative reaction mechanisms are equal so that the discrimination between allowed and forbidden reactions becomes impossible. The origin of this insufficiency was analysed in subsequent studies [46,47] in which we demonstrated that the primary cause lies in the restricted information content of the index rRP. In order to overcome this certain limitation, a solution was proposed based on the use of the so-called second-order similarity index gRP [46]. This... 

If we now look at the values of the above indices, it is possible to see that the prediction of the Woodward-Hoffmann rules is indeed confirmed since the greater values of the similarity index for the conrotatory reaction clearly imply, in keeping with the expectations of the least-motion principle, the lower electron reorganisation. If now the same formalism is applied to a stepwise reaction mechanism, the following values of the similarity indices result (Eq. 21). 

Least Motion Principle and the Mechanisms of Pericyclic Reactions... 

Assuming that in a given case the structures of all the participating molecular species is known, it is possible to begin the practical exploitation of Eq. (22) and aim at the variational formulation of the least-motion principle. For this purpose, it is first necessary to introduce the first order density matrix p(9,

position vector of the i-th electron, its spin coordinate and N the total number of electrons... 

As can be seen, this index attains its maximal value of unity for two identical structures (9 = 9,

monotonously decreases. The use of this index for the formulation of the least motion principle arises from the following simple idea. Let us assume that we are on a reaction path at point characterised by the wave function P(, q> ) and we are looking for such an infinitesimally close structure (9, (p) for which the transformation (9,

requires minimal change in electronic configuration. This condition is equivalent to a search of the direction along which the derivative of K at the point = 9 and q> = q> attains its minimum. This directional derivative can be mathematically described as (27),... 

Let us discuss now the most important conclusions that can be deduced from these figures. First, the most important conclusion concerns the comparison of the values of functional L along the optimal allowed and forbidden reaction paths. As can be seen, the value for the allowed conrotatory cyclisation is lower in absolute value than in the forbidden one. This confirms the intuitive expectation of the least motion principle that the extent of electron reorganisation should be smaller in allowed reactions than in the forbidden ones. On the basis of this primary test of reliability of the proposed model it is, in the next step, possible to start with the analysis and the classification of the reaction mechanisms for both individual reactions. Especially interesting in this connection is again the thermally forbidden disrotatory cyclisation. The reason for this... 

As it was mentioned above the centers of the reacting molecules ideally remain fixed in topochemical reactions. As a consequence the reaction mode which requires the smallest atom displacements will prevail (least motion principle). Reactivity is... 

The postulate of a half-chair transition state may be valid if relative rates may be accounted for by least motion principles (Hine, 1977). The equatorial phosphate isomer, populating a twist-boat ground state, may require less... 

A simple model is proposed in this note. It is shown to incorporate the possibility of including several potential energy curves in a treatment of scattering processes in a linear geometry based on very limited information from electronic structure calculations. The present demonstration of the feasibility will be followed by more detailed examinations and possible comparisons with more elaborate theoretical work and even experimental findings. Extensions to two and higher dimensions appear to be within reach and in conjunction with a geometrical characterization of the molecular conformation space based on a distance concept derived from a quantification of the Rice-Teller least motion principle TO there opens an avenue towards detailed descriptions of chemical processes from basic principles. 

Ring opening proceeds stereospecificaUy in accordance with the least-motion principle (see Sect. 3.3 in [72e] and Sect. 5 in [72f]). 

The application of this index to pericyclic reactivity is the subject of the study [91]. The values of similarity indices for several typical pericyclic processes, taken from this study are given in Table 4. The presented values immediately imply several interesting conclusions confirming the expected parallel between the similarity index and the least motion principle. This parallel is based on a simple intuitive... 

A number of previous examples has demonstrated that the applicability of similarity indices in the theory of pericyclic reactions is indeed very broad. The basis of this exploitation is the intuitively expected relation of the similarity indices to the extent of electron reorganization and the so-called least motion principle. Since in all previous applications this relation has remained only at the level of the intuitively expected parallel it is desirable to analyze the above relation in more details. 

The importance of this index for the variational formulation of the least motion principle arises from the following simple idea. Let us suppose that we are on the... 

We thus believe that the basic features of the least motion principle are indeed well reflected by our model. On the basis of this primary test of the reliability of the proposed model it is, in the next step, possible to start with the analysis and the classification of the reaction mechanisms for both individual reactions. Especially interesting in this coimection is the case of thermally forbidden disrotatory cyclization. 

As can be seen from this comparison, the resulting values are affected by the choice of the critical structure and on going from X(n/4) to X(-7t/4), the systematic shift of the dominant similarity from the zwitterionic state Z + Z2 to the state Zj -Z2 is observed. We can thus see that the predictions for both types of critical structures differ and the problem thus appears which of the above two critical structures should be regarded as a true model of the transition state in forbidden reactions. Similarly as in the case of allowed reactions such a decision does not arise from the approach itself, but some external additional information is generally required. This usually represents no problem since the desired information can be obtained, as in the case of allowed reactions, from the simple qualitative considerations based on the least motion principle [80,81], or from the direct quantum chemical calculations.This is also the case with us here, where the desired information is provided by the quantum chemical study [63] of the thermally forbidden cyclization of the butadiene to cyclobutene. From this shufy it follows that the ground state of the transition state should correspond to the ground state of the cyclobutadiene which is the Zj - Z2 state. 

In previous chapters the broad possibilities of similarity indices in characterizing various mechanistic features of pericyclic reactions were demonstrated. The basis of these applications is a simple, from the least motion principle arising parallel, according to which the extent of electron reorganization in allowed reactions is systematically lower than in the forbidden ones. In most cases this intuitive parallel... 

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