Table of Contents of Rotational Energy Levels and Line Intensities in Diatomic Molecules

1.7. Rotating-Molecule Basis Set

It can be shown [3] (pp. 6-16), [2] (pp. 279-281) that the wave functions | ΩJ 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 Lz + Sz, 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:

eq 1.12 (1.12)

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.

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