Cubic potential

Fig. 8. Arrhenius plot of dissipative tunneling rate in a cubic potential with Vq = Sficoo and r jlto = 0, 0.25 and 0.5 for curves 1-3, respectively. The cross-over temperatures are indicated by asterisks. The dashed line shows k(T) for the parabolic barrier with the same CO and Va-...

Figure 17. Iteration process of the calculation of instanton trajectory in the cubic potential for N = 2 and Ci = 0.6 in Eq. (98). The parameter A iter is the number of iteration. Taken from Ref. [31],...

The only nonvanishing matrix elements of x3 are those with j = n 1 and j = n 3. This result is obtained by repeated application of Eq. (40), as before. Thus, there are four terms that the cubic potential constant contributes to the second-order energy correction, Eq, (35). The final result can be written as... 

According to Bertie and Millen 19) the appearance of the (Vj + ve) tends is due to the cubic potential constant kU3 or, according to the notations used here, kUo. The coupling constants for the hot bands are equal to their spacing (see Sect. 2.1) that is 4.7 — 1.7 cm-1 for the different complexes for v2 and 7.6 — 3.0 for vb. These may seem to be small but they suffice to cause the fine structure and the breadth of the observed vibrational bands ... 

This coupling term kie(xi — Xe)2 is a phenomenological approximation of the lowest-order (cubic) potential energy terms that gives rise to nonlinear effects. These effects can be described by perturbative expansions of x (7,30,32,33) ... 

Quadratic and cubic potential constants have been obtained from IR frequencies, isotopic shifts, inertial defects, Coriolis constants and centrifugal distortion, assuming the geometry from microwave data. The quadratic force field is characterized by four symmetry force constants F which are related to the inner force constants by the following equations... 

In order to calculate Go, one needs to know the potential parameters k, a, and b. These should be obtainable from spectroscopically observed quantities. Thus, k can be obtained from observed Teller has shown that the rotational-vibrational constant ae (in cm. ) can be expressed in terms of the cubic potential constant a and other parameters in the following manner [see Reference 6]. 

Fig. 1.7. Potential energy curves for a diatomic molecule actual potential (solid line), parabolic potential (dashed line), and cubic potential (dotted line).

The Eu coordination is octahedral in CaS and eightfold cubic in Cap2. This difference in coordination leads to a reversal in sign of the fourth-order and some higher terms in the cubic potential. The opposing trends observed can then be interpreted by a sr-electron donation to the otherwise vacant 5d-orbitals of the europium [48]. 

A planar dividing surface might seem to lead to divergences in the case of a cubic potential of mean force. This question has been dealt with at length in Ref. 79. By introducing a kink into the planar dividing surface one can remove the divergence. In... 

The vibration-rotation constants af are complicated functions of the harmonic (quadratic) and an-harmonic (mainly cubic) potential constants [9] and depend on the masses of the component atoms. Since a rotational constant is inversely proportional to a moment of inertia, a/ does not simply represent averaged vibrational contributions. It has, however, been proven [10] that the rotational constant corrected for the harmonic part of af gives the moment of inertia which corresponds to the real vibrational average. The corrected rotational constant is often denoted as B, i. e.. 

For numerical convenience. Lea, Leask and Wolf (LLW) (1962) wrote the cubic potential... 

Diffraction data for the difluorides of manganese, iron, cobalt, nickel and zinc, all of which are linear, have been reanalysed to give equilibrium distances, corresponding to the minima of the cubic potential functions in curvilinear coordinates, and including corrections for dynamic anharmonicity effects and for centrifugal distortion. The equilibrium distances are Mn-F 179.7(6), Fe F 175.5(6), Co-F 173.8(6), Ni F 171.5(7) and Zn-F 172.9(7) pm. Data for copper difluoride are not of the same quality, because there is a large uncertainty in the composition of the vapour (which also contains CuF), but a Cu-F equilibrium distance of 170.0(174) pm has been estimated. 

Fig.6. Classification of cubic potentials according to the roots of the help functions f and g. The potentials corresponding to the points indicated by +,A,o,, are shown in Figs. 7-12.

To demonstrate the applicability of the theory, we consider the particle of mass m in the A-dimensional cubic potential... 

FIGURE 8.2 Iteration process for the instanton trajectory in the separable 20-dimensional cubic potential model. The calculation is carried out within the framework of the nonUnearly transformed coordinates [see Equations (8.54) and (8.55)]. Mier is the number of iterations and 0 is the classical action. Its exact value is So = 3.233859. The inset shows the enlarged scale of the converged trajectory after 17 iterations. (Taken from Reference [44] with permission.)... 

FIGURE 8.3 Behavior of the integrand I z) in the case of cubic potential. Solid squares represent the numerical results obtained by solving the matrix equation, Equation (8.46). Solid line is the exact/(z). (Taken from Reference [44] with permission.)... 

The first term is a harmonic element. For as < 1, the typical situation, the terms diminish in size beyond the harmonic term. Even for as 1, the terms diminish beyond the cubic term. The cubic and quartic terms are the leading sources of anharmonicity near the equilibrium of a Morse oscillator, and it is very often the case that a cubic potential element is the most important contributor to anharmonicity effects in real molecules. 

Infrared measurements yield the vibrational frequencies associated with the various normal vibrational modes, and these data, including isotopic frequency data, can be used to evaluate the force constant matrix F = Since these calculations are often iU conditioned and also since there are usually more force constants than vibrational frequency data, both infrared data and the microwave distortion constant data are often combined to help characterize the force constant matrix. Some examples are given in Table XIV. It may be noted that the F or sextic distortion constants depend on the cubic potential constants, and these data have been employed to obtain information on these anharmonic potential constants. 

To define the rotation-vibration constants in terms of more fundamental parameters or to understand the origin of various nonrigidity effects in the rotational spectrum, the general rotation-vibration Hamiltonian must be employed. This Hamiltonian contains pure rotation and vibration terms as well as interaction terms between rotation and vibration. Perturbation treatments to various orders are required to characterize the different rotation-vibration effects. Space does not permit further discussion of this however, we mention that such a perturbation treatment shows that the a constants depend on the cubic potential energy constants of the molecule. 

In general, the anbarmonic contributions to intensities are small. The cubic d le and mixed cubic potential and quadratic dipole terms have more significant contribution. In the case of H2CO the picture is complicated fir>m the presence of Fermi resonances near some fundamentals. Thus, large terms ear in tiie anbarmonic correction terms. The authors conclude that the prediction of overiapping resonances is a particularly difficult task in intensity analysis. 

Mixed (from P2 and cubic potential) 2.72x10-3 3 2.40x10-3... 

Mechanical (from cubic potential) -5.05x10-4 3 -1.90x10-4... 

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