Trivial resonance

It is convenient to carry out some further calculations with regard to whether u) is commensurable to 27t or not. The non-resonant case where (jj/2n is irrational is pretty simple because only trivial resonances occur there. Therefore, a polynomial transformation brings the map to the form... 

There is no classical analogue of it save the trivial case of the resonance of two similar harmonic oscillators. 

In summary, it is non-trivial to implement magnetic resonance pulse sequences which allow us to monitor unambiguously the decrease in absolute concentration of reactant species and associated increase in product species, but measures of relative concentrations from which conversion and selectivity are calculated are much easier to obtain. However, if such measurements are to be deemed quantitative the spectra must be free of (or at least corrected for) relaxation time and magnetic susceptibility effects. 

The experimental determination of D and E for a dilute single crystal is not trivial, even when the crystal axes are known. Durene, for example, has two molecules per unit cell with different orientations of the molecular plane. Thus for any orientation there are four resonances, two from each type of site. Sorting out the data is a challenging exercise.2... 

If the energy is transferred by trivial emission/reabsorption, it will lengthen the measured lifetime of the donor emission, not shorten it as happens in resonance energy transfer. This comes about because intervening absorption and emission processes take place prior to the final fluorescence emission (the reabsorption cannot take place until the photon has been emitted) the two processes do not compete dynamically, but follow in a serial fashion. In FRET, such an emission/reabsorption process does not occur, and the fluorescence lifetime of the donor decreases. This is an experimental check for reabsorption/reemission. 

Forster (1959) classifies the qualitative features based on which one can distinguish the various modes of energy transfer. Mainly, only collisional transfer depends on solvent viscosity (vide infra), whereas complexing between the donor and acceptor changes the absorption spectrum. On the other hand, the sensitizer lifetime decreases for the long-range resonant transfer process, whereas it should be unchanged for the trivial process. 

Resonance energy transfer between the aromatic amino acids proceeds by very weak coupling between the donor and acceptor.151,52) Very weak coupling implies that the interaction between the donor and acceptor wave functions is small enough so as not to perturb measurably the individual molecular spectra. This transfer process, which is distinct from the trivial process of absorption of an emitted photon, involves radiationless deexcitation of an excited-state donor molecule with concomitant excitation of a ground-... 

By studying the AU55 system by means of PNMR, one would hope to be able to obtain additional microscopic information about the electronic behavior. Unfortunately, NMR experiments proved to be non-trivial [29], since the resonance was extremely weak. This has been taken as an indication that metallic shielding may still be incomplete. 

Therefore, it follows that in a backside trajectory, we obtain both the lowest crossing point as well as the largest TS resonance energy. Computationally, the backside barrier is smaller by 10—20 kcal/mol compared with a front side attack (42). Equation 6.18 defines an orbital selection rule for an Sn2 reaction. Working out this rather trivial prediction is nevertheless necessary since it constitutes a prototypical example for deriving orbital selection rules in other reactions, using FO—VB configurations. Thus, a simple rule may be stated as follows ... 

The olefinic methylene group (protons and carbon) and the three methyl groups (protons and carbons) are trivial assignments, and they correspond with our previous discussion. Of more interest and of greater utility, we assign the three methine protons the doublet of doublets at 2.86 ppm (correlates with the carbon resonance at 63.0 ppm), the apparent quartet at 2.60 ppm (correlates with the carbon resonance at 48.0 ppm), and an apparent triplet at 1.76 ppm (correlates with the carbon resonance at 50.1 ppm). From the COSY and from the known structure, we now are now able to assign all three methine resonances and feed this information back into the COSY to establish other correlations. 

Several of the major peaks in the mass spectrum are difficult to assign since there are two closely spaced functional groups. Although trivial, verification of the assignments of the protons and their multiplicities are left as an exercise for the student. Likewise, verification of the assignments of the resonances in the l3C/DEPT spectra are left for the student. 

With the complete proton assignments and the direct correlations between carbons and attached protons from the HMQC, we are able to assign all of the carbon resonances, except for the quaternary carbon, which is a trivial assignment in this case. An interesting example is found in the inset of the HMQC spectrum, which shows the correlations of the two overlapped protons, H-4 and H-5. Even though they are overlapped in the proton spectrum, they are well resolved in the HMQC spectrum because the carbon resonances are not overlapped. 

Despite the above-mentioned modifications that reduce some of the severe requirements of JCP, the technique as performed in the quoted papers suffers from limitations the 1H and 29Si pulses must be on resonance, and the HaHa condition must be established for full enhancement and polarization time should be optimized. These conditions are not trivial to obtain, and the difficulty in establishing them has prohibited routine applications of the JCP method to 29 Si NMR spectroscopy. 

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