Small molecule limit

These minimalistic peptide scaffolds potentially provide a biologically relevant laboratory in which to explore the details of heme-peptide interactions and, with development, perhaps approach the observed range of natural heme protein fimction. These heme-peptide systems are more complex than typical small molecule bioinorganic porphyrin model compoimds, and yet are seemingly not as enigmatic as even the smallest natural heme proteins. Thus, in the continuum of heme protein model complexes these heme-peptide systems lie closer to, but certainly not at, the small molecule limit which allows for the effects of single amino acid changes to be directly elucidated. 

The state s is not really a physically accessible state, since in order to prepare the molecule initially in radiation with particular coherence between different frequencies which correspond to all the observed spectral lines from the molecular states which contain this zeroth order state initial state cj>0. The molecular eigenstates should then have unit quantum yields at very low pressure, a result which is not inconsistent with experimental extrapolations to zero pressure. This small molecule limit is depicted schematically in Fig. 4a. 

Fig. 4. (a) A schematic representation of the small molecule limit. The states are the same as those represented in Fig. 1. The molecular eigenstates approximately diagonalize the effective molecular Hamiltonian (20), and each carries only a portion of the original oscillator strength to s can therefore decay radiatively to

In the small molecule limit, the physical situation corresponds more closely to that described by the molecular eigenstates 10 I. Each molecular eigenstate contains some admixture of the zeroth order states, the vibrationally cold ... 

The presence of the vibronic states, (< )/, can and generally does modify these simple expectations. This dense manifold of vibronic levels leads to a series of phenomena, which depend, in part, on the densities of levels, p within this set. For instance, the small-molecule limit is characterized by a very low density of levels in this set In this case the non-Bom-Oppenheimer couplings can lead to the observation of perturbations in the spectrum of arising from the vibronic state (,. Such perturbations are associated with the displacement of levels of from their anticipated positions and possibly in the emergence of additional lines in the absorption spectra. This small molecule limit of low-level densities is an ideal case for the assignment and analysis of all spectroscopic lines. 

Here hT, represents the energy uncertainty of a level with a decay rate of r and e, is the average spacing between the levels. The condition for the small-molecule limit is... 

X, < 1, small-molecule limit whereas for the large molecule case it is... 

These limits are rather simply understood by the consideration of even a simple diatomic molecule example. Suppose first that and , represent two bound electronic states of the diatomic molecule. Then under isolated molecule conditions F, corresponds to the radiative decay rate of the individual level. F is typically on the order of 10" -10" sec". However, in the diatomic molecule the average spacing between vibronic levels, /, is on the order of hundreds or thousands of cm". Thus the simple diatomic molecule example is definitely within the small molecule limit of (2.2). 

There are then no possibilities for the occurrence of irreversible radiationless decays in such small-molecule limit triatomics. However, interesting effects, arising from the coherent superposition of many levels, may still appear when (2.14) is violated. The presence of hyperfine structure makes this possibility very likely. For instance, Demtroder has observed nonexponential decays of excited states of NO2 in a molecular beam where the spacing between hyperfine levels is claimed to be sufficient to excite a single hyperfine component with his MHz bandwidth laser. Demtroder then has no recourse but to explain the nonexponential decays in terms of some elusive radiationless decay despite the fact that the conditions (2.2) for the small-molecule limit are obeyed and prohibit irreversible decays. It should, however, be recalled that when traveling along with the molecule in the molecular beam, the molecule encounters a pulse of radiation whose duration is given by the laser spatial extent divided by the molecular velocity. For a laser spot size of 10" cm and a molecular velocity of 10 cms -the pulse duration is 10 s. This yields an effective pulse frequency width of 10 MHz which could yield a coherent superposition of a number of hyperfine levels. The nonexponential decay of such a superposition is discussed in Section II. C. 

On longer time scales, where (2.3 ) is violated, the behavior of the system is then again given by the small-molecule limit and the complicated... 

In (2.1), e, is again taken to be the average spacing between neighboring ((>/ levels at the energy of is again the decay rate of the zeroth order level, , which in isolated molecules is due to infrared emission and has typical values on the order of 10 -10 s . In the small molecule limit the condition (2.2) is obeyed, and the states of the system are characterized... 

As the majority of cases studied by Smalley and co-workers fall into the too many level small molecule limit, the intermediate case, we have generalized the intermediate case relaxation theory to describe the Smalley type experiments. This generalization involves the description of the radiative decay characteristics of the /y states to evaluate the time dependence of the relaxed emission as well as the relaxed and unrelaxed quantum yields. The lack of observation of the rapid intramolecular dephasing in the alkylated benzenes may be due to an excitation pulse too long to coherently excite the zeroth order ring vibration. An alternative... 

Given that the small-molecule limits (2.2) and (2.2 ) for relevant experimental time scales are both satisfied, the real difference between a veiy small diatomic molecule and an intermediate-sized one like glyoxal resides only in the number of coupled levels in (2.5) and (2.6), respectively. Hence it is convenient to pursue the general analysis for the case of a pair of coupled levels, S and T. Then the more general case of many levels (2.6) is readily generated. 

The fortuitous near-degeneracy of the levels poses severe difficulties to a complete quantitative description of the process in the too-many level small-molecule limit, and even in the small-molecule limit itself. The interaction potentials are poorly known. It is still unclear why collisional processes proceed on a much more rapid time scale in electronically excited states than in ground electronic states. The intuitive explanation, that the excited states are larger, is insufficient. An analysis of the excited potential energy surfaces should prove enlightening in this regard. 

Further condensation can be obtained by heating the resole again to continue the reaction of the methylol groups. This further condensation, however, results in the evolution of gas which could result in a porous bondline. This evolution of small molecules limits the utility of resole phenolics in structural adhesives. Novolac phenolic resins have a structure similar to that shown in Pig. 1 and they are obtained by heating formaldehyde in the presence of excess phenol. As one can see, the novolac structure contains no residual methylol groups. Novolacs will not cure with themselves as do resoles, but they can be cross-linked by means of hexamethylene tetramine ( HEXA"). This is also shown in Pig. 1. 

Li = L, i.e., if the simple states include all the nonradiant manifold, we recover the small-molecule limit, as treated in Section II,E,1. On the other hand, if L2 = L, i.e., if the whole nonradiant manifold is complex and treated as a random quasi-continuum, we have the statistical limiting case of Section II,E,2,a, for which the matrix (179c) reduces to the single diagonal element for the special radiant state. 

Fig. 5. Energy level scheme in (a) small-molecule limit, (b) statistical limit, and (c) intermediate case.

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