where the rotational angular momentum is expressed in the second line of (1.10) as the total angular momentum (J) minus the electronic orbital and spin angular momenta (L and S). For the purposes of calculation, is often written in the form
The extremely thorough student will find that the total angular momentum operator J for linear molecules is somewhat peculiar, since its molecule-fixed components do not obey angular-momentum-type commutation relations. Furthermore, the operator for linear molecules does not have precisely the simple form indicated in (1.10). Nevertheless, it can be shown that correct results are obtained by ignoring the peculiarities associated with J in linear molecules and by treating J like the angular momentum operator defined for nonlinear molecules in .
The complete Hamiltonian describing the nonrotational and rotational parts of the problem is of course given by (1.1). It is the matrix of the Hamiltonian (1.1) which will ultimately be diagonalized to obtain molecular energy levels and molecular wave functions.
Loosely speaking, one can see in eq (1.10) the
origin of the various entries in column 2 of
table 2. Forgetting for a moment the
absence of Rz in (1.10), and remembering that the eigenvalue
associated with the sum of the squares of the components of an angular momentum
operator has the form
Since the operators L and S in (1.10) affect the course of the rotational energy levels only through the four cross terms JxLx, JyLy, JxSx, JySy, and since the selection rules for nonvanishing matrix elements of Lx, Ly, and Sx, Sy are ΔΛ and ΔΣ = ± 1, respectively, we see that the effects of L and/or S in (1.10) can be ignored when the separation between states of the nonrotating molecule satisfying the selection rules ΔΛ and/or ΔΣ = ± 1 is large compared to B J.