Zero-point energy determination

Once the phonon frequencies are known it becomes possible to determine various thermodynamic quantities using statistical mechanics (see Appendix 6.1). Here again some slight modifications are required to the standard formulae. These modifications are usually a consequence of the need to sum over the points sampled in the Brillouin zone. For example, the zero-point energy is ... 

That is, the exponential increase of the isotope effect with is determined by the difference of the zero-point energies. The cross-over temperature (1.7) depends on the mass by... 

If H is replaced by D, p will increase, whereas k will not change, because it is determined by the electronic distribution. Therefore Vd q. The implication of this result can be seen graphically in Fig. 6-19 because of the difference in zero-point energies, the bond dissociation energies of R-H and R-D are different, the energy required to break the R-D bond being the greater. 

To determine the zero-point energy for the transition structure. 

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 ... 

Fe(acac2trien)]N03. For the iron(III) complex of the hexadentate Schiff-base ligand acac2trien, the barrier heights have been determined from ultrasonic relaxation [94] as AG[h = 6-28 kcalmoD = 2196 cm and AGJil = 5.85 kcalmol= 2046 cm The difference of zero-point energies has been obtained from equilibrium studies as AG° = 0.43 kcal mol =... 

Fig. 28. Schematic of potential energy surfaces of the vinoxy radical system. All energies are in eV, include zero-point energy, and are relative to CH2CHO (X2A//). Calculated energies are compared with experimentally-determined values in parentheses. Transition states 1—5 are labelled, along with the rate constant definitions from RRKM calculations. The solid potential curves to the left of vinoxy retain Cs symmetry. The avoided crossing (dotted lines) which forms TS5 arises when Cs symmetry is broken by out-of-plane motion. (From Osborn et al.67)...

HF/6-31G(d) geometry optimization for neutral species with vibrational frequencies to determine zero-point energy. 

On this theory, it is assumed (a) that the motion of the reactant-state C-H/D bonds can be thought of as one stretching and two bending motions, and (b) that the bending motions are httle different in the transition state than in the reactant state. Then, the maximum possible isotope effect will be determined by the isotopic zero-point energy difference in the reactant-state C-H/D motion. For a stretching motion with a C-H frequency of 2900 and CD frequency 2130 cm , the isotopic... 

---