Anharmonic oscillators

Treanor C E, Rich J W and Rehm R G 1968 Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions J. Chem. Phys. 48 1798-807... 

Marquardt R and Quack M 1989 Infrared-multlphoton excitation and wave packet motion of the harmonic and anharmonic oscillators exact solutions and quasiresonant approximation J. Chem. Phys. 90 6320-7... 

CHEOPS is based on the method of atomic constants, which uses atom contributions and an anharmonic oscillator model. Unlike other similar programs, this allows the prediction of polymer network and copolymer properties. A list of 39 properties could be computed. These include permeability, solubility, thermodynamic, microscopic, physical and optical properties. It also predicts the temperature dependence of some of the properties. The program supports common organic functionality as well as halides. As, B, P, Pb, S, Si, and Sn. Files can be saved with individual structures or a database of structures. 

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

Figure 6.4 Potential energy curve and energy levels for a diatomic molecule behaving as an anharmonic oscillator compared with those for a harmonic oscillator (dashed curve)...

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared wifh fhose of a harmonic oscillator. Figure 6.6 shows some wave functions and probabilify densify functions (iA A ) for an anharmonic oscillator. The asymmefry in and (iA A ) 5 compared wifh fhe harmonic oscillator wave functions in Figure f.i3, increases fheir magnitude on the shallow side of the potential curve compared with the steep side. 

The vibrational term values for a polyatomic anharmonic oscillator with only nondegenerate vibrations are modified from the harmonic oscillator values of Equation (6.41) to... 

For an anharmonic oscillator with degenerate vibrations the term values are modified from those of Equation (6.88) to... 

Except for the nonlocal last term in the exponent, this expression is recognized as the average of the one-dimensional quantum partition function over the static configurations of the bath. This formula without the last term has been used by Dakhnovskii and Nefedova [1991] to handle a bath of classical anharmonic oscillators. The integral over q was evaluated with the method of steepest descents leading to the most favorable bath configuration. 

One of the more interesting results of these calculations is the contribution to the heat capacity. Figure 10.10 shows the temperature dependence of this contribution to the heat capacity for CH3-CCU as calculated from Pitzer s tabulation with 7r = 5.25 x 10-47 kg m2 and VQ/R — 1493 K. The heat capacity increases initially, reaches a maximum near the value expected for an anharmonic oscillator, but then decreases asymptotically to the value of / expected for a free rotator as kT increases above Vo. The total entropy calculated for this molecule at 286.53 K is 318.86 J K l-mol l, which compares very favorably with the value of 318.94T 0.6 TK-1-mol 1 calculated from Third Law measurements.7... 

Table A4.5 Anharmonic oscillator and nonrigid rotator corrections...

The Warner function has all the desired asymptotical characteristics, i.e. a linear dependence of f(r) on r at small deformation and a finite length Nlp in the limit of infinite force (Fig. 3). In a non-deterministic flow such as a turbulent flow, it was found useful to model f(r) with an anharmonic oscillator law which permits us to account for the deviation of f(r) from linearity in the intermediate range of chain deformation [34] ... 

It was shown above that the cubic term in the potential function for the anharmonic oscillator cannot, for reasons of symmetry, contribute to a first-order perturbation. However, if the matrix elements of = ax3 are evaluated, it is found that this term results in a second-order correction to the... 

This represents a truncated anharmonic oscillator with anharmonicity controlled... 

Figure 2.2 Spectrum of states of the one-dimensional anharmonic oscillator, N = 6.

One can see that these represent the eigenvalues of two local anharmonic oscillators. The spectrum of Eq. (4.21) when the two oscillators are identical, as in H20, where they represent the stretching of the O-H bonds, is shown in Figure 4.1. 

Figure 4.1 Spectrum of two coupled local anharmonic oscillators. Note the inherent degeneracies in the spectrum.

Figure 4.2 Spectrum of two normal coupled anharmonic oscillators. Note how the different levels are almost equispaced.

Figure 6.2 Degeneracies of states for six uncoupled anharmonic oscillators.

Iachello, F., and Oss, S. (1991a), Model of n Coupled Anharmonic Oscillators and Applications to Octahedral Molecules, Phys. Rev. Lett. 66, 2776. 

Kellman, M. E., and Lynch, E. D. (1986), Algebraic Resonance Quantization of Coupled Anharmonic Oscillators, J. Chem. Phys. 85, 5855. 

Lehmann, K. K. (1983), On the Relation of Child and Lawton s Harmonically Coupled Anharmonic-Oscillator Model and Darling-Dennison Coupling, J. Chem. Phys. 79, 1098. 

Recamier, J., and Ortega, J. (1991), Transition Probabilities in Collisions Between an Atom and an Anharmonic Oscillator, Mol. Phys. 73, 635. 

Wu, G. (1991), The Semiclassical Fixed Point Structure of Three Coupled Anharmonic Oscillators Under SU(3) Algebra with Iz = 0, Chem. Phys. Letts. 179, 29. 

Even if one restricts one s attention to vibrations and rotations of molecules, there are a variety of Lie algebras one can use. In some applications, the algebras associated with the harmonic oscillator are used. We mention these briefly in Chapter 1. We prefer, however, even in zeroth order to use algebras associated with anharmonic oscillators. Since an understanding of the algebraic methods requires a comparison with more traditional methods, we present in several parts of the book a direct comparison with both the Dunham expansion and the solution of the Schrodinger equation. 

The interest in efficient optical frequency doubling has stimulated a search for new nonlinear materials. Kurtz 316) has reported a systematic approach for finding nonlinear crystalline solids, based on the use of the anharmonic oscillator model in conjunction with Miller s rule to estimate the SHG and electro optic coefficients of a material. This empirical rule states that the ratio of the nonlinear optical susceptibility to the product of the linear susceptibilities is a parameter which is nearly constant for a wide variety of inorganic solids. Using this empirical fact, one can arrive at an expression for the nonlinear coefficients that involves only the linear susceptibilities and known material constants. 

The Impedance Z can Increase to very high values. If this happens, the oscillator prefers to oscillate In resonance with an anharmonic frequency. Sometimes this condition Is met for only a short time and the oscillator oscillation jumps back and forth between a basic and an anharmonic oscillation or It remains as an anharmonic oscillation. This phenomenon Is well known as "mode hopping". In addition to the noise of the rate signal created, this may also lead to Incorrect termination of a coating because of the phase jump. It Is Important here that, nevertheless, the controller frequently continues to work under these conditions. Whether this has occurred can only be ascertained by noting that the coating thickness Is... 

To obtain the allowed energy levels, Ev, for a real diatomic molecule, known as an anharmonic oscillator, one substitutes the potential energy function describing the curve in Fig. 3.2c into the Schrodinger equation the allowed energy levels are... 

When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are Av = +1 that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to Av = +2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and... 

---