Subject zero-point energy

It is not possible to discuss this complex subject in any detail here. It is necessary, however, to indicate its nature. The major factor involved is the difference in zero point energies of bonds between an element and the various isotopes of another element. For example a C—D bond has a lower zero point energy than a C—H bond. Consequently when proceeding along a reaction co-ordinate from reactants to products over an energy barrier (Fig. 1), rupture of a C—D bond requires a higher activation energy than rupture of a C—H bond. Thus, the kinetic isotope effect manifests itself in a smaller rate constant for C—D bond rupture. 

As with most other isotope effects the literature on the subject is enormous the reader is therefore referred to a comprehensive review by Halevi [30c] for further information. When an organic molecule ionises to yield a carbenium ion the adjacent C-H bond (the a-hydrogen) to the ionising atom weakens thus reducing the contribution of vibrational modes involving this bond to the zero-point energy of the transition state. An effect of some 10-15% is observed in an S l reaction (Eqn. 51). 

For polymer chains in a crystal lattice, however, acoustic vibrations of polymer chains are subject to interchain interactions, yielding the crystal vibrations of the acoustic and optical branches. Accordingly, for vibrational analyses of neutron-scattering spedra in the low-frequency region, it is required to treat the normal vibrations of the crystal, on the basis of the interchain force field as well as the intrachain force field. Treatments of crystal vibrations are also necessary for the theoretical study of specific heat, zero-point energy and temperature factor of x-ray diffraction. 

More sensitive to the level of theory is the vibrational component of the interaction energy. In the first place, the harmonic frequencies typically require rather high levels of theory for accurate evaluation. It has become part of conventional wisdom, for example, that these frequencies are routinely overestimated by 10% or so at the Hartree-Fock level, even with excellent basis sets. A second consideration arises from the weak nature of the H-bond-ing interaction itself. Whereas the harmonic approximation may be quite reasonable for the individual monomers, the high-amplitude intermolecular modes are subject to significant anharmonic effects. On the other hand, some of the errors made in the computation of vibrational frequencies in the separate monomers are likely to be canceled by errors of like magnitude in the complex. Errors of up to 1 kcal/mol might be expected in the combination of zero-point vibrational and thermal population energies under normal circumstances. The most effective means to reduce this error would be a more detailed analysis of the vibration-rotational motion of the complex that includes anharmonicity. 

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