Potential energy parabolic

One-Particle Model with Parabolic Potential-Energy Wells... 

Figure 7.2 Parabolic potential-energy well for one-diinensional particle jumps. Unlike...

Fig. 2. The model of an electron transfer reaction as the intersection of two parabolic potential energy wells. The abcissa x represents a reaction coordinate such a diagram is only one two-dimensional section through a multidimensional space...

It was recently shown (Ratner and Levine, 1980) that the Marcus cross-relation (62) can be derived rigorously for the case that / = 1 by a thermodynamic treatment without postulating any microscopic model of the activation process. The only assumptions made were (1) the activation process for each species is independent of its reaction partner, and (2) the activated states of the participating species (A, [A-], B and [B ]+) are the same for the self-exchange reactions and for the cross reaction. Note that the following assumptions need not be made (3) applicability of the Franck-Condon principle, (4) validity of the transition-state theory, (5) parabolic potential energy curves, (6) solvent as a dielectric continuum and (7) electron transfer is... 

The density of reactive states p°( ) was fit by a sum of terms KTpT( ), as given in Eqs. (14) and (15), appropriate to scattering by parabolic potential energy barriers. (Note that the use of the parabolic barrier is the simplest barrier shape for understanding p°( )... 

Here, we provide formulas that will enable the calculation of the Condon locus in terms of molecular constants for parabolic potential energy fiinctions. Figure 8.1 shows schematically the parabolic energy curves of two simple harmonic oscillators and their discrete vibrational energy levels. 

A parabolic potential energy well (harmonic oscillator) reduces this tendency, and the energy levels are equidistant. The distance decreases if the parabola gets wider (less restrictive). 

Fig. 12.7. Sadlej relation. The electric field mainly causes a shift of the electronic charge distribution toward the anode (a). A GTO represents the eigenfunction of a harmonic oscillator. Suppose that an electron oscillates in a parabolic potential energy well (with the force constant I ). In this situation, a homogeneous electric field corresponds to the perturbation x, that conserres the hannonicity with unchanged force constant k (b).

Indeed, inserting jc = x + j leads to d/dx = d/dx and d /dx = d /dx which gives a similar Schrddinger equation except that the harmonic potential is shifted. Therefore, the solution to the equation can be written as simply a zero-field solution = ijf x + j) shifted by — This is quite understandable because the operation only means adding to the parabolic potential energy kx /2 a term proportional to x i.e., a parabola potential again (though it is a displaced one see Fig. 12.7b). 

The wavefunctions and associated energies are shown in Figure 4.5, together with the parabolic potential energy curve. Each wavefunction has been given a baseline which corresponds to the total energy of the particle. 

Thus, in Figure 4.5 the maximum classical extension of the bond occurs at the point where the horizontal, total energy line crosses the parabolic potential energy curve. 

Representation of a reaction coordinate diagram as the sum of two parabolic potential energy surfaces for a thermoneutral reaction. 

One problem inherent in any Marcus-type equations, based on parabolic potential energy curves, lies in their limiting behaviour at very high endo- or exo-thermicities for very endothermic reactions the theory leads to AG =AG and for very high exothermic processes, which avoid the so called "inverted region", there is a cut-off AG =0. Both forms of limiting behaviour have been criticized as being physically unrealistic [24,25]. However, this problem does not arises within ISM AGl for very exothermic reactions... 

Morse potential and harmonic (parabolic) potential-energy surfaces for a diatomic molecule. The dissociation energy, Df, represents the energy required to sever the chemical bond. 

FH 9.37 The parabolic potential energy characteristic of a harmonic oscillator. Positive displacements correspond to extension of the spring negative displacements correspond to compression of the spring. 

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