Vibration of a Diatomic Molecule

Through quantum mechanical considerations, the vibration of two nuclei in a diatomic molecule can be reduced to the motion of a single particle of mass fx, whose displacement q from its equilibrium position is equal to the change of the internuclear distance. The mass /tx is called the reduced mtiss and is represented by 

Here K is the force constant for the vibration. Then the Schrodinger wave equation becomes 

If this equation is solved with the condition that 0 must be single valued, hnite, and continuous, the eigenvalues are 

Here v is the vibrational quantum number, and it can have the values 0, 1, 2, The corresponding eigenfunctions are 

As Fig. 1-5 shows, actual potential curves can be approximated more exactly by adding a cubic term  

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Figure 1.11(b) illustrates the ball-and-spring model which is adequate for an approximate treatment of the vibration of a diatomic molecule. For small displacements the stretching and compression of the bond, represented by the spring, obeys Hooke s law ... 

The vibration of a diatomic molecule, or any vibrational mode in a polyatomic molecule, may be approximated by two atoms of mass m and m2 joined by a Hooke s law bond that allows vibration relative to the centre of mass. The frequency of such a two-body oscillator is given by... 

Figure 1. Translation, rotation, and vibration of a diatomic molecule. Every molecule has three translational degrees of freedom corresponding to motion of the center of mass of the molecule in the three Cartesian directions (left side). Diatomic and linear molecules also have two rotational degrees of freedom, about rotational axes perpendicular to the bond (center). Non-linear molecules have three rotational degrees of freedom. Vibrations involve no net momentum or angular momentum, instead corresponding to distortions of the internal structure of the molecule (right side). Diatomic molecules have one vibration, polyatomic linear molecules have 3V-5 vibrations, and nonlinear molecules have 3V-6 vibrations. Equilibrium stable isotope fractionations are driven mainly by the effects of isotopic substitution on vibrational frequencies.

The frequency, y of vibration of a diatomic molecule is well known to be given by... 

The type of interaction envisaged above between a high and a low frequency motion (vXH and rXR Y or yXH and <5 or y RXH YR ) is closely similar to that commonly envisaged between the (low frequency) vibration of a diatomic molecule and a (high frequency) electronic transition the explanation advanced for the broad bands is also closely analogous in the two cases. Although such interactions cause features which are most pronounced in the vibrational spectra of H-bonded systems, they are probably also more generally a common cause of the finite widths of vibrational bands of molecules in the condensed states. 

A parallel band of a linear molecule has no (7-branch lines. (The single vibration of a diatomic molecule is a parallel mode.) See Fig. 6.6. In C02, we see from Fig. 6.2 that v3 changes the dipole-moment component along the symmetry axis hence the v3 fundamental band is a parallel band in contrast, the v2 fundamental is a perpendicular band. 

In order to obtain an analytic solution they subsequently took the Wvfl belonging to a quantized harmonic oscillator, which is not unreasonable for the vibrations of a diatomic molecule ). Thus their M-equation has an artificial boundary ... 

The internal motion of a polyatomic molecule is rather complex. In order to understand its principles, it is convenient and customary to refer to the vibration of a diatomic molecule. The internal motion in molecules has above all been studied experimentally and theoretically by spectroscopists. The approach towards the study of polyatomic molecules through diatomic molecules has been excellently demonstrated by the two classical books of G. Herzberg, the first one dealing with diatomic molecules2, the second one with polyatomic molecules3). Although these two books should by now be hopelessly out of date, and in spite of the fact that they have been succeeded by a series of up to date works, they can still be recommended to to-day s students. 

Consider the vibration of a diatomic molecule in which two atoms are connected by a chemical bond. 

In quantum mechanics (18,19) the vibration of a diatomic molecule can be treated as a motion of a single particle having mass n whose potential energy is expressed by (1-21). The Schrodinger equation for such a system is written as... 

The vibration of a diatomic molecule may be of only one kind, an alternate expansion and contraction of the interatomic distance. The simplest mathematical treatment (useful, but approximate) of such a vibration assumes the molecule to be a harmonic oscillator, roughly analogous to a... 

Some representative examples of common zero-temperature VER mechanisms are shown in Fig. 2b-f. Figures 2b,c describe the decay of the lone vibration of a diatomic molecule or the lowest energy vibrations in a polyatomic molecule, termed the doorway vibration (63), since it is the doorway from the intramolecular vibrational ladder to the phonon bath. In Fig. 2b, the excited doorway vibration 2 lies below large molecules or macromolecules. In the language of Equation (4), fluctuating forces of fundamental excitations of the bath at frequency 2 are exerted on the molecule, inducing a spontaneous transition to the vibrational ground state plus excitation of a phonon at Fourier transform of the force-force correlation function at frequency 2, denoted C( 2). 

The microwave experiment studies rotational structure at a given vibrational level. The spectra are analyzed in terms of rotational models of various symmetries. The vibration of a diatomic molecule is, for instance, approximated by a Morse potential and the rotational frequencies are related to a molecular moment of inertia. For a rigid classical diatomic molecule the moment of inertia I = nr2 and the equilibrium bond length may be calculated from the known reduced mass and the measured moment, assuming zero centrifugal distortion. 

Thus, we obtain an equation describing the frequency of the vibrations of a diatomic molecule ... 

If atomic mass units are employed and the force constants are measured in N/cm (= mdyne/A), the frequency of the vibration of a diatomic molecule is given by ... 

The absorptions just described seem to be correctly identified as the stretching modes, v of the H bond between water molecules. Assuming the simplest possible model, that v, is equivalent to the vibration of a diatomic molecule with atoms of mass 18, the absorption at 212 cm" corresponds to a force constant of 0.2-10 dynes/cm. This is in acceptable accord with the value obtained for formic acid (0.3-10 dynes/cm). 

The quantum oscillator is a good model to describe the vibrations of a diatomic molecule. The frequency is given by the familiar equation but using the reduced mass of the two nuclei /t = [mxm- / mx + m ) in place of m. 

EXERCISE 8.3 The vibration of a diatomic molecule resembles that of a harmonic oscillator. Since both nuclei move, the mass must be replaced by the reduced mass,... 

The vibration of a diatomic molecule involves the compression and extension of its covalent bond. The number of to and fro movements per second is called the vibrational frequency of the molecule. The greater is the vibrational energy of a molecule, the higher is the amplitude of its vibration i.e. the greater is the distance travelled by the atoms during the vibration. 

FIGURE 4.5 Potential energy for vibration of a diatomic molecule (solid curve) and for a 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... 

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