Vibrations of diatomic molecules

We shall see in Section 13.1 that to an excellent approximation one can treat separately the motions of the electrons and the motions of the nnclei of a molecnle. (This is due to the much heavier mass of the nuclei.) One first imagines the nnclei to be held stationary and solves a Schrddinger equation for the electronic energy U. U also includes the energy of nuclear repulsion.) For a diatomic (two-atom) molecule, the electronic energy U depends on the distance R between the nuclei, U = U R), and the U versus R curve has the typical appearance of Fig. 13.1. 

The internal motion of a diatomic molecule consists of vibration, corresponding to a change in the distance R between the two nuclei, and rotation, corresponding to a change in the spatial orientation of the line joining the nuclei. To a good approximation, one can usually treat the vibrational and rotational motions separately. The rotational energy levels are found in Section 6.4. Here we consider the vibrational levels. 

FIGURE 4.6 Potential energy for vibration of a diatomic molecule (solid curve) and fora harmonic oscillator (dashed curve). Also shown are the bound-state vibrational energy levels for the diatomic molecule. In contrast to the harmonic oscillator, a diatomic molecule has only a finite number of bound vibrational levels. 

The harmonic-oscillator force constant k in Eq. (4.26) is obtained as k = (fiV/dx, and the harmonic-oscillator curve essentially coincides with the U R) curve at / = Rg, so the molecular force constant is k = d U/dR i =K (see also Prob. 4.28). Differences in nuclear mass have virtually no effect on the electronic-energy curve U R), so different isotopic species of the same molecule have essentially the same force constant k. 

We expect, therefore, that a reasonable approximation to the vibrational energy levels of a diatomic molecule would be the harmonic-oscillator vibrational energy levels Eqs. (4.45) and (4.23) give 

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FIGURE 3.2 (a) Vibration of diatomic molecule, HC1, (b) potential energy of an ideal harmonic oscillator, and (c) an anharmonic oscillator described by the Morse function. 

The vibrations of diatomic molecules being approximately simple harmonic, the quantum law takes the second... 

The Vibration of Diatomic Molecules.—In addition to their rotation, we have seen that diatomic molecules can vibrate with simple harmonic motion if the amplitude is small enough. We shall use only this approximation of small amplitude, and our first stop will be to calculate the frequency of vibration. To do this, we must first find the linear restoring force when the interatomic distance is displaced slightly from its equilibrium value / ,. We can get this from Eq. (1.2) by expanding the force in Taylor s series in (r — rt). We have... 

Potential energy diagrams for diatomic molecules were introduced in Section 3.5, and you can see that they are not parabolic over the entire region 0 < r < 00 (for example, see Fig. 3.9). Near the equilibrium internuclear separation the potential appears to be well approximated by a parabola. This similarity suggests that the harmonic oscillator should be a good model to describe the vibrations of diatomic molecules. The dependence of the vibrational frequency v on the force constant k and the mass has the same form as Equation 4.44, but now the mass is the reduced mass /t of the two nuclei... 

In inharmonic approximation of the vibration of diatomic molecules of the selection rule, relating to the variation in Hvib allowed for the quantum number, it is not so strict as in the case described harmonics. The model does not exclude the possibility inharmonic transitions between the status of vibration to which variation nvib quantum number to be 2,3, etc., in practice IR spectrophotometry. 

W. Holzer, Y. Le Duff, The depolarization ratio of the Raman bands of the vibration of diatomic molecules. iaAdv. Raman Specrosc, vol. 1 (Heyden, London, 1973), pp. 109-112... 

If electrons and other particles act like waves, they should obey a wave equation. In 1926, Erwin Schrodinger published a series of four articles containing a wave equation for de BrogUe waves, which we now call the Schrodinger equation. The first three articles presented the time-independent version of the wave equation and applied it to the hydrogen atom, rotation and vibration of diatomic molecules, and the effect of an external electric field on energy levels. The time-dependent version of the equation was reported in the fourth article. ... 

The vibration of diatomic molecules may be related to the motion of two masses joined by a spring of negligible mass Tq in length and a force constant of k, such as a simple harmonic oscillator. A small displacement of the particle from its equilibrium position will require a restoring force that obeys Hooke s law ... 

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