## 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
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:

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

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.