## 1.7. Rotating-Molecule Basis Set

It can be shown [3] (pp. 6-16),
[2] (pp. 279-281) that the wave functions
| Ω*J M* ⟩ associated with the rotational part
of the problem can be characterized by one parameter and two good quantum
numbers: Ω, *J*, *M*. The quantum number *J* specifies the
*total* (not the rotational) angular momentum in the molecule. The quantum
number *M* specifies the projection of the total angular momentum along
some laboratory fixed *Z* axis, and takes on the values *J*,
*J* - 1, ..., -*J*. The parameter Ω,
which helps to characterize the rotational wave functions of a diatomic
molecule, is somewhat peculiar. It is convenient to consider the parameter
Ω in the rotational wave functions to be the quantum number associated
with the projection of the total angular momentum **J** along the
internuclear axis. Table 1 indicates that
the projection of **J** along the axis is actually equal to the projection
of **L** + **S**, i.e., to the quantity represented in the
nonrotating molecule by the symbol Ω. (Hence the rule
*J* ≥ | Ω |.) Actually, however, Ω is
not a *quantum number* for the rotational wave functions, since it does
not correspond to an eigenvalue of some operator acting on the rotational wave
functions. It arises in the rotational problem because the expression for the
differential operator (1.10)
[3] p. 13 contains the operator
*L*_{z} + *S*_{z}, and this latter
operator, when acting on a nonrotating molecule basis set function, can be
replaced by Ω.
Basis set functions for the complete problem consist of products of the basis
set functions for the nonrotational problem and basis set functions for the
rotational problem. Such functions are represented by one of the symbols:

The quantities Ω, *J*, *M* must be integers for molecules with
an even number of electrons and half-integers for molecules with an odd number
of electrons.