Anharmonic oscillator motion

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... 

Figure 4. Still shots from movies of the motion of an anharmonic oscillator with a total energy approximately 50% of the dissociation energy. The system starts at the inner turning point and travels to the outer turning point. The students make animations of these plots to compare the observed motion with the calculations plotted in Figures 2.

This rate of energy exchange between an oscillator and the thermal environment was the focus of Chapter 13, where we have used a quantum harmonic oscillator model for the well motion. In the y -> 0 limit of the Kramers model we are dealing with energy relaxation of a classical anharmonic oscillator. One may justifiably question the use of Markovian classical dynamics in this part of the problem, and we will come to this issue later. For now we focus on the solution of the mathematical problem posed by the low friction limit of the Kramers problem. 

The phase space of a coupled, two-identical-anharmonic oscillator system is four-dimensional. Conservation of energy and polyad number reduces the number of independent variables from four to two. At specified values of E and N = vr + vl = vs+ v0 (in classical mechanics, N need no longer be restricted to integer values nor E to eigenenergies), accessible phase space divides into several distinct regions of regular, qualitatively describable motions and (for more general dynamical systems) chaotic, indescribable motions. Systematic variation of E and N reveals bifurcations in the number of forms of these describable motions. Examination of the classical mechanical form of the polyad Heff often reveals the locations and causes of such bifurcations. 

In order to illustrate some of the basie aspeets of the nonlinear optieal response of materials, we first discuss the anharmonic oscillator model. This treatment may be viewed as the extension of the classical Lorentz model of the response of an atom or molecule to include nonlinear effects. In such models, the medium is treated as a collection of electrons bound about ion cores. Under the influence of the electric field associated with an optical wave, the ion cores move in the direction of the applied field, while the electrons are displaced in the opposite direction. These motions induce an oscillating dipole moment, which then couples back to the radiation fields. Since the ions are significantly more massive than the electrons, their motion is of secondary importance for optical frequencies and is neglected. 

While the Lorentz model only allows for a restoring force that is linear in the displacement of an electron from its equilibrium position, the anharmonic oscillator model includes the more general case of a force that varies in a nonlinear fashion with displacement. This is relevant when the displacement of the electron becomes significant under strong driving fields, the regime of nonlinear optics. Treating this problem in one dimension, we may write an appropriate classical equation of motion for the displacement, x, of the electron from equilibrium as... 

So far we have used the models of the rigid rotor, and the harmonic or anharmonic oscillator to describe the internal dynamics of the diatomic molecnle. Since the period for rotational motion is of the order 10 " s, and that for vibrational motion is 10 s, the... 

In our discussion so far, we have assumed that the motions of atoms in a vibrating molecule are harmonic. Although making this assumption made the mathematics easier, it is not a realistic view of the motion of atoms in a real vibrating molecule. Anharmonic motion is the type of motion that really takes place in vibrating molecules. The energy levels of such an anharmonic oscillator are approximately given by... 

The vibrations of a polyatomic molecule can be considered as a system of coupled anharmonic oscillators. If there are N atomic nuclei in the molecule, there will be a total of 3N degrees of freedom of motion for all the nuclear masses in the molecule. Subtracting the pure translations and rotations of the entire molecule leaves (3N-6) vibrational degrees of freedom for a non-linear molecule and (3N-5) vibrational degrees of freedom for a linear molecule. These internal degrees of freedom correspond to the number of independent normal modes of vibration. Note that in each normal mode of vibration all the atoms of the molecule vibrate with the same frequency and pass through their equilibrium positions simultaneously. 

The key observation is that the higher-order corrections to the energy, in powers of 1/D, arise from anharmonic corrections to the normal mode harmonic oscillator motion. Now a given anharmonic correction to the energy, as we all learned long ago when we studied quantum mechanics, can be computed exactly from a finite number of excited harmonic oscillator functions. This means that a truncated basis which contains properly scaled harmonic oscillator functions can be used to compute exactly a finite number of anharmonic corrections. One simply pre-determines to which order one wants to compute the anharmonic corrections, calculates how many excited... 

So far we have considered the potential energy W of our system only, and found it to have a minimum at P = 0 and at P = Pq if P is the value of P at which the metastable state occurs. The system can of course oscillate harmonically if the displacement from P = 0 or P = Pq is small. More generally, however, the system will carry out anharmonic oscillations. In fact it is shown in Ref. 8 that to a good approximation P satisfies the equation of motion... 

