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Additional resources for Algebraic Theory of Molecules (Topics in Physical Chemistry Series)
The 4x4=16 bilinear products b^b^ span the algebra of U(4) [Eq. 21)]. This form of the algebra, called the uncoupled form, is not well suited for the analysis of the problem since one wants states of good angular momentum. The corresponding operators must then have definite transformation properties under rotations. This is equivalent to saying that Cartesian coordinates are not particularly useful to solve the Schrodinger equation with a spherically symmetric potential and that one prefers to use spherical coordinates.
Instead of the quantum numbers N, M we can introduce as before, except that we have Fx instead of Fz, in order to emphasize the difference with the previous case. This difference is important for threedimensional problems, as discussed in the following sections. , U(l) and O(2)] are isomorphic, the two cases (I) and (II) can be related to each other. Using Eq. 46) as where Again note the similarity with the usual angular momentum. Dynamic symmetries for chain (II) correspond to an expansion of the Hamiltonian in terms of invariant operators of O(2).
11 Herman-Wallis expansion of intensities Intensities of transition can be also analyzed by means of fitting formulas. An expansion quite often used for infrared transitions is the Herman—Wallis (1955) form The quantity m in Eq. 74), which should not be confused with the projection of the angular momentum / on the z axis, is given by 20 Chapter 1 One can see by comparing with Eq. 45) that \m\ is the line strength S/. 7. Notes 1. The material in this section is discussed in all texts of quantum mechanics.