Energy, activation zero-point

For accurate comparison of relative energies, one must add to the BO-optimized energy the zero-point vibrational energy (ZPVE), which in the harmonic approximation is half the sum of the fundamental frequencies. This correction is most critical for the calculation of activation energies. The contribution of the ZPVE of the mode corresponding most closely to the reaction coordinate is lost completely. Processes that involve breaking of a bond to H are the most seriously affected torsional changes are the least affected. 

Fig. 8.2.2 An illustration of Eq. (8.11) for the activation energy. The zero-point energy levels of the activated complex and the reactants are indicated by solid lines and the associated average internal energies (relative to the zero-point levels) are given by dashed lines. Note that the zero-point energy in the activated complex comes from vibrational degrees of freedom orthogonal to the reaction coordinate.

Finally, the activation energies (without zero point corrections) calculated for the 1,5-hydrogen shift in cycloheptatriene by MROPT2, CASSCF, and B3LYP are 38.7, 60.2, and 40.6 kcal/mol, respectively, and a zero point correction would lower these energies by about 4 kcal/mol. Clearly, the CASSCF method without dynamic correlation is suspect just as in the Cope rearrangement (see Chapter 7, Section 4.1). 

The activation energies and zero point energies are different and lead to unequal available energies see (20). 

The activation energy, is defined as tlie minimum additional energy above the zero-point energy that is needed for a system to pass from the initial to the final state in a chemical reaction. In tenns of equation (A2.4.132). the energy of the initial reactants at v = v is given by... 

The unique feature in spontaneous Raman spectroscopy (SR) is that field 2 is not an incident field but (at room temperature and at optical frequencies) it is resonantly drawn into action from the zero-point field of the ubiquitous blackbody (bb) radiation. Its active frequency is spontaneously selected (from the infinite colours available in the blackbody) by the resonance with the Raman transition at co - 0I2 r material. The effective bb field mtensity may be obtained from its energy density per unit circular frequency, the... 

Even at 0 K, molecules do not stand still. Quantum mechanically, this unexpected behavior can be explained by the existence of a so-called zero-point energy. Therefore, simplifying a molecule by thinking of it as a collection of balls and springs which mediate the forces acting between the atoms is not totally unrealistic, because one can easily imagine how such a mechanical model wobbles aroimd, once activated by an initial force. Consequently, the movement of each atom influences the motion of every other atom within the molecule, resulting in a com-... 

B synchronously moving away from and toward H the H atom does not move (if A and B are of equal mass). If H does not move in a vibration, its replacement with D will not alter (he vibrational frequency. Therefore, there will be no zero-point energy difference between the H and D transition states, so the difference in activation energies is equal to the difference in initial state zero-point energies, just as calculated with Eq. (6-88). The kinetic isotope effect will therefore have its maximal value for this location of the proton in the transition state. 

Reaction path computations allow you to verify that a given transition structure actually connects the starting and ending structures that you think it does. Once this fact is confirmed, you can then go on to compute an activation energy for the reaction by comparing the (zero-point corrected) energies of the reactants and the transition state. 

Determine the reaction path connecting trans hydroxycarbene and H2 + CO. Predict the activation energy, referring to the values for the SCF and zero-point energies for the products and reactants summarized at the conclusion of this problem. This reaction occurs via a two step process ... 

Table 3 contains the enthalpies, zero point energies, entropies and free enthalpies of the activation and reaction steps (3)—(5). The enthalpies are the pure differences of the enthalpies of formation calculated by MINDO/3 at T = 298 K in the gas phase. The free enthalpies were calculated with the help of enthalpies corrected by the zero point energies and of the entropies given in Table 3. 

Table 3. Enthalpies, free enthalpies, zero point energies (ZPE) (all in kJ mol-1) and entropies (J K 1 mol-1) of the activation and reaction steps for Eqs. (3)-(5)...

Table 3 shows that the small activation enthalpies of the reactions (3) and (4) are clearly affected by the zero point energy corrections. But the relative order of the activation enthalpies remains the same with or without the corrections. The activation entropies have great negative values, which is of mechanistic interest (see part 4.3.1). However, because of their similarity, when comparing the three reactions to one another they have only small importance, e.g. for estimation of copolymerization parameters (see part 4.3.2). 

Arrhenius activation energy potential energy zero-point energy degrees of freedom... 

Here qi2D-vib refers to the electronic ground state of the transition state and q -vih to the vibrational ground state of the transition state. We have combined the t vo zero-point vibrations vith AE into an effective activation energy A act- We shall later explain how this important quantity can be measured. 

Based on C-H versus C-D zero point vibrational differences, the authors estimated maximum classical kinetic isotope effects of 17, 53, and 260 for h/ d at -30, -100, and -150°C, respectively. In contrast, ratios of 80,1400, and 13,000 were measured experimentally at those temperatures. Based on the temperature dependence of the atom transfers, the difference in activation energies for H- versus D-abstraction was found to be significantly greater than the theoretical difference of 1.3kcal/mol. These results clearly reflected the smaller tunneling probability of the heavier deuterium atom. 

Table 13-2. Computed activation (AEa) and reaction energies (AEr) for the concerted gas-phase cycloaddition of ethylene to Irans-butadiene [kcal/mol]. The HF and DFT calculations were performed with the 6-311+G(d,p) basis set and include zero-point vibrational contributions.

A clear-cut dependence of the activation energy on the heat (enthalpy) of the reaction, which is equal, in turn, to the difference between the dissociation energies of the ruptured (Z> ) and the formed (D j bonds, was established for a great variety of radical abstraction reactions [1,2,16]. In parabolic model, the values of Dei and Def, incorporating the zero-point energy of the bond vibrations, are examined. The enthalpy of reaction AHe, therefore, also includes the difference between these energies (see Equation [6.7]). 

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