Electronic excited states state averaging

Fig. 21. Top The general Jablonski diagram for the flavin chromophore. The given wavelengths for absorption and luminescence represent crude average values derived from the actual spectra shown below. Due to the Franck-Condon principle the maxima of the peak positions generally do not represent so-called 0 — 0 transitions, but transitions between vibrational sublevels of the different electronically excited states (drawn schematically). Bottom Synopsis of spectra representing the different electronic transitions of the flavin nucleus. Differently substituted flavins show slightly modified spectra. Absorption (So- - S2, 345 nm S0 -> Si,450nm 1561) fluorescence (Sj — S0) 530 nm 156)) phosphorescence (Ty Sq, 605 nm 1051) triplet absorption (Tj ->Tn,...

The two most important nonradiative relaxation methods that compete with fluorescence are illustrated in Figure 27-lb. Vibrational relaxation, depicted by the short wavy arrows between vibrational energy levels, takes place during collisions between excited molecules and molecules of the solvent. Nonradiative relaxation between the lower vibrational levels of an excited electronic state and the higher vibrational levels of another electronic state can also occur. This type of relaxation, sometimes called internal conversion, is depicted by the two longer wavy arrows in Figure 27-lb. Internal conversion is much less efficient than vibrational relaxation, so that the average lifetime of an electronic excited state is between 10 and 10 s. The exact mechanism by which these two relaxational processes occur is currently under study, but the net result is a tiny increase in the temperature of the medium. 

The underlying theoretical approach is characterized by the type of the wavefunction and the choice of the basis set. Most current general-purpose semiempirical methods are based on molecular orbital theory and employ a minimal basis set for the valence electrons. Electron correlation is treated explicitly only if this is necessary for an appropriate zero-order description (e.g., in the case of electronically excited states or transition states in chemical reactions). Correlation effects are often included in an average sense by a suitable representation of the two-electron integrals and by the overall parametrization. 

The difference from Eq.5 is the additional averaging over final states. Most experiments, in which product state distributions are measured, are done at one particular wavelength. In contrast, the total absorption cross section is typically measured as a function of wavelength. Although the absorption cross section is a higher averaged quantity, it contains important information about electronically excited states. 

Deactivation of electronic excited states may also involve phosphorescence. After intersystem crossing to the triplet state, further deactivation can occur either by internal or external conversion or by phosphorescence. A triplet — singlet transition is much less probable than a singlet-singlet conversion. Transition probability and excited-state lifetime are inversely related. Thus, the average lifetime of the excited triplet state with respect to emission is large and ranges from 10 to 10 s or more. Emission from such a transition may persist for some time after irradiation has ceased. 

Analogously, the study of charged species, especially anions, in their electronic excited states requires, on average, more extended basis sets than that of their neutral counterpart. In this case, for example, the energy ordering obtained at the 6-31G(d) level is often misleading. 

When class of compounds that are known to be ill-treated by TD-DFT, for example, cyanine, are included in the set of the experimental data ( 500 compounds and more than 700 excited states), the average error of TD-PBEO VEE increases up to 0.24 eV, and a similar value is obtained when the comparison is made with the best theoretical estimates computed for a smaller set of compounds in the gas phase (104 singlet state). In any case, such a value is an average between that expected for electron transitions with a monodeterminantal nature, for which PBEO (and other hybrid functionals including 20/30% of HF exchange) is remarkably accurate (expected error 0.15 eV), and those with a strong multideterminantal nature (e.g., cyanine, triphenylmethane, and acridine derivatives) for which TD-DFT is inadequate [54]. 

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