Murrell-Laidler theorem

Stanton and McIver (63) have given a more detailed discussion of the symmetries of transition states from the standpoint of the Murrell-Laidler Theorem. 

Moreover, Fig. 2 does not contradict the well-known Murrel-Laidler theorem [35,36] which forbids taking the locally symmetric intermediate for the transition state because for the reason of symmetry at least two independent paths exist for its isomerization, i.e., for the insertion of the monomer into the polymer chain. In other words, the bifurcation of the reaction coordinate proceeds in the locally symmetric intermediate. Nevertheless, this is forbidden for the transition state by the Murrel-Laidler theorem which asserts that the matrix of force constants in the transition state has a single negative value, i.e., that the transition state corresponds to a single reaction coordinate. 

Although Fig. 2 is not quite correct in the above sense, the conclusion is probably physically justified that the free energies of the locally symmetric intermediate and the nearest transition state are very close to each other. First, the Jahn-Teller system, not being at the minimum of the adiabatic potential, is hardly long-lived in the condensed liquid phase, in other words it will hardly be at any considerable minimum of free energy. Secondly, this system cannot be at the maximum of free energy (in this case it would be the transition state) because of the Murrel-Laidler theorem [36], Hence, the concept of a plateau in Fig. 2 is actually physically meaningful. 

In essence a similar rationale underlies the Murrell-Laidler theorem on the structure of potential energy surfaces near transition states [74,75]. The assertion that a chemical transition state should always be a saddle point between only two valleys - a reactant valley and a product valley - indeed is based on the assumption that the potential near the transition state is dominated by second-order interactions. Further interesting analogies can be drawn with the maxinudity principle for phase transitions in the solid state [14], the principle of least motion [9] in chemical reaction theory and the related principle of maximal symmetry expressed by Rodger and ScMpper [76]. 

Theorem 1 is simply a group-theoretical reformulation of the Murrell-Laidler theorem, it implies that a PES cannot have more than one negative curvature direction at the transition state point. 

Table 1.4 lists calculation data on activation and thermodynamic parameters as well as kinetic isotopic effects for three reactions of the retroene type. In the first stage, a search for the transition state structure was conducted and its compatibility with the demands of the Murrell-Laidler theorem verified. Afterwards the vibration frequencies of the reactants and the transition state structure were calculated whose values were used in the corresponding equations. Underestimation of the kinetic isotopic effect in the last two reactions is related to underestimation of the role of the tunnel mechanism (see Sect. 1.5). An exact reproduction of the values of kinetic isotopic effects is a more reliable check on the accuracy of the calculated transition state structures than that of the values of activation entropies. This is explained by the fact that the calculated values of normal vibration frequencies, corresponding to the negative force constants, are directly included into Eqs. (1.24)-(1.26) that determine the magnitude of the kinetic isotopic effect. 

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