Matrix elements [5] (pp. 90-102 and 176-188) of the nonrotating-molecule Hamiltonian operator in the basis set |Ω⟩ have the following form

(1.6a) | |

(1.6b) |

Since Ω is always a good quantum number in the nonrotating molecule, the exact eigenfunctions of the operator can be characterized by a value of Ω. It is convenient to choose the set of exact eigenfunctions of the nonrotating-molecule Hamiltonian to be the nonrotating-molecule basis set. Such exact eigenfunctions obey the relations given in (1.6). For the purposes of calculating rotational energy levels and rotational line intensities, the energy of each state |Ω⟩ is represented by an appropriate parameter, to be fit by comparison with experimental data.

Matrix elements of the nonrotating-molecule Hamiltonian operator in the basis set can be taken to have the following form

(1.7a) | |

(1.7b) | |

(1.7c) |

The two exact equalities in (1.7) are satisfied only if the functions involved
in the matrix elements are exact eigenfunctions of . However, the basis set functions
defined above are *not* exact
eigenfunctions of the nonrotating-molecule Hamiltonian, because spin-orbit
coupling destroys the goodness of the three quantum numbers Λ, *S*,
and Σ. In other words, because the spin-orbit interaction operator
_{i} ξ(*r _{i}*)

Nevertheless, when spin-orbit splittings between the various components of a
given multiplet e.g., ^{2}Π_{1/2},
^{2}Π_{3/2}) are small compared to separations between
different multiplet groups (e.g., ^{2}Σ, ^{4}Σ,
^{2}Π, ^{4}Π, ^{2}Δ, etc.), then the basis
set functions are very
good approximations to the exact eigenfunctions of the nonrotating-molecule
Hamiltonian. Under these circumstances *it is convenient to shift our point
of view a bit,* and to consider the functions to be these exact eigenfunctions. It is then
necessary, however, to remember that the quantum numbers Λ, *S*,
Σ are no longer perfectly good, i.e., the appropriate eigenvalue
equations are only approximately satisfied. For example,

(1.8a) | |

(1.8b) |

where |δ_{1}⟩ and |δ_{2}⟩ are small
functions which vanish when the quantum numbers Λ and Σ are
perfectly good. Because of the presence of small "left-over"
functions like these |δ_{i}⟩, the diagonal matrix elements
of the spin-orbit operator *A***L · S** are only
approximately equal to *A*ΛΣ. Thus, precise energies
of the multiplet components represented by the wave functions
deviate somewhat from
the expression: constant +*A*ΛΣ. However, the exact
energies of the nonrotating- molecule problem can always be represented by a
set of adjustable parameters in the rotational calculations.

A shift in point of view similar to that above is also desirable for the basis set functions . If the functions are taken to be the exact eigenfunctions of the nonrotating-molecule Hamiltonian, then we can write

(1.9a) | |

(1.9b) |

where the quantum numbers *L*, Λ, *S*, Σ in these exact
eigenfunctions of the nonrotating-molecule Hamiltonian are all slightly bad.
The energies of these exact eigenfunctions can again be represented by
appropriate adjustable parameters in the rotational calculations.