The anharmonic oscillator

The true potential energy curve can be determined from experiment if sufficient data about the vibrational levels are obtained we will describe how this is accomplished later in this chapter. As we mentioned earlier, improvements over the Morse potential have been described, a particularly important one being due to Hulburt and Hirschfelder [64]  

Here x = R - Re and b and c are constants which depend upon both vibrational and rotational constants, as we will describe later. This form of the potential works very well for a large number of molecules and electronic states. 

Most of our discussion of vibrational eigenstates and selection rules has been centered on the harmonic approximation. Of course, the effective vibrational potential Uj R) is not well approximated by a parabola for energies corresponding to large vibrational quantum numbers v (cf. Fig. 3.7), and considerable work has been expended to find alternative expressions for either l7kk(K) or the vibrational energy levels which are both compact and accurate. It has become conventional to fit experimentally determined vibrational levels Gp(cm ) to expressions of the form [6] 

If the vibrational energy level expression (3.81) is cut off after its leading anharmonic term 

This is known as the Morse potential. At R = Rg, the first and second derivatives of the Morse potential are [7] 

Marion, Classical Dynamics of Particles and Systems, Academic, New York, 1965. 

Nonadiabatic processes in molecular collisions. In W. H. Miller (Ed.), Dynamics of Molecular Collisions, Part B, Plenum, New York, 1976. 

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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... 

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. 

Energy levels in the anharmonic oscillator are not equal, although they become slightly closer as energy increases. This phenomenon can be seen in the following equation , ... 

For deriving an expression for the frequency-time correlation function the formulation of Oxtoby is followed. If V is the anharmonic oscillator-medium interaction, then by expanding V in the vibrational coordinate Q using Taylor s series we obtain... 

In the estimation of Acon(t), only the first two terms are considered, neglecting the higher-order terms. (Q - Goo) and (Q m - Goo) 810 die quantum mechanical expectation values of the anharmonic oscillator. They can be calculated using perturbation theory and is given by... 

The anharmonic oscillator - molecular orbital theory connection... 

If there was no interaction between vibration and rotation, the energy levels would be given by the simple sum of the expression giving the vibrational levels for the anharmonic oscillator, equation (6.188), and that describing the rotational levels of the rigid rotor, equation (6.162). There is an interaction, however during a vibration the moment of inertia of the molecule changes, and therefore so also does the rotational constant. We may therefore use a mean value of Bv for the rotational constant of the vibrational level considered, i.e. 

In this case, the zeroth-order Hamiltonian is chosen to represent the vibrational energy of the anharmonic oscillator ... 

The first of these two terms is zero, since the wavefunctions i . and j are defined as orthogonal hence vibrations are only infrared active when Q 7 0. If the harmonic model is assumed, the transition moment is only nonzero for transitions where An = 1 although this restriction is lifted for the anharmonic oscillator, transitions where An = 2, 3, etc., are still much weaker than An = 1. 

Problem 40-3. Using first-order perturbation theory, find perturbed wave functions for the anharmonic oscillator with V = + ax2,... 

These attempts have considered simple problems like the anharmonic oscillator problem in a harmonic oscillator basis or the hydrogen atom in a Gaussian-type basis, generally with rather poor results. The reason for these difficulties is rather clear. Given N basis functions i>, there are basis operators iX7 - Consequently, the equations for the eigenvalues and eigenvectors of L represent equations of rank as compared with... 

Figure Bl.2.3. Comparison of the harmonic oscillator potential energy curve and energy levels (dashed lines) with those for an anharmonic oscillator. The harmonic oscillator is a fair representation of the true potential energy curve at the bottom of the well. Note that the energy levels become closer together with increasing vibrational energy for the anharmonic oscillator. The anharmonicity has been greatly exaggerated.

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... 

Figure Bl.5.3 Magnitude of the second-order nonlinear susceptibility x versus frequency co, obtained from the anharmonic oscillator model, in the vicinity of the single- and two-photon resonances at frequencies coq and cOq/2, respectively.

Case 2. Av 1. For the anharmonic oscillator, the selection rule requiring that Ai = 1 is no longer a rigid requirement. There is a small probability of transitions with Av = 2 and an even smaller probability of transitions with Av = 3. If we insert these conditions in Eq. (25.19), that is, v = 2, v = 0, we can develop expressions analogous to Eqs. (25.21) and (25.22) for the R and P branches of a band centered on the first overtone frequency, 2vq(1 — 3Xg), or approximately twice the fundamental vibrational frequency. This overtone band is much weaker than the fundamental band. If i = 3,v = 0, there is a second overtone band that is much weaker than the first overtone band. The requirement, AJ = 1, still applies. 

This oscillator has a fixed number of states given by l/(a — 1). Suppose that the molecule has j, independent anharmonic oscillators, and S2 harmonic oscillators. We first generate a two-dimensional array, K(7, J) for the anharmonic oscillators, in which I identifies the particular oscillator in question, and J represents the energy levels of the / oscillator. The index I will thus run from 1 to Sj, while J will run from 0 to l/(ai — 1). The anharmonic density of states is generated with program steps 16 through 23, the output of which is then used as the initial vector for the harmonic part. 

Note added in proof In view of the failure of the harmonic oscillator model to account for the observed rate of activation in unimolecular dissociation reactions (the dissociation lag problem) these calculations have been repeated for a Morse anharmonic oscillator with transition between nearest and next-nearest neighbor levels [S. K. Kim, /. Chem. Phys. (to be published)]. The numerical evaluation of the analytical results obtained by Kim has not yet been carried out. From the results obtained by us and our co-workers [Barley, Montroll, Rubin, and Shuler, /. Chem. Phys. in press)] on the relaxation of vibrational nonequilibrium distributions of a system of Morse anharmonic oscillators it seems clear, however, that the anharmonic oscillator model with weak interactions (i.e., adiabatic perturbation type matrix elements) does not constitute much of an improvement on the harmonic oscillator model in giving the observed rates of activation. The answer to tliis problem would seem to lie in a recalculation of the collisional matrix elements for translational-vibrational energy exchange which takes account of the strong interactions in highly energetic collisions which can lead to direct dissociation. 

NIRS is based on absorption of radiation between polar bonds in molecules, and in particular the bonds between light atoms, such as those in the first period of the periodic table. The molecular bonds vibrate in a manner similar to a diatomic oscillator, and the simplest model explaining these absorptions is that of the harmonic oscillator. However, the harmonic oscillator does not explain overtone transitions, and the model of the anharmonic oscillator is more precise (Fig. 1 Bokobza, 1998). 

Infrared spectroscopy is based on the interaction of electromagnetic radiation with a molecular system, in most cases in the form of absorption of energy from the incident beam. The absorption of infrared light induces transitions between the vibrational energy levels given by Eq. (4.7). As shown in Fig. 4.2, the energy levels of the anharmonic oscillator are not equidistant. 

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