Motion Principle

Lactam antibiotics, mechanisms of reactions, 23, 165 Least nuclear motion, principle of, 15, 1... 

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... 

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