In the standard approximation, an anharmonic oscillator model is used to describe the vibrational motion. The vibrational amplitude and energy are largely dictated by the potential energy. From a descriptive point of view the simplest form of a potential is a so-called Morse potential, which requires only three mathematical parameters ... 

As already stated, the Morse potential is our first example of a potential surface that describes a particular motion. The bond vibrates within the constraints imposed by this potential. One may ask, "At any given moment, what is the probability of having a particular bond length " This is similar to questions related to the probability of finding electrons at particular coordinates in space, which we will show in Chapter 14 is related to the square of the wave-function that describes the electron motion. The exact same procedure is used for bond vibrations. We square the wavefunction that describes the wave-like nature of the bond vibration. Let s explore this using the potential surface for a harmonic oscillator (such as with a normal spring), instead of an anharmonic oscillator (Morse potential). For the low energy vibrational states, the harmonic oscillator nicely mimics the anharmonic oscillator. 

The local mode description with only one bond such as X-H being in motion at a time leads us to consider the polyatomic molecule as a single diatomic molecule M-H where H is hydrogen and M denotes the full rest of the molecule. The main advantage of this description is the dramatic reduction of the number of coupling constants. The anharmonic oscillator potential adopted in the local mode model is the Morse potential, which takes account of the dissociation energy De (53). 

The Ml manifestation of chaotic behavior requires that at least Mee equations of motion are coupled. This is not a strong requirement for a mechanical system. Each degree of freedom gives rise to two (Hamilton) equations of motion (or one, but second-order, Newton equation). So two coupled (anharmonic) oscillators can already exhibit chaotic behavior. Solving the trajectory for an atom colliding with a diatom requires six equations. If there are only two variables, one can get oscillatory solutions but not chaos. 

These harmonic-oscillator solutions predict evenly spaced energy levels (i.e., no anharmonicity) that persist for all v. It is, of course, known that molecular vibrations display anharmonicity (i.e., the energy levels move closer together as one moves to higher v) and that quantized vibrational motion ceases once the bond dissociation energy is reached. 

This is a check on the reasonableness of the method chosen. For example, it would not be reasonable to select a method to investigate vibrational motions that are very anharmonic with a calculation that uses a harmonic oscillator approximation. To avoid such mistakes, it is important the researcher understand the method s underlying theory. 

According to Bartell (1961a), the relative motion of the interacting non-bonded atoms is described by means of a harmonic oscillator when the two atoms are bonded to the same atom, and by means of two superimposed harmonic oscillators when the atoms are linked to each other via more than one intervening atom. It is the second case which is of interest in connection with the biphenyl inversion transition state. The non-bonded interaction will of course introduce anharmonicity, but since a first-order perturbation calculation of the energy only implies an... 

While the idea of LF explained the H2 data quite well [28], we were surprised by the magnitude of the oscillations in our I2 data [16], as, unlike H2,12 is not vibrationally cold at room temperature - the conditions for our experiment. Generally, thermal motion is detrimental to observing coherent motion. Thus, we took a long time scale run to get a more accurate measurement of the frequency of the vibrations, shown in Fig. 1.5. These data also exhibit a vibrational revival, from which the anharmonicity of the potential well can be determined. Indeed, the vibrational frequency accurately matched that of the ground state. 

The Morse oscillator was discussed by Levine and Wulfman (1979) and by Berrondo and Palma (1980). Levine (1982) is a review of algebraic work on onedimensional anharmonic vibrations. Additional work on one dimensional motion is found in Alhassid, Gtirsey, and Iachello (1983a, 1983b) and Levine (1988). 

The model fundamental to all analyses of vibrational motion requires that the atoms in the system oscillate with small amplitude about some defined set of equilibrium positions. The Hamiltonian describing this motion is customarily taken to be quadratic in the atomic displacements, hence in principle a set of normal modes can be found in terms of these normal modes both the kinetic energy and the potential energy of the system are diagonal. The interaction of the system with electromagnetic radiation, i.e. excitation of specific normal modes of vibration, is then governed by selection rules which depend on features of the microscopic symmetry. It is well known that this model can be worked out in detail for small molecules and for crystalline solids. In some very favorable simple cases the effects of anharmonicity can be accounted for, provided they are not too large. 

Oscillations of atoms and their groups, typically occurring on a time scale of 10 12-10 14 s, are much faster motions. However, there is evidently no sharp distinction between these motions and other slower motions. When damping and anharmonicity arise, the oscillations become diffusive and have the properties of transitions between microstates. It is natural to suppose (as... 

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