Vibrations and Rotations of Diatomic Molecules
With the electronic part of the problem treated in the previous chapter, the nuclear motion shall occupy our attention in this one. In many ways the motion of two nuclei in a potential well formed by the electron cloud is among the simpler of all quantum
- PDF / 555,277 Bytes
- 24 Pages / 439.36 x 666.15 pts Page_size
- 32 Downloads / 186 Views
Vibrations and Rotations of Diatomic Molecules
With the electronic part of the problem treated in the previous chapter, the nuclear motion shall occupy our attention in this one. In many ways the motion of two nuclei in a potential well formed by the electron cloud is among the simpler of all quantum mechanical problems. The reason is that it readily reduces to the motion of a single particle in a potential well. The derivation of how that comes about is given in detail in case some have not seen it before. A single particle in a potential well is also what the hydrogen atom is, and it is useful to reflect on the differences. For hydrogen, one is interested in the motion of the electron in the coulombic field of the nucleus. It too is a two-body problem, but the reduced mass is almost the same as the electron mass. For the diatomic molecule the reduced mass is half the mass of either nucleus for homonuclear molecules but always on the order of nuclear masses, not the electronic mass. More importantly, the attractive force is not coulombic but rather harmonic; it becomes stronger as the distance between the nuclei gets larger. As a field theory professor of mine remarked to his class, all of quantum mechanics is the simple harmonic oscillator! He was not thinking of the diatomic molecule when he said that, but the diatomic molecule affords an excellent example of a system for which the simple harmonic oscillator is a good approximation. When studying an electron in the hydrogen atom one can readily imagine the spherical coordinates used to locate the electron from the origin which is positioned at the proton (more properly at the center of mass but that is very close to where the proton is). For a diatomic molecule one should picture a dumbbell with the center of mass located midway between the nuclei for a homonuclear molecule, otherwise closer to the more massive nucleus. The dumbbell can now rotate about the center of mass, and only two angular parameters are needed to describe that. They are the same two as used for the angular description of the electron in hydrogen, and hence the rotational motion is a solved problem. Imagining a diatomic molecule as a rigid dumbbell when rotating and as two masses connected to a spring when vibrating is simplistic on the one hand and remarkably good on the other. Your task is to keep in mind when and how these R.L. Brooks, The Fundamentals of Atomic and Molecular Physics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-1-4614-6678-9 6, © Springer Science+Business Media New York 2013
139
140
6 Vibrations and Rotations of Diatomic Molecules
approximations break down and what can be done about that. One of the easier ways to picture the situation is to realize that the simple harmonic oscillator demands a potential curve that looks like a parabola while the real potential curve looks more like the Morse potential, to be treated subsequently. The real potential curve supports only a finite number of vibrational levels (often fewer than 20) while a parabola supports
Data Loading...
