Harmonic Picture of Polyatomic Vibrations

In low-resolution infrared spectroscopy of diatomic molecules, the rotational fine structure is lost, and some feature in a spectrum assigned to be a fundamental transition is but a single peak. The frequency of that peak is taken to be the vibrational frequency in the absence of any more precise experiments, and that is the extent of the information obtained. This low-resolution information corresponds mostly with a nonrotating picture or else a rotationally averaged picture of the molecule s dynamics to the extent that we can analyze the data, we need only consider pure vibration. To understand the internal dynamics of polyatomic molecules, it is helpful to start with a low-resolution analysis. This means neglecting rotation, or presuming the molecules to be nonrotating. 

A force field refers to any potential for the vibrations of a molecule expressed in terms of some chosen set of internal coordinates. In principle, we arrive at complete xmderstanding 

The simplest force field for a molecule is one that is harmonic. This means that the potential energy has only linear and quadratic terms involving the 3N - 6 coordinates. Some of these terms may be cross-terms, for example, a product of two coordinates such as r r2- As discussed in Chapter 7, a potential that is harmonic in all internal coordinates can be written so that there are no cross-terms provided that the original coordinates are transformed to the normal coordinates. In this section, we will designate normal coordinates as qi, q2, qs. and then the classical Hamiltonian for vibration is 

This Hamiltonian is that of a separable problem, and as we have already considered in Chapter 7, it is equivalent to a problem of 3N - 6-independent harmonic oscillators. The vibrational frequencies of the oscillators are the (o/s. The Schrodinger equation that develops from the quantum mechanical form of this Hamiltonian is also separable. The energy level expression comes from the sum of the eigenenergies of the separated harmonic oscillators or modes. For each, there is a quantum number, n,. 

A vibrational state of the polyatomic is specified by a set of values for the 3N - 6 quantum numbers in Equation 9.57. The lowest energy state is the state that has all the quantum 

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