Exponential potential

It is not necessary to restrict ourselves to bonds that are described by Morse potentials. We can regard eqn. (56) as a quadratic equation in x, use any form of the potential energy V(R) with the usual shape (i.e., a minimum, a repulsive barrier at short distances, and a monotonical increase at large distances), and determine x to get another definition of the bond order. This is called the unity bond index quadratic exponential potential (UBI QEP) method by Shustor-ovich and Sellers. ... 

In contrast, a drastic change in the ion selectivity due to kinetic limitations must be expected if cations of different charge are present. The reason is that the Eyring-type description for the interfacial ion fluxes involves an exponential potential dependency.55 This term gives a strong preference for the species of the highest valence when the applied voltage is increased within the kinetic domain. ... 

What are the implications of Eq. (6.132) The Stem synthesis of the two models implies a synthesis of the potential-distance relations characteristic of these two models [Fig. 6.66(b)] a Z/ncar variation in the region from. v = 0 to the position of the OHP according to the Helmholtz-Perrin model (see Section 6.6.2), and an exponential potential drop in the region from OHP to the bulk of solution according to the Gouy-Chapman model (see Section 6.6.4), as shown in Fig. 6.67. 

LJl) and van der Waals (LJ2) potentials were used for nonpolar bead-bead and bead-wall interactions, respectively. For polar interactions, exponential potential functions (EXP 1,2) were added to both bead-bead and bead-wall cases. For the bonding potential between adjacent beads in the chain, a finitely extensible nonlinear elastic (FENE) model was used. For example, PFPE Zdol... 

In the hypothesis of an exponential potential, the situation is even worse because Jeans instability is always reduced by the presence of quintessence (i.e., o ia is never equal to 1). More detailed calculation give the following value of the matter power spectrum normalization as as a function of Wq in Fig. 8.2. 

Nonadiabatic transitions play crucial roles in various fields of physics and chemistry [1. 2. 3. 4. 5, 7, 8 9. 10], and it is quite important to develop basic analytical theories so that we can understand fundamental mechanisms of various dynamics. The most fundamental models among them are the Landau-Zener type curve crossing and the Rosen-Zener-Demkov type non-curve-crossing. Furthermore, there is an interesting intermediate case in which two diabatie exponential potentials arc... 

V is the collision velocity and a is the logarithmic derivative of the repulsive exponential potential [3]. 

This expression coincides with a general expression (9) provided T is identified with the transmission probability through the exponential potential step, which corresponds to... 

Two interesting non-quasiclassical limits of Eq.(33) correspond to purely repulsive potential and to the case when the potential well supports a single bound state, i) If kd=0, the Morse potential becomes a purely repulsive exponential potential U R) oc exp(-2R/fl). For this case, Eq.(33) yields ... 

Besides using the DFT approach, other high-level theoretical models have also been developed to probe the kinetic parameters, relevant to SOFC operation. For example, the unity bond index-quadratic exponential potential (UBI-QEP) [104,105] was successfully performed to study several elementary reactions on various transition metal surfaces [106]. This approach generated activation barriers and enthalpy changes for forward and reverse reactions for the formation and dissociation of H20 and OH" species. 

UBI-QEP, See Unity bond index-quadratic exponential potential (UBI-QEP) Ubiquitinated proteins, folding of, 271-273 protein degradation, 271-272... 

A one channel example the resonances associated with a repulsive exponential potential. 

Unity Bond Index-Quadratic Exponential Potential... 

Because of the compensation of the surface charge by the positively charged counterions, an exponential potential drop in the diffuse layer was discussed by Gouy (1910, 1917) and Chapman (1913). The potential at the particle surface is the Nernst potential if/0 the extension of the diffuse layer is equal to 1 Ik (Debye-Hiickel parameter). 

Thus, it is adequate to determine the energetic characteristics of the elementary reactions based on the Unity Bond Index-Quadratic Exponential Potential (UBI-QEP) method developed by Shustorovich [2], while the pre-exponential factors may be estimated simply from the transition-state theory [4,26]. Here we employ, for illustrative purposes, a simplified version of a microkinetic WGSR model developed by us earlier [14],... 

Landau-Teller expression P °c exp(-2M /ft v) for a head-on collision. Here V is the collision velocity and a is the logarithmic derivative of the repulsive exponential potential [3],... 

A Morse exponential potential [2] can also be used for the bond-stretching term in the form... 

The 6th order term in the LJ expression represents dispersion (long-range) interactions while the 9th (or 12th) order term represents short-range repulsion. Sometimes an exponential potential may be used as the short-range term in combination with a 6th order dispersion term in the form... 

In the same formula, h is the step size of the integration and n is the number of steps, i.e. w is the approximation of the solution in the point x and Xn = XQ + n h and Xq is the initial value point. The authors, based on the theoretical background developed in [164], obtained four RKN methods (1) two methods based on the fifth order method [175] and (2) two methods based on the sixth order method [144] developed by Dor-mand, Mikkawy and Prince. For the methods based on [175] the authors constructed two methods with first and second exponential order. For the methods based on [144] the authors constructed two methods with first and second exponential order. They have applied the new developed methods to the radial time-independent Schrodinger equation for the calculation of the eigenvalues. The authors are used the following potentials the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential. From the results the authors proved the efficiency of the new developed methods. 

For illustration, in Figure 4.5.58 impedance spectra measured over the whole performance range of a PEFC from open circuit to full load were presented. The nature of the impedance spectra varied with cell potential, between open circuit potential (1024 mV) and 840 mV an exponential potential dependency of the impedance due to the high activation energy of the oxygen reduction reaction is observed in the low frequency range (/< 1 Hz). 

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