## 1.8. Rotating-Molecule Matrix Elements

Most of the matrix elements of the rotational Hamiltonian (1.10) can be obtained from general considerations of the matrix elements of an angular momentum operator in a basis set characterized by a quantum number specifying the total magnitude of the angular momentum and a quantum number specifying the projection along the z axis [5] (pp. 103-109), [7] (pp. 45-78). For example, the only nonvanishing matrix elements of the components of the spin angular momentum operator S in such a basis set are the following

 (1.13)

The nonvanishing matrix elements of the components of the orbital angular momentum L can be obtained from (1.13) by replacing S by L, and Σ by Λ everywhere. The nonvanishing matrix elements of the total angular momentum J can be obtained from (1.13) by replacing S by J, and Σ by Ω everywhere, except that S± must be replaced by J in the third equation:

 (1.14)

This somewhat surprising difference in behavior of J from L and S is discussed by Van Vleck [8]. (footnote 3)

In actual calculations, two principal kinds of complications arise. First, it is possible, as discussed in sect. 1.5, that the angular momentum quantum numbers characterizing the basis set are not perfectly good. (Note, however, that J and Ω are always good quantum numbers in the basis sets defined above.) In that case, it is common to introduce an additional parameter in matrix elements like (1.13) to allow for the fact that eqs (1.13) are not exact. For example, part of the third equation of (1.13) might be written

 (1.15)

where γ is a small dimensionless parameter much less than unity. This parameter can in principle be determined from accurate electronic wave functions. In practice it is usually determined by fitting the final calculated energy expressions to observed levels.

The second complication arises because sometimes a projection quantum number is used to characterize the basis set while the total-magnitude quantum number is not, e.g., the quantum number L is missing in the basis set . Under these circumstances, the first and third equations of (1.13) cannot be used to obtain the matrix elements of the angular momentum concerned. Nevertheless, and this will be important below, as long as the projection quantum number is good, the selection rules on it implied by (1.13) for the components of an angular momentum operator are still valid [7] (pp. 45-78). Matrix elements which cannot be obtained from (1.13) are usually represented by a parameter, which either drops out of the calculation, or is determined from a fit to experimental data. Symmetry arguments (see chapter 2) are helpful in such cases to determine exactly how many different parameters can (or must) be used.

To illustrate the procedures described above we now consider two examples.