Quartic potential

Bond stretching is most often described by a harmonic oscillator equation. It is sometimes described by a Morse potential. In rare cases, bond stretching will be described by a Leonard-Jones or quartic potential. Cubic equations have been used for describing bond stretching, but suffer from becoming completely repulsive once the bond has been stretched past a certain point. 

It is intuitively obvious that this phenomenon should also exist in systems having steady states (e.g., in a system described by quartic potential that has been intensively studied in the context of stochastic resonance), but it is more natural to investigate the resonant properties of signal-to-noise ratio (SNR) in those cases [113]. 

A useful trial variational function is the eigenfunction of the operator L for the parabolic barrier which has the form of an error function. The variational parameters are the location of the barrier top and the barrier frequency. The parabolic barrierpotential corresponds to an infinite barrier height. The derivation of finite barrier corrections for cubic and quartic potentials may be found in Refs. 44,45,100. Finite barrier corrections for two dimensional systems have been derived with the aid of the Rayleigh quotient in Ref 101. Thus far though, the... 

Figure 2 Position time correlation functions for the quartic potential at two different temperatures o/P = 1 and P = 8. Shown are the exact (dots), CMD (solid line), and classical MD (dashed line) results.

Crowell discovered a variety of effects numerically, including modified Rabi flopping, which has an inverse frequency dependence similar to that observed in the solid state in reciprocal noise [73]. The latter is also explained by Crowell [17] using a non-Abelian model. A variety of other effects of RFR on the quantum electrodynamical level was also reported numerically [17]. The overall result is that the occurrence, classically, of the B V> field means that there is a quantum electrodynamical Hamiltonian generated by the classical term proportional to 3 2. This induces transitional behavior because it contributes to the dynamics of probability amplitudes [17]. The Hamiltonian is a quartic potential where the value of determines the value of the potential. The latter has two minima one where B = 0 and the other for a finite value of the B i) field, corresponding to states that are invariants of the Lagrangian but not of the vacuum. 

V.l.2 Quartic potential. The second model potential studied is given by... 

Recent theoretical treatments of the soft-mode behaviour include a detailed study by Onodera using classical mechanics, and a theory of hydrogen-bond mechanics, including tunnelling effects, by Stamenkovic and Novakovic. ° Onodera assumes a quartic potential function for his individual oscillators, with a bilinear interaction which reduces to c x, where x is the displacement, under the Weiss-molecular-field approximation. The model is soluble without further approximation (in series of elliptic functions), yielding the temperature variation of frequency and damping. If the quartic potential has a central hump larger than kTc,... 

The problem considered is the one-dimensional harmonic oscillator perturbed by cubic and quartic potential terms. Thus, the unperturbed Hamiltonian operator is... 

For large molecules, however, the computer requirements become increasingly prohibitive, especially when conformationally flexible compounds are tackled. Alternative approaches to quantum-mechanical methods are known, based on potential functions and parameters derived from detailed analysis of vibrational spectra. These so-called force field methods are now joined in what is called molecular mechanics, an empirical method that considers the molecule as a collection of spheres (possibly deformable) bound by harmonic forces (eventually corrected with cubic and quartic potentials). The energy... 

In this model, a two-level system is coupled to a classical nonlinear oscillator with mass Mo and phase space coordinates (Pq,Pd)- This coupling is given by hyoRo- The nonlinear oscillator, which has a quartic potential energy function Vn Ro) = aPg/4 — Moco Ro/", is then bilinearly coupled... 

In the case of oxetanone-3, the coefficients in the rotational constant expansions [Eq. (4.2)] were treated as empirical parameters and the potential function was taken from a previous vibrational study10). Figure 2.5 shows the smooth variation, with a definite curvature, of the B rotational constant with ring-puckering vibrational state. Table 4.2 lists the observed and calculated values of the rotational constants. The smooth variation indicates a single-minimum potential with a definite curvature due to the quartic potential term and the quartic terms in the expansion [Eq. (4.3)]. 

Again, this latter effect is of some importance. Referring to Table 4.10 and Eq. (4.7), we see that the vibrational dependence of the quartic term in the effective potential function is quite small, indeed within the quoted uncertainty. For cyclobutane, the reduced quartic potential constant is 26.15 0.07 cm-1 for the ground state and 26.12 0.07 cm-1 for the first excited state of the i>14 mode. On the other hand, the effect on the quadratic term is more noticeable, as expected from Eq. (4.7). For the ground state of p14, it is - 8.87 0.03 cm-1 compared to - 8.76 0.04 cm-1 for the excited state. From these data, we may conclude that the sign of the coefficient of the interaction term Q24Z2 is positive. 

Let us first consider a stationary quartic potential with c 4 for the Cauchy-Levy flight with a = 1 that is, the solution of the equation... 

The nonlinear friction coefficient y(V) thereby takes on the role of a confining potential while for y0 = y(0) the drift term Fy0, as mentioned before, is just the restoring force exerted by the harmonic Omstein-Uhlenbeck potential, the next higher-order contribution y2V3 corresponds to a quartic potential, and so forth. The fractional operator 0a/0 V a in Eq. (125) for the velocity coordinate for 1 < a < 2 is explicitly given by [20,64]... 

First we consider the case where b is so large that the correlation between the two protons is negligible. Then we can write the transfer potential as the sum of two double minimum potentials, which we represent by quartic potentials... 

The one-dimensional potential along the tunneling coordinate, represented by t/c(x) in Eq. (29.20), is a crude-adiabatic potential evaluated with the heavy atoms fixed in the equilibrium configuration, i.e. with y = Ay ,y5 = Ay it is equivalent to the potential along the linear reaction path. This symmetric doubleminimum potential has a maximum U (0) = Ug at x = 0, minima U( ( Ax) = 0 at X = +Ax, and a curvature in the minima given by the effective frequency Qq which accounts for the contribution of the normal modes of the minima to the reaction coordinate [27]. Eor the shape of the potential in the intermediate points we use an interpolation formula based on the calculated energies and curvatures near the stationary points. We have found that in many cases the simple quartic potential of the form... 

Figure 3 Schematic of a Morse function and the related harmonic, cubic, and quartic potentials (Eqs. [3] and [4]). When the bond length is increased beyond the point of the minimum, the harmonic potential rises too steeply. The cubic term corrects for the anharmonicity locally, but at longer distances turns and goes catastrophically to negative infinity. The quartic potential remains a good approximation over a relatively large range and is always attractive at large distances.

This approach gives very good results for the quartic anharmonic double well oscillator V x) = —5x + x (i.e. ground and first excited states), and the quartic potential V(x) = x. The first has all real turning points whereas the second has real and pure imaginary turning points. The results of this analysis are given in Tables 8 and 9, respectively (Handy and Brooks (2000)). 

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