Delocalization continuum

As we have reviewed here, the linear region is not fully repulsive, and transitions of the ground-state, linear conformer access vibrationally excited intermolecular levels that are delocalized in the angular coordinate. As depicted in Fig. 1, however, the internuclear distance is significantly longer in the excited state at the linear geometry. Consequently, there is favorable Franck-Condon overlap of the linear conformer with the inner-repulsive wall of the excited-state potential. It is therefore possible for the linear Rg XY conformers to be promoted to the continuum of states just above each Rg - - XY B,v ) dissociation limit. 

In the electron-sea model, a metal crystal is viewed as a three-dimensional array of metal cations immersed in a sea of delocalized electrons that are free to move throughout the crystal (Figure 21.6). The continuum of delocalized, mobile valence electrons acts as an electrostatic glue that holds the metal cations together. 

Figure 1.8 The probability density of several continuum eigenstates of the Hamiltonian in Eq. (24) plotted on the baseline of their corresponding energy. The potential is also plotted for convenience. Note that most continuum states (dashed lines) are delocalized and have a very small amplitude inside the potential well between the two barriers, whereas there are continuum functions that are localized inside the well. The localized eigenstate (solid line) is the same as shown in Figure 1.1.

Not long before this writing, partially delocalized structures 65, which lie on a continuum between 63 and 64ex/64en, had not been reported. 

The properties of a number of these photosensitizers have been studied in aqueous and sodium lauryl sulfate micellar media,139 their electronic and emission spectra, electrochemistry and quenching of their luminescence by HgCl2 in both media.138 In some cases solvent dependence of MLCT excited state lifetimes has been observed, e.g. in [Os(bipy)2py2]2+, [Os(bipy)2(NCMe)2]2+ and [Os(bipy)j(Ph2PC6H4PPh2]2+. For these, dielectric continuum theory has been invoked to explain such solvent dependence, and it is necessary to assume that the excited electron in the excited state is localized on one ligand rather than delocalized over all three.140... 

It is clear that the introduction of a critical volume fraction is a step toward dealing with percolation on a continuum. To this end Zallen and Scher (1971) considered the motion of a classical particle in a random potential, V r), and introduced a function, which detines the fraction of space accessible to particles of energy E. The connection with percolation is in the fact that, for energies such that 4>(E) > c(.E), there are infinitely extended volumes of allowed (V < T) space. The critical value c is identified with 0.15 for d = 3, and delocalized states appear above c ... 

In Figure 5a, the paraboloids correspond to electronic states aj b, a/ b, 2bj2a, and 2b/2a. This situation arises when the potential along the rp coordinate (shown in Figure 5b) has a low barrier (or no barrier) so that all vibrational states are mixtures of a and b PT states and are delocalized. Note that the minima for the Xa/Xb and Xa/Xb states are virtually identical. These examples illustrate that the multistate continuum theory provides a valuable framework for the analysis and interpretation of the wide range of behavior exhibited by PCET reactions. 

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