Classical breaking points

A curious effect, prone to appear in near degeneracy situations, is the artifactual symmetry breaking of the electronic wave function [27]. This effect happens when the electronic wave function is unable to reflect the nuclear framework symmetry of the molecule. In principle, an approximate electronic wave function will break symmetry due to the lack of some kind of non-dynamical correlation. A typical example of this case is the allyl radical, which has C2v point group symmetry. If one removes the spatial and spin constraints of its ROHF wave function, a lower energy symmetry broken (Cs) solution is obtained. However, if one performs a simple CASSCF or a SCVB [28] calculation in the valence pi space, the symmetry breaking disappears. On the other hand, from the classical VB point of view, the bonding of the allyl radical is represented as a superposition of two resonant structures. 

This structure of the traveling front of a low-temperature reaction exhibits features utterly atypical of classical thermal self-propagation. They are (1) a weak or nonexistent stage of inert preburst heating (2) a jumplike switching on and off of the reaction—characteristic break points b and c (Fig. 8), the temperature where the reaction switches on at point b being far below that for... 

The validity of equation (6.379) is restricted to classically accessible regions, where p(R) is real. It breaks down catastrophically at any classical turning points where p(R) is zero but methods are available to correct JWKB solutions around such singularities. Proper real combinations of i/f (R) in the region R < R < R2 may be expressed as [79, 80]... 

There is a lot of delocalization in this structure, and usually these ligands are represented with a curved line to show the donation of both nitrogen lone pairs to the carbene C atom. You might prefer to include the formal + and - charges, but these compounds really do stretch the classical valence bond representation almost to breaking point, and conventionally the charges are not shown as they cancel out. 

However, this approximation clearly breaks down at the classical turning points E = V(x), i.e., at the points were the classical particle (or the manifestation of the wave packet) stops, since p(x) = 0 there, and turns due to the potential equalizes its (eigen-value) energy. This is the WKB approximation holds for the cases where is much smaller than equivalently with the requirement that quantum Hamilton-Jacobi equation to collapse on its classical variant through the action condition ... 

It should be noted that, however, the treatment here implicitly assumes each elementary process as independent to each other, which is not always correct. For example, in case of a predissociation problem, there exist a curve crossing and classical turning points, both break the applicability of the Wentzel-Kramers-Brillouin (WKB) approximation. 

The bifurcational diagram (fig. 44) shows how the (Qo,li) plane breaks up into domains of different behavior of the instanton. In the Arrhenius region at T> classical transitions take place throughout both saddle points. When T < 7 2 the extremal trajectory is a one-dimensional instanton, which crosses the maximum barrier point, Q = q = 0. Domains (i) and (iii) are separated by domain (ii), where quantum two-dimensional motion occurs. The crossover temperatures, Tci and J c2> depend on AV. When AV Vq domain (ii) is narrow (Tci — 7 2), so that in the classical regime the transfer is stepwise, while the quantum motion is a two-proton concerted transfer. This is the case when the tunneling path differs from the classical one. The concerted transfer changes into the two-dimensional motion at the critical value of parameter That is, when... 

In a recent analysis carried out for a bounded open system with a classically chaotic Hamiltonian, it has been argued that the weak form of the QCT is achieved by two parallel processes (B. Greenbaum et.al., ), explaining earlier numerical results (S. Habib et.al., 1998). First, the semiclassical approximation for quantum dynamics, which breaks down for classically chaotic systems due to overwhelming nonlocal interference, is recovered as the environmental interaction filters these effects. Second, the environmental noise restricts the foliation of the unstable manifold (the set of points which approach a hyperbolic point in reverse time) allowing the semiclassical wavefunction to track this modified classical geometry. 

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