Polyatomic oscillator

Most of the molecules we shall be interested in are polyatomic. In polyatomic molecules, each atom is held in place by one or more chemical bonds. Each chemical bond may be modeled as a harmonic oscillator in a space defined by its potential energy as a function of the degree of stretching or compression of the bond along its axis (Fig. 4-3). The potential energy function V = kx j2 from Eq. (4-8), or W = ki/2) ri — riof in temis of internal coordinates, is a parabola open upward in the V vs. r plane, where r replaces x as the extension of the rth chemical bond. The force constant ki and the equilibrium bond distance riQ, unique to each chemical bond, are typical force field parameters. Because there are many bonds, the potential energy-bond axis space is a many-dimensional space. 

Polyatomic molecules vibrate in a very complicated way, but, expressed in temis of their normal coordinates, atoms or groups of atoms vibrate sinusoidally in phase, with the same frequency. Each mode of motion functions as an independent hamionic oscillator and, provided certain selection rules are satisfied, contributes a band to the vibrational spectr um. There will be at least as many bands as there are degrees of freedom, but the frequencies of the normal coordinates will dominate the vibrational spectrum for simple molecules. An example is water, which has a pair of infrared absorption maxima centered at about 3780 cm and a single peak at about 1580 cm (nist webbook). 

In an approximation which is analogous to that which we have used for a diatomic molecule, each of the vibrations of a polyatomic molecule can be regarded as harmonic. Quantum mechanical treatment in the harmonic oscillator approximation shows that the vibrational term values G(v ) associated with each normal vibration i, all taken to be nondegenerate, are given by... 

The vibrational term values for a polyatomic anharmonic oscillator with only nondegenerate vibrations are modified from the harmonic oscillator values of Equation (6.41) to... 

The vibrational temperature, defined for a diatomic harmonic oscillator by the temperature in Equation (5.22), is considerably higher because of the low efficiency of vibrational cooling. A vibrational temperature of about 100 K is typical although, in a polyatomic molecule, it depends very much on the nature of the vibration. 

We have seen earlier that for a linear polyatomic molecule, the vibrational motions can be divided into (3rj — 5) fundamentals, where rj is the number of atoms. For a nonlinear molecule (3rj - 6) fundamentals are present. In either case, each fundamental vibration can be treated as a harmonic oscillator with a partition function given by equations (10.100) and (10.101). Thus. 

We may also apply the above expressions to polyatomic molecular ions. However, the effective reduced mass M of the oscillator in Equation 5 is a function of the definition of normal coordinate, and it is diffi-... 

The eigenvalue problem was introduced in Section 7.3, where its importance in quantum mechanics was stressed. It arises also in many classical applications involving coupled oscillators. The matrix treatment of the vibrations of polyatomic molecules provides the quantitative basis for the interpretation of their infrared and Raman spectra. This problem will be addressed tridre specifically in Chapter 9. 

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... 

Polyatomic molecules, with the many layers of vibrational levels, can be treated, to a first approximation, as a series of diatomic, independent, and harmonic oscillators. This general equation may be expressed as... 

The formulation of the preceding section is very general. We are interested, however, in rotations and vibrations of polyatomic molecules. We therefore discuss now specific applications of the algebraic method beginning with the simple case of one-dimensional coupled oscillators, presented in Section 3.3 in the Schrodinger picture. In the algebraic theory, as mentioned, one associates to each coordinate, x, and related momentum, px = — iti d/dx, an algebra. For... 

The harmonic oscillator energies and wavefunctions comprise the simplest reasonable model for vibrational motion. Vibrations of a polyatomic molecule are often characterized in terms of individual bond-stretching and angle-bending motions each of which is, in turn, approximated harmonically. This results in a total vibrational wavefunction that is written as a product of functions one for each of the vibrational coordinates. 

When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are Av = +1 that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to Av = +2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and... 

In the case of polyatomic molecules, the energy levels become quite numerous. Ideally, one can treat such a molecule as a series of diatomic, independent, harmonic oscillators and the above equation can be generalized ... 

The fundamental frequencies 9t (t = 1, 2,... 3tf—6) are related to and since Xt are the roots of det B—XE) — 0, r, are related to the matrix B and to the molecular force constants Bif. Hence the vibrational energy levels for a non-linear polyatomic molecule in the harmonic oscillator approximation are given by... 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

The potential energy of such an oscillator can be plotted as a function of the separation r, or, for a normal mode in a polyatomic molecule, as a function of a parameter characterizing the phase of the oscillation. For a simple harmonic oscillator, the potential energy function is parabolic, but for a molecule its shape is that indicated in Figure 2.6. The true curve is close to a parabola at the bottom, and it is for this reason that the assumption of simple harmonic motion is justified for vibrations of low amplitude. 

With polyatomic molecules many more fundamental vibrational modes are possible. A qualitative illustration of the stretching and bending modes for the methylene group is shown in Fig. 3.1. Arrows indicate periodic oscillations in the directions shown the and signs represent, respectively, relative move-... 

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