Zero-point energy effects

Barriers for reaction (7.1), calculated at a wide variety of levels, are presented in Table 6.14. The theoretical results [41] are compared with the experimental barriers obtained from condensed phase (21.3 kJ/mol) [40, 42] and gas-phase (25.7 kJ/mol) [43] studies, back-corrected for temperature and zero-point energy effects [41, 44],... 

The first factor is responsible for normal isotope effects, which arise because the bonds being affected by deuteriation are weakened in the transition state, but the absolute effect is greater on the bonds to deuterium rather than protium because the former have higher vibrational frequencies (typically by a factor of ca 1.37). This factor essentially reflects zero-point energy effects, so it becomes progressively more important at lower internal energies. 

Several theoretical studies have addressed the problem of the relative stabilities of vinylidene and acetylene, one of the most recent concluding that the classical barrier to Eq. (1) is 4 kcal zero-point energy effects lower this to 2.2 kcal (4). Tunneling through this barrier is extremely rapid the calculated lifetime of vinylidene is ca. 10 " sec, which agrees with a value of ca. 10-10 sec deduced from trapping experiments (5). 

As interesting as these special "exit-channel" effects are in their own right, it has been shown that because of a cancellation, they have no bearing on the MIF phenomenon [15]. We stress this point, since occasionally it is assumed in the literature that the special exit channel effect in the ratios is a key to understanding the MIF. Instead, the mass-independent effect of "scrambled" systems and the anomalously large mass-dependent effect for reactions of the type Q -F OO QOO QOO and QO -F O, have very different origins and are unrelated. Perhaps these remarks may seem paradoxical. The various rate constants for these "isotopically unscrambled" reactions can be used to compute the observables for the isotopically scrambled system, and so compute and 5 0. However, the detailed analysis [15] showed that there is much cancellation, summarized below, and that the theoretical expression for the MIF conditions is now simpler than would appear from fhe expression for the MIF in terms of the individual rate constants [15]. In particular, the zero-point energy effect, important for the individual isotope rate constants, disappears when the combination of them that determines the MIF is calculated. 

At even lower temperatures, some unusual properties of matter are displayed. Consequently, new experimental and theoretical methods are being created to explore and describe chemistry in these regimes. In order to account for zero-point energy effects and tunneling in simulations, Voth and coworkers developed a quantum molecular dynamics method that they applied to dynamics in solid hydrogen. In liquid helium, superfluidity is displayed in He below its lambda point phase transition at 2.17 K. In the superfluid state, helium s thermal conductivity dramatically increases to 1000 times that of copper, and its bulk viscosity drops effectively to zero. Apkarian and coworkers have recently demonstrated the disappearance of viscosity in superfluid helium on a molecular scale by monitoring the damped oscillations of a 10 A bubble as a function of temperature. These unique properties make superfluid helium an interesting host for chemical dynamics. 

It should be noted that in the present case the trajectoiy calculations were carried out for coplanar collisions with the vibration of HCl (DCl) fiozen [47], A trajectory was assumed to be reactive if it reached a point on the top of the electronic barrier. There are no corrections of the barrier height due to the vibrational zero-point energy effect in this case. These trajectory results are therefore more directly comparable with the results of the kinematic mass model than the 3-dimensional quasiclassical trajectroiy calculations discussed in the previous case. 

Zero-point energy effects can be traced to the light masses of the constituents in quantum clusters and to the extremely low temperatures in the optical molasses and condensates. The zero-point energy effects can be described in terms of the ratio A of the quantum lengths... 

Quantum Zero-Point Energy Effects for Ultracold Finite Systems... 

Another characterization of zero-point energy effects pertains to the increase in the actual average volume vq occupied by a particle in the ultracold system, relative to the reference volume v that the particles would occupy in a classical lattice. For nearly classical and for quantum clusters we have Vc = [11],... 

Isotope effects are used to probe chemical processes, as isotopic substitution generally alters only the mass of the reacting groups without changing the electronic properties of the reactants. In this fashion, isotope effects can be used as subtle probes of mechanism in chemical transformations. This section will discuss how to use isotope effects to probe for tunneling effects on enzymes. The basic criteria for tunneling are experimental isotope effects that have properties that deviate from those predicted within the semi-classical transition state model, which includes only zero-point energy effects (we refer to this as the bond stretch model ). 

Bond Stretch KIE Model Zero-point Energy Effects... 

The bond-stretch model provides an upper limit for kinetic isotope effects that arise solely from ground state zero-point energy effects. Observations that deviate from this model imply a nonclassical effect. Provided that potential artifacts are controlled, the observation of KlEs that disobey the bond-stretch predictions calls into question the basic theory. 

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