Complex resonance

Demidov A A 1999 Use of Monte-Carlo method in the problem of energy migration in molecular complexes Resonance Energy Transfer e6 D L Andrews and A A Demidov (New York Wiley) pp 435-65... 

Rigorous design reviews must include the often highly complex resonance behavior of impellers and blading to ensure vibration-free or vibration-tolerant design of these critical turboexpander components. In other words, the manufacturer must perform comprehensive theoretical and experimental studies of the blade oscillations in the rotating system. 

For kaolinite the sample permeability was very low and the solution was poorly removed. The spectra (Figure 3C) are consequently complex, containing peaks for inner and outer sphere complexes, CsCl precipitate from resMual solution (near 200 ppm) and a complex spinning sideband pattern. Spectral resolution is poorer, but at 70% RH for instance, inner sphere complexes resonate near 16 ppm and outer sphere complexes near 31 ppm. Dynamical averaging of the inner and outer sphere complexes occurs at 70% RH, and at 100% RH even the CsCl precipitate is dissolved in the water film and averaged. 

The surface behavior of Na is similar to that of Cs, except that inner sphere complexes are not observed. Although Na has the same charge as Cs, it has a smaller ionic radius and thus a larger hydration energy. Conseguently, Na retains its shell of hydration waters. For illite (Figure 6), outer sphere complexes resonate between -7.7 and -1.1 ppm and NaCl... 

The importance of pi-complex resonance structures such as (3.181) can also be directly assessed with NRT analysis (Section 1.6). For the equilibrium C3H6 structure corresponding to the electron-density map in Fig. 3.81(a), the optimal NRT resonance weightings are found to be... 

Copper (II) -acetylacetonate complex (Resonating ring compound)... 

Despite the obvious difficulties associated with the acquisition of good quality spectra, Mg NMR has been usefully applied to the study of organomagnesium compounds . The Mg NMR parameters reported for organomagnesium complexes are hsted in Table 12. Examination of Table 12 shows that the total solution chemical shift range is relatively narrow —85 to +110 ppm. The -cyclopentadienyl complexes resonate at significantly... 

Some of the earliest applications of MQDT dealt with vibrational and rotational autoionization in H2 [21-25]. One concept that emerged from these studies is that of complex resonances [26], which are characterized by a broad resonant distribution of photoionization intensity with an associated rather sharp fine structure. These complex resonances cannot be characterized by a single decay width they are the typical result of a multichannel situation where several closed and open channels are mutually coupled. The photoionization spectrum of H2 affords a considerable number of such complex resonances. 

Compound X, of formula C3H5Br3, with methyllithium formed bromocyclopro-pane and 3-bromopropene. The nmr spectrum of X showed a one-proton triplet at 5.9 ppm, a two-proton triplet at 3.55 ppm, and a complex resonance centered at 2.5 ppm downfield from TMS. What is the structure of X Account for the products observed in its reaction with methyllithium. 

Thus we leam three things 1) the non-crossing rule is not obeyed in the present picture of unstable resonance states, 2) complex resonances may appear on the real axis and 3) unphysical states may appear as solutions to the secular equation. Thus avoided crossings in standard molecular dynamics are accompanied by branch points in the complex plane corresponding to Jordan blocks in the classical canonical form of the associated matrix representation of the actual operator. 

In the Liouville formulation one obtains the well-known connection between x and the imaginary part T of the complex resonance eigenvalue... 

The spontaneous Raman spectrum is then described by the sum of the imaginary parts of the complex resonant susceptibility,... 

Indirect photodissociation involves two more or less separate steps the absorption of the photon and the fragmentation of the excited complex. Resonances, which mirror the quasi-bound states of the intermediate complex in the upper electronic state, are the main features. They have an inherently quantum mechanical origin. If we consider — in very general terms — the inner region, before the fragments have obtained their identities, as the transition state, then the resolution of resonance structures in the absorption spectrum manifests transition state spectroscopy in the original sense of the word (Foth, Polanyi, and Telle 1982 Brooks 1988). 

The H-NMR spectrum of the 1 1 adduct of f-butyllithium-d9 and normal butadiene (DP = 1) in benzene was characterized by complex resonances this spectrum was interpreted by these authors as being a mixture of cis- and trans-1,4-addition products. 

