Truncated harmonic oscillator potential

We postulate the potential energy curve of a dissociating harmonic oscillator reactant as that shown in Fig. 1. It is a truncated harmonic oscillator potential with a finite number of equally spaced energy levels such that the level N is the last... 

Fig. 1. Potential energy curve for the truncated harmonic oscillator.

The product J Eg)coQ is equal to Eg for a harmonic oscillator potential truncated at = Eg, and to 2Eg for a Morse potential with dissociation energy equal to Eg. Equation (2.41) is the low-friction limit result of Kramers. There are other methods to derive the results obtained in the previous section. One is to look for the eigenvalue with smallest positive real part of the ojjerator L defined so that dP/dt = — LP is the relevant Fokker-Planck or Smoluchowski equation. Under the usual condition of time scale separation this smallest real part is the escapie rate for a single well potential. Another way uses the concept of mean passage time. For the one-dimensional Fokker-Planck equation of the form... 

To begin with, we recall that in certain cases, the algebraic model has been already put in a one-to-one correspondence with a specific potential function for the usual space coordinates. We have already studied dynamic symmetries providing exact solutions for the one-, two-, and three-dimensional truncated harmonic oscillators, the Morse and Poschl-Teller potential functions. When we consider more complicated algebraic expansions in terms of Casimir operators, or when we deal with coupled... 

Despite its simplicity, the model was applied with great success to, e.g., the STM-driven transfer of a xenon atom on a nickel surface. This pioneering experiment was the very first example of an STM-controlled atomic switch, where the xenon atom was moved from the nickel surface to the tungsten STM tip. Figure 6 shows the comparison between theoretical and experimental transfer rates for different values of the ratio The computed transfer rates are inferred from an implicit dynamics between truncated harmonic oscillators, with the initial conditions chosen as the fifth excited vibrational state located on the surface. This corresponds to a situation where above-threshold dynamics dominates. By construction, the rates exhibit the proper power-law dependence with increasing potential bias. It is found that, for this... 

When the theoretical magic numbers are compared with the discontinuities measured in the abundance spectrum (fig. 12.5) or in the ionisation potentials (fig. 12.6), they line up pretty well for Kre, the observed numbers are 2, 8, 18, 20, 58, 92, and this is taken to imply that the original assumption of a spherical short range potential was a reasonable one. In fact, the result is not too critically dependent on the form of the potential. Even a square well is a fair approximation, and one can also adapt a harmonic oscillator well, truncated at an appropriate radius, by adding to it a term proportional to ( , all of which generate very similar orderings of shells and magic numbers. 

Coefficients of the harmonic-oscillator expansion for the ground state and first excitation with / = 0. It includes up to = 8 quanta, but the oscillator strength is determined by minimizing the ground state energy when the expansion is truncated at the N = d level. Quark masses are /n, = l, and the interquark potential J E... 

Tbus the anharmonicity in the vibration is captured by the coefficient x, which adopts values typically less than 1 % of ft>e for bond stretches, but may be up to 5% for those involving hydrogen. It can be shown that any function capable of representing the variation of potential energy with displacement will lead to energy levels given by a power series in (v + 1/2), so the simple harmonic and Morse oscillators are particular cases of this general anharmonic oscillator, with the power series truncated after the first and second terms, respectively. 

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