Classical threshold energies

A saddle point is a stationary point on the multidimensional potential energy surface. It is a stable point in all dimensions except one, where the second-order derivative of the potential is negative (see Appendix E). The classical energy threshold Eci or barrier height of the reaction corresponds to the electronic energy at the saddle point relative to the electronic energy of the reactants. 

For instance, H -t H2 — Hj -I- H was studied in 3-dimensions within a model where the vibrational states were reduced to a single one for each of the three possible product molecules ° At low collision energy (less than the classical energy threshold) the reaction cross section is non zero because of tunneUing. For the same reaction studied colinearly the following conclusions emerge... 

The energy threshold of a reaction corresponds to the minimum relative translational energy that must be supplied to the reactants in order to produce products. This energy threshold will differ from the classical barrier height, even when the reac-... 

It is important to notice that the energy threshold that enters the expression in Eq. (6.8) is Eq and not the classical threshold Ec, see Fig. 6.1.1. Within the normalmode description, that is, the local harmonic approximations to the potential energy... 

Fig. 2.7. Plot of the ratio of the anharmonic to harmonic state density versus energy relative to the bottom of the well for a range of total angular momentum in CH2CO considering only the anharmonicities in the intermolecular modes of the CH2 + CO channel. The classical dissociation threshold is at about 33,000 cm. The curve for J=Q includes Monte Carlo integration uncertainty error bars.

Table IV. Adlabatlcally quantized semlclasslcal results for energies of doubly excited states of HOD, In a two degrees of freedom model, as obtained by Skodje and Reinhardt (27) using the technique of Refs. (24), (25). All of these states are above the classical dissociation threshold. AE Is the uncertainty In the adiabatic quantization, as Illustrated In Fig. 6. Eo Is the "unperturbed energy, and E the result of the quantization procedure In this weakly coupled system.

ABSTRACT. After reviewing the time dependent wavepacket method as applied to collision induced dissociation processes,we report accurate quantum results for reactive and non reactive collinear A+BC systems. Both systems display a vibrational enhancement effect in the low energy region. While the non reactive systems exhibit a vibrational inhibition effect at higher energies,a more complex behavior is observed in the reactive case. Below the classical dissociation threshold,the non reactive systems display tunnelling tails which decrease with the initial vibrational excitation of the diatomic molecule. The reactive system displays important quantum effects at energies well above the classical dissociation threshold. 

The most satisfactory situation for making an extrapolation of rate data to the true threshold arises when the threshold is uncertain, but we can confidently calculate the functional form of the rate-energy curve from accurate kinetic theory. For small systems, it is feasible to calculate dissociation rates by quantum methods, but this is not yet feasible for the systems of interest to us. Various approaches to variational transition-state theory (VTST) provide classical or semiclassical calculations that are feasible for large systems and seem to be accurate when carefully... 

The Maxwell-Heaviside theory seen as a U(l) symmetry gauge field theory has no explanation for the photoelectric effect, which is the emission of electrons from metals on ultraviolet irradiation [39]. Above a threshold frequency, the emission is instantaneous and independent of radiation intensity. Below the threshold, there is no emission, however intense the radiation. In U(l), electrodynamics energy is proportional to intensity and there is, consequently, no possible explanation for the photoelectric effect, which is conventionally regarded as an archetypical quantum effect. In classical 0(3) electrodynamics, the effect is simply... 

At energies slightly above the saddle energy, there exists a single unstable classical periodic orbit. This periodic orbit corresponds in general to symmetric stretching motion (or an equivalent mode in XYZ-type molecules). The Lyapunov exponent of this periodic orbit tends to the one of the equilibrium point as the threshold energy is reached from above. 

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