Even if the equations arrived at seem familiar there are some obvious fundamental differences. First of all, the ansatz singles out two mirror spaces, where the particle and its mirror image may be located, respectively. Second, the expanded complex symmetric representation takes into account complex resonance states. Note also, as said above, that the complex symmetry can be obtained from the hermitean representation via a non-positive definite metric, i.e. 

We will not analyse the situation further here except pointing that the present resonance model, under appropriate environmental perturbations, admits primary complex resonance energies commensurate with rigorous mathematics and precise boundary value conditions, i.e. 

In the case of adiabatic electron transfer reactions, it is found that the potential energy profiles of the reactant and product sub-systems merge smoothly in the vicinity of the activated complex, due to the resonance stabilization of electrons in the activated complex. Resonance stabilization occurs because the electrons have sufficient time to explore all the available superposed states. The net result is the attainment of a steady, high, probability of electron transfer. By contrast, in the case of -> nonadiabatic (diabatic) electron transfer reactions, resonance stabilization of the activated complex does not occur to any great extent. The result is a transient, low, probability of electron transfer. Compared with the adiabatic case, the visualization of nonadiabatic electron transfer in terms of potential energy profiles is more complex, and may be achieved in several different ways. However, in the most widely used conceptualization, potential energy profiles of the reactant and product states... 

The association of nitrobenzene and nitronaphthalene in non-polar solvents, such as hexane and carbon tetrachloride, forms an exception. Here the association depends in the first place on interaction of the Keesom type with the very large moment (4.20 D). The stronger association of the last-mentioned compound points, however, in addition to an interaction due to complex resonance as observed between nitro compounds and aromatic hydrocarbons in general. It is plausible that when dissolved in benzene this association gives way to a solvation. This interaction between solute and solvent molecules is closely related to the association between like molecules. In benzene etc. no stoichiometric association is observed but, owing to the anisotropy of the polarizability, a more or less parallel... 

Brackman5 formed the conception of complex resonance in which the stationary state is pictured as one of resonance between a no-bond and a bonded configuration. 

There is thus a complex resonance in this case also... 

The two extremes of the Dewar-Chatt-Duncanson model for olefin coordination can also be applied to describe aldehyde, ketone, and imine complexes. Resonance structure A is an rf complex of Zr(II), while its resonance structure B is a zirconaaziridine containing Zr-C and Zr-N bonds (Fig. 6). X-ray structural studies of zirconaaziridines and their observed reactivity suggest that resonance structure B is more important. 

Wertz provides a good inorganic example of Raman spectroelectrochemistry in which a series of ruthenium polyazine complexes were studied in varying redox states. The series of complexes and redox states studied were [Ru(bpm)3] " (n = 0 4), [Ru(bpz)(bpy)2] " [n = 0-3), [Ru(bpy)2(bpz)] " (n = 0-3), [Ru(bpz)3] " (n = 0 3), (bpm = 2,2 -bipyrimidine, bpz = 2,2 -bipyrazine, bpy = 2,2 -bipyridine) where n is the number of electrons added to the complex. Resonance Raman spectra are recorded at each redox state to investigate the identity of the redox orbital for the series of complexes. Figure 19 shows a... 

The method defined in (6.2.58) turns the real bound state energies into complex resonances, where the width of the resonances, A /2, is assumed... 

Fig. 6.10. Ionization rates computed by turning the SSE energies into complex resonances with widths corresponding to the one-photon decay rates.

The first nitrogen NMR work for nitrosyl compounds was reported in the 1960s for N. This work showed that linear complexes resonate at medium field but due to the problems of quadrupolar broadening, it... 

Two different kinds of JT effect are distinguished in the literature the static and dynamic JT effects. In the static JT effect the molecule or complex remains distorted in a particular way long enough for the distortion to be detected experimentally. In the dynamic JT effect, the molecule or complex resonates between two or more equivalent modes of distortion, and the distortion is not directly observable [9]. 

Figure 1.16 A silver ion pi-complex. Resonance forms describe the orbital overlap shown at left.

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