The vibrations of diatomic molecules

We need to know how to treat the vibrations of diatomic moiecuies quantitativeiy because the vibrationai characteristics of even the iargest bioiogicai moiecuies can be understood in terms of the harmonic motion of each atom reiative to its neighbors. 

We base our discussion on Fig. 12.14, which shows a typical potential energy curve of a diatomic molecule as its bond is lengthened by pulling one atom away from the other or pressing it into the other. In regions close to the equilibrium bond length (at the minimum of the curve) we can approximate the potential energy by a parabola (a curve of the form y = x ) and write 

The potential energy in eqn 12.12 has the same form as that for the harmonic oscillator, so we can use the solutions of the Schrodinger equation given in Section 9.6. The only complication is that both atoms joined by the bond move, so the mass of the oscillator has to be interpreted carefully. Detailed calculation shows that for two atoms of masses Ma and joined by a bond offeree constant kf, the energy levels are  

At first sight it might be puzzling that the effective mass appears rather than the total mass of the two atoms. However, the presence of y is physically plausible. If atom A were as heavy as a brick wall, it would not move at all during the 

We have previously warned about the importance of distinguishing between the quantum number V (vee) and the frequency v (nu). 

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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... 

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 ... 

This procedure assumes that the translational, rotational, vibrational, and electronic energy levels are independent. This is not completely so. In the instance of diatomic molecules, we will see how to correct for the interaction. For more complicated molecules we will ignore the correction since it is usually a small effect. 

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. 

To investigate the spectra of diatomic molecules, we need the selection rules for radiative transitions. We now investigate the electric-dipole selection rules for transitions between vibration-rotation levels belonging to the same 2 electronic state. (Transitions in which the electronic state changes will be considered in Chapter 7.)... 

Apart from the influence of coupling, which usually must be taken into account, there is another more basic reason why the frequencies of diatomic molecules are not directly comparable. One has to distinguish between singly and multiply bonded systems. While the vibrational frequency in the N2 molecule lies at 2330 cm, molecules containing N-N single bonds exhibit vibrations as low as 930 cm. ... 

The vibration of a molecule does not affect its center of mass. The vibration is equivalent to that of a single particle of reduced mass /x moving in a field of force constant k. A diatomic molecule AB (such as CO) will vibrate with a frequency given by the equation... 

The application of this result to the determination of bond stretching force constants in molecules encounters two difficulties. First, real molecules are not exactly harmonic oscillators. Secondly, although the only mode of vibration possible in diatomic molecules is a bond stretching motion, the vibrations of polyatomic molecules are much more complicated, and cannot be expressed as consisting only of a combination of bond stretching motions. We discuss these two problems in turn in the next two sections. 

Such a mechanism has been proposed by Slanger and observed for the first time by Bondybey et al. in the case of diatomic molecules inbedded in rare-gas matrices. In their subsequent work, similar effects have been found for collisional processes in the gas phase. The vibrational relaxation of CO excited to the v = 3 and c = 2 levels of the A w state induced by collisions with He is more efficient by 4-5 orders of magnitude than in the ground-state CO + He system.Moreover, the form of the fluorescence decay from the o = 1 level observed under v = 2 excitation cannot be fitted if a direct v = 2 v=l relaxation path is assumed the induction time of the relaxed emission being much longer than the decay of the resonance fluorescence. 

In addition to the bands centered on the fundamental frequencies, other bands appear in the spectra of polyatomic molecules. We have mentioned overtone bands in the spectrum of diatomic molecules due to violation of the selection rule, Ap = +1, that is permitted because of anharmonicity. But in polyatomic molecules, combination bands also appear. For example, in the case of water if the absorbed quantum splits to raise from 0 to 1 and V2 from 0- 1, there will be a vibration-rotation band centered on the combination frequency, + V2 This process is relatively less probable than the absorbtion of a single quantum at either fundamental frequency, so the intensity of the band is relatively weak. Nonetheless, combination bands appear with sufficient intensity to be an important feature of the infrared spectra of polyatomic molecules. Even in the case of a simple molecule like water, there are a large number of prominent bands, several of which are listed in Table 25.2. 

Dissociation of Molecules Through Electronic Excitation by Direct Electron Impact. Why does the electron energy threshold of dissociation through electronic excitation almost always exceed the actual dissociation energy in contrast to dissociation, stimulated by vibrational excitation, where the threshold is usually equal to the dissociation energy Use the dissociation of diatomic molecules as an example. 

El < Eo) non-dissociative states (dissociative states are rapidly depopulated by the fast intramolecular dissociation process). As is well known, the time evolution of the populations [A(i)] is given by a series of exponentially decaying terms which ctHTiespond to an initial rovibrational relaxation, a subsequent incubation period with overlap of vibrational rriaxation of upper levels and dissociation, and the final dissociation period with steady-state of all populations [A(i)]. Explicit solutions of the master equation for the dissociation of diatomic molecules have been extensively reviewed by H. O. Pritchard in Volume 1 of this series. Such... 

Of much greater importance than the rotation spectra are the spectra of diatomic molecules corresponding to simultaneous changes of both the vibrational and the rotational quantum number, the vibration-rotation spectra. In analogy with the classical result they are present only in molecules for which the oscillation of the nuclei is accompanied by an oscillation of the dipole moment so that homonuclear molecules have no vibration-rotation spectra. The quantum... 

It Las been pointed out in section 20 that in the case of diatomic molecules the investigation of the chemical binding falls into two parts viz. on the one hand the collection from band spectra of numerical data, characteristic for the different molecules, such as the equilibrium distances, the vibration frequencies, the dissociation energies of the different electronic states, and more especially of the ground state on the other hand the interpretation, on the basis of wave mechanics, of the energy curves F, derived from these data, and their relation to the concepts of chemistry. For polyatomic molecules it is advisable to adhere to this same division of labour, although there are some notable differences compared with the case of diatomic molecules. 

It remains to inquire after the effect of rotation of the molecule upon the spectra. If only the valence vibrations are excited, there is no essential difference compared with the case of diatomic molecules so that we can take over the conclusions arrived at in sections 22 and 30. If, however, the deformation vibrations are excited, then the oscillating electric moment will also have a component at right angles to the intemuclear line and hence along the axis of rotation. This brings about that in the emission or absorption spectrum of the molecule not only the frequencies oj Vj ot appear but also the unmodified vibrational frequency o). By similar arguments in the Raman effect there will not only appear the frequency shifts also Vi-ot ... 

The molecule nitrous oxide NgO, contains just as many electrons as CO2. Hence, one would at first be inclined to expect that it too is rectilinear and symmetrical. The investigations of Plyler and Barker(i39), however, have shown conclusively that the molecule is rectilinear and unsymmetrical, very probably of the type NNO, although an unsymmetrical configuration NON cannot be excluded in principle. The rectilinear character is confirmed again by the simple type of rotational structure of all the vibration-rotation bands, already discussed in the case of diatomic molecules like HOI (see section 23 and fig. 33) and also found for CO2 (see previous section, fig. 43). That the molecule is unsymmetrical follows in the first place from the fact that all three normal frequencies ct>2, CO3 are active and, more generally, that the selection rule of Dennison for symmetrical triatomic molecules, according to which for the observable combination bands AtJg + A 3 must be odd, is not obeyed, in contrast to CO2. In addition, if... 

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