Isotope potential energy wells

There are two geometric isotope effects A/ (D — H), which refers to the change in bond length / . .. g of a hydrogen bond AHB on deuteration, and 8/ , which is defined as the change in the distance between the two minima of the potential energy well of the hydrogen bond when it is deuterated. The isotope effect, A/ (D—H), was dealt with previously (see p. 263). 

The second isotope effect, 87 , requires the proton and deuteron to be accurately located. The distance between the equilibrium positions of the potential energy well of double minima, symmetrical hydrogen bonds, which Ichikawa calls 7 h/h defined as q — 2i o . This distance can... 

Kreevoy and coworkers have devised a method for obtaining the shape of the potential energy well of a hydrogen bond118,119). The method is based on isotopic fractionation factors, 0, defined as the ratio of D to H in the compound in question, compared to the D to H ratio of the surrounding medium. For HX DX in HiO-D20 then [Pg.178]

Fig. 8 Potential energy wells for the isotope-exchange equilibrium in (3), L = H or D.

Given the last statement, classical mechanics predicts no energy differences between two molecules that differ only in their isotopic composition. At a temperature of absolute zero, both molecules should have energies corresponding to the bottom of their identical potential energy wells. Quantum theory, however, indicates that the vibrational energy, E, is quantized and given by ... 

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...

The use of isotopic substitution to detennine stmctures relies on the assumption that different isotopomers have the same stmcture. This is not nearly as reliable for Van der Waals complexes as for chemically bound molecules. In particular, substituting D for H in a hydride complex can often change the amplitudes of bending vibrations substantially under such circumstances, the idea that the complex has a single stmcture is no longer appropriate and it is necessary to think instead of motion on the complete potential energy surface a well defined equilibrium stmcture may still exist, but knowledge of it does not constitute an adequate description of the complex. 

Here Tq are coordinates in a reference volume Vq and r = potential energy of Ar crystals has been computed [288] as well as lattice constants, thermal expansion coefficients, and isotope effects in other Lennard-Jones solids. In Fig. 4 we show the kinetic and potential energy of an Ar crystal in the canonical ensemble versus temperature for different values of P we note that in the classical hmit (P = 1) the low temperature specific heat does not decrease to zero however, with increasing P values the quantum limit is approached. In Fig. 5 the isotope effect on the lattice constant (at / = 0) in a Lennard-Jones system with parameters suitable for Ne atoms is presented, and a comparison with experimental data is made. Please note that in a classical system no isotope effect can be observed, x "" and the deviations between simulations and experiments are mainly caused by non-optimized potential parameters. 

In highly exothermic reactions such as this, that proceed over deep wells on the potential energy surface, sorting pathways by product state distributions is unlikely to be successful because there are too many opportunities for intramolecular vibrational redistribution to reshuffle energy among the fragments. A similar conclusion is likely as the total number of atoms increases. Therefore, isotopic substitution is a well-suited method for exploration of different pathways in such systems. 

This situation applies with weak hydrogen bonds at one extreme and very strong hydrogen bonds at the other with H and D confined to the same potential well. However, when the potential energy barrier has fallen sufficiently to allow the proton to escape the confines of its parent well, but leaves the deuteron trapped, then different values of the isotopic ratio can be observed (Fig. 7). The effect of isotopic exchange is now much more than merely one of doubling the reduced mass of the vibrating bond. When the proton is above the barrier, the force constant of the A—H bond, k A.—H),... 

Quantum chemical calculations need not be limited to the description of the structures and properties of stable molecules, that is, molecules which can actually be observed and characterized experimentally. They may as easily be applied to molecules which are highly reactive ( reactive intermediates ) and, even more interesting, to molecules which are not minima on the overall potential energy surface, but rather correspond to species which connect energy minima ( transition states or transition structures ). In the latter case, there are (and there can be) no experimental structure data. Transition states do not exist in the sense that they can be observed let alone characterized. However, the energies of transition states, relative to energies of reactants, may be inferred from experimental reaction rates, and qualitative information about transition-state geometries may be inferred from such quantities as activation entropies and activation volumes as well as kinetic isotope effects. 

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