Hopping, electron

Graaf and R. Broer, Magnetic Interactions in Molecules and Solids, 

The probability for these hopping processes is normally condensed into a single parameter referred to as tab, but Vdb and p (Hiickel theory) are also used. Intuitively one would say that the hopping parameter is the same for all three processes, since one electron moves from a to b, while the rest of the occupations stay the same in all cases. But the calculation of the ( Pj H Pp) matrix element shows that this is not exactly the case. The interaction matrix elements of the initial and final states are easily determined with the Slater-Condon rules. In the first case, the hopping of an electron to an empty orbital is defined by 

The contribution of the inactive doubly occupied orbitals is the same in the three cases as is the one-electron term hat- However, the appearance of two-electron integrals for those cases with more than one electron in the magnetic orbitals introduces differences in the interaction matrix elements. 

Numerical estimates of the hopping parameter are relatively easy to obtain with the different computational schemes discussed in Chap. 4. Starting with in a centrosymmetric two-site system, two electronic states can be defined with doublet spin coupling  

In practice, an effective hopping parameter can be obtained from accurate ab initio energies for the two doublets. For non-centrosymmetric systems, the calculation is slightly more involved. The two doublets are now defined as 

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Figure C3.2.15. Schematic diagram showing (A) electron hopping between electron reservoirs via empty states of an intervening bridge, (B) tunnelling, and (C) hole hopping via filled states of an intervening bridge. From...

The electron-lattice interaction is introduced through the dependence of the electron hopping amplitude on the carbon-carbon bond length ... 

For explicitness, let us assume that off-diagonal disorder is caused by chain twists, which randomly diminish the overlap between the n-orbitals of neighboring carbon atoms (see Fig. 3-7). The electron hopping amplitudes that depend both on the interatomic distances and on the relative orientation of the electronic orbitals on neighboring atoms can then be written in the form ... 

Electron-hopping is the main charge-transport mechanism in ECHB materials. There is precedence in the photoconductivity Held for improved charge transport by incorporating a number of redox sites into the same molecule. A number of attempts to adapt this approach for ECHB materials have been documented. Many use the oxadiazole core as the electron-transport moiety and examples include radialene 40 and dendrimer 41. However, these newer systems do not offer significant improvements in electron injection over the parent PBD. 

The alternation of the bond lengths results in an alternation of the electron hopping amplitudes ... 

Figure 3-7. A simplified picture of a //wi-v-polyacelylcne chain with conformational disorder the right part of the chain is rotated with respect to its left part over the angle .

Cerril, R.H., Posdethwaite, T.A., Chen, C.H., Poon, C.D., Tzerzis, A., Hutchinson, A.D., Clark, M.R., Wignall, G., Londono, J.D., Superfine, R., Calvo, M., Johson, C.S., Samulski, E.T. and Murray, R.W. (1995) Monolayers in Three Dimensions NMR, SAXS, Thermal, and Electron Hopping Studies of Alkanethiol Stabilized Gold Clusters. Journal of the American Chemical Society, 117, 12537-12548. 

Bates et al. reported the construction and characterization of a gold nanoparticle wire assembled using Mg -dependent RNA-RNA interactions for the future assembly of practical nanocircuits [31]. They used magnesium ion-mediated RNA-RNA loop-receptor interactions, in conjunction with 15 nm or 30 nm gold nanoclusters derivatized with DNA to prepare self-assembled nanowires. A wire was deposited between lithographically fabricated nanoelectrodes and exhibited non-linear activated conduction by electron hopping at 150-300 K (Figure 16). 

For highly ordered 3D systems, Terril et al. showed that the electron-hopping conductivity depends on the activation energy of the electron transfer and the electronic coupling term [3 [71]. They took the latter as a... 

The charge propagates in the film by electron hopping between the polymer Red/Ox couples. This is controlled by the electrode potential only in a close proximity of the electrode in more distant sites, the charge transport is driven by a concentration gradient of reduced or oxidized mediators. The observed faradaic current density, jF, is a superposition of... 

The individual macromolecular chains of conducting polymers agglomerate into more complicated structures, usually fibrous. The electronic conductivity of this system is a superposition of the conductivity of the individual fibres (chains) and that due to electron hopping between these domains. The latter is usually much lower, i.e. it controls the total conductivity of the system. 

The reduced poly-[Fe(II)TPP] porphyrin site now finds itself next to a fresh poly-[Fe(III)TPP(X)] site one polymer lattice unit further into the polymer. An electron hopping - or self exchange - reaction can then ensue, repeatedly, in successive layers and sites ... 

The above mechanistic aspect of electron transport in electroactive polymer films has been an active and chemically rich research topic (13-18) in polymer coated electrodes. We have called (19) the process "redox conduction", since it is a non-ohmic form of electrical conductivity that is intrinsically different from that in metals or semiconductors. Some of the special characteristics of redox conductivity are non-linear current-voltage relations and a narrow band of conductivity centered around electrode potentials that yield the necessary mixture of oxidized and reduced states of the redox sites in the polymer (mixed valent form). Electron hopping in redox conductivity is obviously also peculiar to polymers whose sites comprise spatially localized electronic states. 

In the present case, the electron hopping chemistry in the polymeric porphyrins is an especially rich topic because we can manipulate the axial coordination of the porphyrin, to learn how electron self exchange rates respond to axial coordination, and because we can compare the self exchange rates of the different redox couples of a given metallotetraphenylporphyrin polymer. To measure these chemical effects, and avoid potentially competing kinetic phenomena associated with mobilities of the electroneutrality-required counterions in the polymers, we chose a steady state measurement technique based on the sandwich electrode microstructure (19). 

Secondly, Fig. 5 shows that the polymeric rate constants parallel values of heterogeneous rate constants that have been observed for the electrochemical reactions of solutions of the corresponding dissolved porphyrin monomers. (The slope of the line is 0.5). This re-emphasizes what was said above, that measurements of electron hopping in polymers can give rate constants that are meaningful in the context of the metalloporphyrin s intrinsic electron transfer chemistry. 

Much theoretical work went into trying to understand these results more completely. Newns constructed a simple ID model that incorporated the idea of electrons hopping in and out of the tt anti-bonding orbital of NO as the means by which hot electron-hole pairs could transfer energy to the NO molecule.26 Using reasonable assumption, he was able to quantitatively reproduce the experimentally-observed surface temperature and incidence energy dependence. 

In order to explain the changing optical properties of AIROFs several models were proposed. The UPS investigations of the valence band of the emersed film support band theory models by Gottesfeld [94] and by Mozota and Conway [79, 88]. The assumption of nonstoichiometry and electron hopping in the model proposed by Burke et al. [87] is not necessary. Recent electroreflectance measurements on anodic iridium oxide films performed by Gutierrez et al. [95] showed a shift of optical absorption bands to lower photon energies with increasing anodic electrode potentials, which is probably due to a shift of the Fermi level with respect to the t2g band [67]. 

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