Quantum mechanical resonance energy

S. S. Shaik, E. Duzy, A. Bartuv, J. Phys. Chem. 94, 6574 (1990). The Quantum Mechanical Resonance Energy of Transition States An Indicator of Transition State Geometry and Electronic Structure. 

The third parameter of the avoided crossing diagram, the B factor, is the quantum-mechanical resonance energy (QMRE) of the CHJ transition state. While... 

FIGURE 2.1 Valence bond curve-crossing diagram, for the Jt-component, interconversion of two Kekule structural forms along the bond alternation coordinate (bj mode). VG and QMRE correspond to vertical gap and quantum mechanical resonance energy, respectively. The two putative possibilities represent that in one case (a) the vertical gap is high and therefore a localized structure is preferred in the other ease (b) a small vertical gap results in bringing the stability to a fully delocalized symmetric structure. 

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

The sin7o term reflects the fact that the intramolecular vector r is expressed in polar coordinates it is especially important for the dissociation of linear molecules and must not be forgotten The delta-function S(Hf — Ef) selects only those points (i.e., trajectories) in the multidimensional phase-space that have the correct energy Ef. It ensures that the quantum mechanical resonance condition Ef = Ei + Ephoton is fulfilled. 

The quantization of transition state energy levels is not simply a mathematical device to add quantum effects to the partition functions. The quantized levels actually show up as structure in the exact quantum mechanical rate constants as functions of total energy [51]. The interpretation of this structure provides clear evidence for quantized dynamical bottlenecks, both near to and distant from the saddle points, as reviewed elsewhere [52]. Quantized variational transition states have also been observed in molecular beam scattering experiments [53]. Analysis of the reactive flux in state-to-state terms from reactant states to transition state levels to product states provides the ultimate limit of resolution allowed by quantum mechanics [53,54]. Quantized energy levels of the variational transition state have been used to rederive TST using the language of quantum mechanical resonance scattering theory [55]. 

The correspondence between classical and quantum mechanics tells us that this trajectory corresponds to a resonance state with a localized wave function in the H-C-C vibration continuum. The quantum mechanical resonance state (discussed in detail in Section 15.2.4) will have nearly the same energy and be assignable with the semiclassical quantum numbers. It is also expected to have a very long lifetime as a result of the classical quasiperiodic motion. 

There are two types of ener transfer from a point in a lattice to a luminescent center. For semi-conducting phosphors like ZnS, migration of electrons in the conduction band, or holes in the valence band, conve3rs the excitation energy to localized luminescent centers. Excltons (bound electron-hole pairs) is another mechanism. For insulators (which are generally oxygen dominated compositions), the excitation energy can be transferred from an excited point in the lattice, or from an excited luminescent center to other unexcited centers in the lattice by Quantum Mechanical resonance. 

Closer approach is prevented by the repulsive forces then arising out of quantum-mechanical resonance effects. The column headed in the table contains those minimum distances which prevail at room temperature in consequence of thermal agitation. They are somewhat smaller than those considered heretofore because the repulsive forces are overcome to a slight extent by the inherent kinetic energy of the separate parts of the molecule. 

Comparison of the HO2 -> H + O2 unimolecular dissociation rates as obtained from the quantum mechanical resonances (kqm, open circles) and from variational transition state RRXM, theory (kuRKM, step function). E,i,f is the threshold energy for dissociation. Also shown is the quantum mechanical average of the as in a AF = 0.075 eV energy interval (kq, solid line) and the experimental prediction (dashed line). Adapted from ref. 122. 

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