Volume elements, vibration-rotation

The term containing Jp in the first composite brackets on the right side of Eq. (5.2) can be removed by changing the volume element dp to Ipp dp. The terms in the second composite brackets represent the effective potential function for each vibrational—rotational state. 

Here, Vexm is a purely multiplicative operator called the extra potential term. The volume elements used to normalize the wavefunction reads dV = sin (3dad0d f dqi... dqsN-6- This ensures that the momentum operators Pi are Hermitian. The matrices G, C and F, which stand for internal vibrations, Coriolis couplings and overall rotation, respectively, are, in general, functions of the internal coordinates. 

As determining elements of molecules, therefore, besides the total charge e, the nuclear distances and the molecular volume, we have also now before us the dipole and quadrupole moments, as well as the polarizability. All these quantities (with the exception of e), however, also depend more or less on the state of excitation of the molecule. Just as in the case of atoms, so also in molecules there exist different electronic states, characterized by quantum numbers which in the following sections we shall denote collectively by n. Moreover, the nuclear motion has an effect, consisting on the one hand of a rotation of the whole molecule, determined by the rotational quantum number y, and on the other hand of a vibration of the nuclei relative to one another, described by a vibrational quantum number s. With these different states of motion we shall deal in detail in the pages which follow. 

In recent years a very considerable bod of theory has been developed which makes it possible to understand and utilize the infrared and Raman spectra of a large number of polyatomic molecules. The purpose of this book is to develop essential elements of this theory, starting from its simplest form and advancing to fairly elaborate and powerful theorems useful in more complicated applications. In order to hold the volume lo a reasonable size, only vibrational spectra are treated in detail. However, vibrational (as opposed to rotational) spectra constitute the great bulk of published infrared and Raman data. The theory herein presented applies to gases and is a useful approximation for liquids but requires some extension for crystals. 

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