Spectral density equation substitutions

By substituting the expressions for spectral densities in Eq. 8.11, we obtain an equation that is algebraically cumbersone in general but that can be simplified in either of two regimes (1) homonuclear spins (I = S) or (2) rapid tumbling (extreme narrowing limit). 

On substituting Equation 30 into Equation 28, we get in an expression for the spectral density... 

The spectral density for the output sliver can be obtained by substituting Equations [1.54] and [1.55] into Equation [1.47] ... 

Explicit indices are inserted to denote site dependence, e.g., r,y is the / j distance in conformer v. Substitution of equation (II) into equation (8) yields the desired spectral densities ... 

If the perturbations thus caused are relatively slight, the accepted perturbation theory can be used to interpret observed spectral changes (3,10,39). The spectral effect is calculated as the difference of the long-wavelength band positions for the perturbed and the initial dyes. In a general form, the band maximum shift, AX, can be derived from equation 4 analogous to the weU-known Hammett equation. Here p is a characteristic of an unperturbed molecule, eg, the electron density or bond order change on excitation or the difference between the frontier level and the level of the substitution. The other parameter. O, is an estimate of the perturbation. 

Spin densities (p) are theoretical quantities, defined as the sum of the squared atomic orbital coefficients in the nonbonding semi-occupied molecular orbital (SOMO) of the radical species (Hiickel theory). For monoradical species, the spin density is connected to the experimental EPR hyperfine coupling constant a through the McConnell equation [38]. This relation provides the opportunity to test the spin density dependence of the D parameter [Eq. (8)] for the cyclopentane-1,3-diyl triplet diradicals 10 by comparing them with the known experimental hyperfine coupling constants (ap) of the corresponding substituted cumyl radicals 14 [39]. The good semiquadratic correlation (Fig. 9) between these two EPR spectral quantities demonstrates unequivocally that the localized triplet 1,3-diradicals 9-11 constitute an excellent model system to assess electronic substituent effects on the spin density in cumyl-type monoradicals. 

The present results clearly establish the D parameter of localized triplet diradicals as a reliable spectral tool to probe for electronic factors that control spin delocalization and radical stabilization in the benzyl-type radicals 14. Equation (8) offers us the opportunity to interpret the experimental results through semiempirical MO calculations in terms of the theoretically accessible a-spin-density variations. The spectroscopic AD scale does not suffer from the limitations (polar vs. radical contributions) inherent in the kinetic chemical scales and provides us with over 40 aryl substituents, which is the most comprehensive collection of electronic effects on radicals. Its extension to heteroaryl-substituted diradicals (12) provides for the first time a quantitative experimental measure of delocalization in aromatic n systems. The salient features of this novel method are... 

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