Quadratic potential

The formulas derived in the time-independent framework can be easily transferred into the corresponding time-dependent solutions. The formulas in the time-independent linear potential model, for example, provide the formulas in the time-dependent quadratic potential model in which the two time-dependent diabatic quadratic potentials are coupled by a constant diabatic coupling [1, 13, 147]. The classically forbidden transitions in the time-independent framework correspond to the diabatically avoided crossing case in the time-dependent framework. One more thing to note is that the nonadiabatic tunneling (NT) type of transition does not show up and only the LZ type appears in the time-dependent problems, since time is unidirectional. 

Here Ho includes not only the quadratic potential term but also some contribution from nonlinear terms a la the Hartree-Fock or mean-field approximation. Introducing a Green function for Ho... 

For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch s theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). 

On the negative side, the exact time dependent centroid Hamiltonian in Eq. (44) is a constant of motion and the CMD method does not satisfy this condition in general except for quadratic potentials. 

Req 19 cm. The quadratic potential R can be expressed in terms of displacements along the bonds,... 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

The thermodynam ic force X that generates the dissipation will be assumed to be a standard friction force. This means that X = hp. For the sake of simplicity, we shall limit ourselves only to small forces and introduce the quadratic dissipation potential 5 = jAX2, where A>0 is a kinetic coefficient. [We can easily consider also non-quadratic potentials as, for example, S = A(exp X + exp(—X) — 2)]. 

Calculated Frequencies. Table II contains the normal-mode vibrational frequencies vu of the light isotopic species, and the frequency shifts A Vi = vii — V2i upon isotopic substitution, calculated with the force fields listed in Table I. The force field for NOa" reproduces the observed frequencies and frequency shifts very well, whereas the calculated frequencies and shifts for N02 differ somewhat from those observed. However, we consider the general quadratic potential used in the calculation the best fit to the observed frequencies. The discrepancy is caused by a disagreement of the observed (2) frequencies with the Teller-Redlich product rule, which is, of course, assumed in the calculations. 

Having found the independent linear combinations of bond vectors [see Eqs. (2.1.17), (2.1.27), and (2.1.28)], through the central-limit theorem it is easy to construct the Gaussian joint distribution and the associated quadratic potential. Adopting for simplicity the periodic-chain transform, we have... 

All three curves are anharmonic potential wells with energy minima corresponding to the equilibrium bond angles. A quadratic potential with a Lorentzian hump, Vu-o-uip), was fitted to the calculated data (Eq. 1). 

The one-dimensional quadratic potential V = kx2 has been used for the description of covalent binding. The ground-state wave functions for a simple harmonic oscillator, /t and iR, have been used to describe the proton in the left and right wells. The force constant k has been determined from the stretch-mode vibrational transitions for water occurring at 3700 cm-1. The ground-state energy for the proton is 0.368 x 10-19 J. The tunneling barrier is AE = 4 x 10-19 J. 

Symmetric single minimum quartic-quadratic potential function... 

In contrast to the case of cyclobutanone, the addition of two more adjustable parameters does not seem warranted in the case of silacyclobutane in that only a small improvement in the fit results. The barrier determined is 442 cm-1, within 2 cm"1 of the barrier determined from the simpler quartic-quadratic potential function. As pointed out by Pringle, the tendency is to weight the microwave data heavily because of the precision of the rotational data compared to that of the measurement of the vibrational intervals in the far-infrared or Raman spectrum. However, in doing so, one fails to recognize the limitations of the Hamiltonian. If the potential func-... 

Each monomer is constrained to stay fairly close to the primitive path, but fluctuations driven by the thermal energy kT are allowed. Strand excursions in the quadratic potential are not likely to have free energies much more than kT above the minimum. Strand excursions that have free energy kT above the minimum at the primitive path define the width of the confining tube, called the tube diameter a (Fig. 7.10). In the classical affine -and phantom network models, the amplitude of the fluctuations of a... 

A chain or network strand (thick curve) is topologically constrained to a tube-like region by surrounding chains. The primitive path is shown as the dashed curve. The roughly quadratic potential defining the tube is also sketched. 

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