Resonances discrete

Zhang FJ, Zheng ML (1992) Generalized Hexagonal Systems with Each Hexagon Being Resonant. Discrete Appl Math 36 67... 

The energy spectrum of the resonance states will be quasi-discrete it consists of a series of broadened levels with Lorentzian lineshapes whose full-width at half-maximum T is related to the lifetime by F = Fn. The resonances are said to be isolated if the widths of their levels are small compared with the distances (spacings) between them, that is... 

Figure Bl.6.11 Electron transmission spectrum of 1,3-cyclohexadiene presented as the derivative of transmitted electron current as a fiinction of the incident electron energy [17]. The prominent resonances correspond to electron capture into the two unoccupied, antibonding a -orbitals. The negative ion state is sufficiently long lived that discrete vibronic components can be resolved.

Leforestier C and Museth K 1998 Response to Comment on On the direct complex scaling of matrix elements expressed in a discrete variable representation application to molecular resonances J. Chem. Phys. 109 1204... 

Figure Cl.5.8. Spectral jumping of a single molecule of terrylene in polyethylene at 1.5 K. The upper trace displays fluorescence excitation spectra of tire same single molecule taken over two different 20 s time intervals, showing tire same molecule absorbing at two distinctly different frequencies. The lower panel plots tire peak frequency in tire fluorescence excitation spectmm as a function of time over a 40 min trajectory. The molecule undergoes discrete jumps among four (briefly five) different resonant frequencies during tliis time period. Arrows represent scans during which tire molecule had jumped entirely outside tire 10 GHz scan window. Adapted from...

The heightened appreciation of resonance problems, in particular, has been quite recent [63, 62], and contrasts the more systematic error associated with numerical stability that grows systematically with the discretization size. Ironically, resonance artifacts are worse in the modern impulse multiple-timestep methods, formulated to be symplectic and reversible the earlier extrapolative variants were abandoned due to energy drifts. 

Atomic and Molecular Energy Levels. Absorption and emission of electromagnetic radiation can occur by any of several mechanisms. Those important in spectroscopy are resonant interactions in which the photon energy matches the energy difference between discrete stationary energy states (eigenstates) of an atomic or molecular system = hv. This is known as the Bohr frequency condition. Transitions between... 

Reciprocating compressor pulsations were covered in Chapter 3, but neeti to be mentioned with the discussions on reliability. Problems with reciprocating compressor pulsations and the potential for acoustic and mechanical resonances are very similar to those experienced with helical-lobe compressors. The significant difference is the frequencies are much lower and the number of discrete frequencies per compressor aie much less. However, piping vibrations can occur and there is always a... 

However, the duality of substituent constants and the attempt to deal with crossconjugation by selecting cr+, a or a in any given case is somewhat artificial. The contribution of the resonance effect of a substituent relative to its inductive effect must in principle vary continuously as the electron-demanding quality of the reaction center is varied, i.e. whether it is electron-rich or electron-poor. A sliding scale of substituent constants would be expected for each substituent having a resonance effect and not just a pair of discrete values a and a for — R, or o and a for + R substituents55. 

Finally, we return to the specific heat. The effects of the phonon coupling on the ripplon spectrum can be taken into account in the same fashion as in the conductivity case. Here we replace the discrete summation in Eq. (38) by integration over the broadened resonances, as prescribed by Eq. (57). The bump, as shown in Fig. 15, is also predicted to be nonuniversal depending on Tg/oio-The predicted bump for Tg/(Od = 2 seems to match well the available data for... 

While the di- and triphenylmethanide anions are resonance stabilized, there is a tendency for isolating the 7T-system with a systematic localization of the double bonds this discrete arrangement of single and double bonds has been observed and studied in detail in alkali bisdipyridylmethanides.43-45... 

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