Electronic-energy gap

In Eq. (77), x = h(o/2kg T is the reduced internal frequency, q = EJhoi the reduced solvent reorganization energy, p = hElha> the reduced electronic energy gap and / (z) the modified Bessel function of order m. The quantity S is a coupling parameter which defines the contribution of the change in the internal normal mode ... 

For the case in which the electronic energy gap is much larger than the difference in vibrational energies, we can apply the Plazcek approximation EIV — En[l ... 

In the case of displaced harmonic surfaces with inhomogeneity in the electronic energy gap, (0 (00)) becomes... 

The rate of internal conversion between electronic states is determined by the magnitude of the energy gap between these states. The energy gaps between upper excited states (S4, S3, S2) are relatively small compared to the gap between the lowest excited state and the ground state, and so the internal conversion between them will be rapid. Thus fluorescence is unable to compete with internal conversion from upper excited states. The electronic energy gap between Si and S0 is much larger and so fluorescence (Si —> S0) is able to compete with Si(v = 0) So(v = n) internal conversion. 

Jortner, J. Navon, G. J. Phys. Chem. 1981, 85, 3759]. On the other hand, if one considers the variation of activation energy with electronic energy gap, one finds a very large discrepancy between the quantum-mechanical and classical approaches for very exothermic reactions at room temperature (see Figure 1). 

Figure 1. Activation energy of electron-transfer process as a function of electronic energy gap of a reaction. Er = Eg + Ec is the total reorganization energy where Es is the classical solvent reorganization energy and Ec is the reorganization energy of an intramolecular mode, l

Figure 2. Deuterium isotope effect for electron transfer between ammine complexes as a function of the reduced electronic energy gap AE/EC where Er is the total reorganization energy E, = Eg + Ec. Key for parameters — —, hci)H/kBT — 2.0 and Es/Ec = 0 ------------------, Es/Ec = 1 and-------, Es/Ec = 2.

To summarize, we find that for two very different systems coherent nuclear motion can survive surface-hopping events and persist in condensed-phase systems for comparatively long times. We now turn to a discussion of how nuclear motion influences electronic energy gaps. 

Fig. 5. The dependence of the nonradiative lifetime for intersystem crossing (from the first excited triplet to the ground state) on the electronic energy gap (O) CxHy ( ) CxDy. All data were obtained for aromatic hydrocarbons in solid solutions. The nonradiative lifetime (j3 = r r) was calculated by Siebrand30 from eq. (3-1) taking r, = 30 sec-1 for all the aromatic hydrocarbons. This figure is reproduced from Siebrand s paper.30...

Fig. 4.2 A short segment of t rans -polyacetylene is shown with an abrupt (idealized) reversal of the bond alternation pattern (see text). Top- a neutral soliton with an unpaired spin and an energy state near the middle of the electron energy gap. Middle the addition of an electron results in the formation of a spinless negatively charged soliton. Bottom the extraction of an electron from the top results in the formation of a spinless positive soliton. The optical transitions associated with the charged solitons are indicated as arrows on the right.

This indicates that the electronic energy gap is reduced by to, one quantum of the promoting mode i.e., the promoting mode can accept at least one vibration quantum. From Eq. (65) one can see that the symmetry argument can be used to determine the promoting mode. For example, for formaldehyde [35 42] the n-> ti transition corresponds to Aj A2 for the C2v symmetry in this case the promoting mode for the IC A2-> A should possess the A2 symmetry. 

In this manner, the electronic transitions of some molecules are sensitive probes of the solute environment. Since the probe molecules selected by Kamlet and Taft have n electronic states which are more polar than the ground state, a change in the polar-ity/polarizability of the solvent medium changes the electronic energy gap, and thus the position of the absorption band. Kamlet and Taft have developed an empirical relationship between measured solute absorption maxima in a solvent and the polarity/polar-izability of that solvent ... 

Figure 16-12. Left normalized non-equilibrium response function for the electron energy gap in SCA at different densities and 450 K. Right equilibrium spatial correlations between the center of the first excited state rj and the nitrogen site of ammonia for the supercritical states. Solid and dashed lines correspond to adiabatic trajectories with forces taken from the ground and first excited electronic states, respectively. Adapted from Ref. [28]...

The electronic energy gap thus serves as a collective reaction coordinate X reflecting the strength of coupling of the nuclear modes to the electronic states of the donor and acceptor. The point of intersection of Pi (X) and Pi (X) sets up the ET transition state, X = 0. 

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