ABSTRACT
In this chapter, we continue our effort to treat topics in physical chemistry in a way that does not skip over the main details but still tries to simplify the presentation. We need to present the derivation for at least a partial answer if a few students ask, ‘‘Where did that come from?’’ However, let us be clear at the outset of this chapter that the desired goal is for students to understand pure rotational and vib-rotational spectra of molecules and know where the H atom orbital shapes come from. If you follow the derivation with pencil and paper that may help you understand the
derivation, but if not at least learn El ¼ l(lþ 1)h 2
2mr2 and the selection rule Dl¼1. Here we find a
shortcut that makes this problem simpler than the harmonic oscillator. The classic text by Pauling and Wilson [1] is one of the best sources of the full derivation but they still gloss over a few details requiring additional steps from a text on differential equations (Figure 13.1). Even the excellent text by Eyring, Walter, and Kimball [2] does not give the full derivation. Perhaps this is due to the fact that the key equation was worked out by Legendre in 1785 (!) and many previous mathematics texts expand on the Legendre equation. If you follow the derivation in this chapter, it will really pay off in making the next chapter on the H atom easier. Since the H atom solution is the model for the entire periodic chart, it really is essential! In this chapter, quantum chemistry [3-5] is only part of the list of essential topics but we need to solve the rigid rotor problem prior to the H atom solution because the angular wave functions are the same for both problems. We have struggled to make this chapter understandable using only basic calculus but in a correct way [6]. However, a wise student will look beyond the derivation to study the applications to spectroscopy, which probe the quantized behavior of molecules.
