Zero point vibrational energy transition state theory

The reaction with Fe (Fig. 3) is somewhat more complicated as it also involves participation of an intermediate-spin (IS, S = 1) state between the LS (S = 0) and the HS (S = 2) states in the course of the reaction. From the initial complex 4, the reaction proceeds virtually without barriers until the final complex 6 is formed. In the cases of both ]TS3 and 1TS4, the activation energies with respect to x4 and ]5 were found to be 2.9 and 0.5 kcal mol-1, respectively, without zero-point vibrational energy (ZPVE) correction. With ZPVE, both 1TS3 and ]TS4 become lower on the potential energy surface than the corresponding complexes 4 and 5 by 0.3 and 1.3 kcal mol-1, respectively. In some cases, we were unable to locate transition states and local minima at all three levels of theory. 

Fortunately, it is relatively simple to estimate from harmonic transition-state theory whether quantum tunneling is important or not. Applying multidimensional transition-state theory, Eq. (6.15), requires finding the vibrational frequencies of the system of interest at energy minimum A (v, V2,. . . , vN) and transition state (vj,. v, , ). Using these frequencies, we can define the zero-point energy corrected activation energy ... 

Figure 3.5 Models accounting for kinetic isotope effects (KIE). (A) Principle of the KIE according to the semi-classical transition state theory (TST ref. 242). The vibrational ground states within the potential energy well of the reactant state are presented as horizontal bars for hydrogen (H), deuterium (D) and tritium (T) indicating the differences in their zero-point energies (ZPE= l/2Av). According to the TST, the KIE is due to the different ZPE and the semi-classical limit for the KIE is 7. In the semi-classical regime, the Arrhenius pre-factors ( h/ d) typically yield a ratio of 1 and the difference in the activation energies (A a = - a) can...

One point of interest deriving from the equations of TST (and Arrhenius theory) is that the upper limit for the 298 K rate constant of a unimolecular reaction that takes place with zero activation energy (of whatever sort) is roughly 10 sec . This is, in some sense, a conceptually obvious result since that is on the order of a molecular vibrational frequency, which is thought of as the mechanism by which a transition state goes to its products. 

---