Potential fluctuation

As a fiinction of the inter-electron distance, the fluctuation potential decays to zero more rapidly than does the mean-field potential. Flowever, the magnitude of Fis quite large and remains so over an appreciable range of inter-electron distances. The corrections to the mean-field picture are therefore quite large when measured in... 

Here Uj and Uj are Cartesian unit vectors, a) and j3) are localized orbitals that are doubly occupied in the HF ground state, jm) and n) are virtual orbitals. Rq is the position vector of the local gauge origin assigned to orbital a) and = (r — R ) x p is the angular momentum relative to Re- Superscript 1 denotes terms to first order in the fluctuation potential, and = [A — is the principal propagator at the zero energy... 

The perturbative analysis in Section II.D showed that single-reference L-CTSD is exact through third order in the fluctuation potential, much like L-CCSD theory, and our results are consistent with this analysis. This suggests that one of the things we... 

All the preceding mechanisms of the carrier packet spread and transit time dispersion imply that charge transport is controlled by traps randomly distributed in both energy and space. This traditional approach completely disregards the occurrence of long-range potential fluctuations. The concept of random potential landscape was used by Tauc [15] and Fritzsche [16] in their models of optical absorption in amorphous semiconductors. The suppressed rate of bimolecular recombination, which is typical for many amorphous materials, can also be explained by a fluctuating potential landscape. 

Formulating appropriate rate laws for CO adsorption, OH adsorption and the reaction between these two surface species, a set of four coupled ordinary differential equations is obtained, whereby the dependent variables are the average coverages of CO and OH, the concentration of CO in the reaction plane and the electrode potential. In accordance with the experiments, the model describes the S-shaped I/U curve and thus also bistability under potentiostatic control. However, neither oscillatory behavior is found for realistic parameter values (see the discussion above) nor can the nearly current-independent, fluctuating potential be reproduced, which is observed for slow galvanodynamic sweeps (c.f. Fig. 30b). As we shall discuss in Section 4.2.2, this feature might again be the result of a spatial instability. 

Driving all these fluctuations is thermal energy kT, which is available to pay for transitory changes in the numbers of the various species of ions. These species are all at the same average potential, an average over a region that is small compared with the distances of variation of electrostatic fluctuation potential. 

Again, in a very diluted electrolyte solution, the major contribution will make the self-image interaction. The self-image interaction potential can be defined on the basis of the fluctuation potential as [7,18]... 

The binary (fluctuation) potential and its derivative are continuous at the interface... 

Now it is possible to seek solutions for the electrostatic and fluctuation potentials in the form of an expansion into series in terms of the parameter X (see, for example, [23,24])... 

Treating the fluctuation potential, we will depart a bit from the scheme used for the (unary) electrostatic potential. As a zero iteration the modified Onsager-Samaras approximation will be taken [25], that preserves a proper asymptotic behavior of this potential along the p-axis (see for details elsewhere [24]). 

They may be denoted ATi, AfDT, AfTD, and ACIT, respectively, where D and T standing for doubles and triples manifolds in which the perturbation corrections are gathered. The fluctuation potential V is defined as H — H0 with... 

Another deficiency of the Poisson-Boltzmann equation is that as Kirkwood showed [8], it neglects the effects of the self-atmosphere (or fluctuation) potential. The... 

The mean potential and the fluctuation potential derived above are expressed in terms of correlation (or distribution) functions. Hence we need next a procedure to calculate these functions. There exist several statistical thermodynamical procedures to express Into p, p Into etc. According to the... 

While the use of effective energy surfaces defined in terms of free energies is commonplace in the electron transfer literature, we note an equivalent, alternative viewpoint, in which s expressed in terms of probability distributions of the fluctuating potential-energy gap specified by // (as in Eqs. 20 and 21) [95]. 

The zeroth-order component of the electronic Hamiltonian is taken to be the Fock operator such that the perturbation operator (sometimes called the fluctuation potential) is then the remaining two-electron operator,... 

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