## 3.5. Example: Rotational Intensity Distribution in a 3Σ- - 1Σ+ Transition

The wave functions and symmetry properties for the rotational levels of a 1Σ+ state have been given in eqs (3.10a) and (3.11a).

The rotational energy levels of a 3Σ state have been discussed in sect. 1.10; the symmetry properties of the rotational levels of a 3Σ+ state have been discussed in sect. 2.5. It is easy to show, by arguments analogous to those of sect. 2.5, that the parities of the rotational levels of a 3Σ- state are just the opposite of those of a 3Σ+ state, i.e., states of even N are of odd parity and states of odd N are of even parity.

The wave functions for the rotational levels of a 3Σ state were not determined in sect. 1.10. These wave functions can be determined, however, by finding the eigenvectors of the sum of the matrices given in (1.24) and (1.27). We consider a 3Σ- state which is very near case (b); for the purposes of calculating intensities, we thus set λ = 0. The three normalized eigenfunctions of given J then become:

 (3.22a)

 (3.22b)

 (3.22c)

These three functions are eigenfunctions of the matrix sum (1.24) plus (1.27) when λ = 0, and belong to the eigenvalues E + B ⟨⟩ + BN(N + 1), where N = J + 1, N = J, and N = J-1, respectively.

We must next calculate all matrix elements of µZ allowed by the selection rules on J and by the parity selection rule. (Alternatively, the parity selection rule can be used as a check on the calculations.) The selection rule ΔJ = 0, µ1, and the fact that N = J + 1, J, or J - 1 in the triplet state, indicates the possibility of nine rotational branches. In standard notation, i.e., ΔNΔJ, these would be called SR, RR, QR, RQ, QQ, PQ, QP, PP, OP, branches. Let us consider in detail the SR branch. The intensity expression for such a branch depends on a matrix element between an upper state wave function of the form (3.22a) with J replaced by (J + 1) everywhere (J′ = J + 1, N′ = J + 2) and a lower state wave function of the form (3.10a) with J unchanged (J″ = N″ = J). Just as in eq (3.13) above, this matrix element can be separated into parts

 (3.23)

Furthermore, by applying the symmetry operation συ (see sect. 2.9), we find that the first and second terms in (3.23) are equal.

The next step is to replace µZ by the right-hand side of (3.12). Before doing this, however, we examine the matrix elements of the molecule-fixed components of the dipole moment operator in the nonrotating-molecule basis set under consideration. The nonrotating-molecule basis set for the 1Σ+ state consists of one singlet function |0+ 0 0⟩; that for the 3Σ- state consists of three triplet functions: |0- 1 1⟩, |0- 1 0⟩, and |0- 1 -1⟩. There are no spin-allowed matrix elements of the dipole moment operator between singlet functions and triplet functions, since nonvanishing dipole moment operator matrix elements obey the selection rule ΔS = 0 when S is a good quantum number [see eqs (3.6) and (3.7)]. However, if we assume that the singlet state is slightly contaminated by triplet states, and that the triplet state is slightly contaminated by singlet states, then any matrix elements allowed by the selection rules ΔΩ = 0, ±1 are nonvanishing [see eq (3.3)]. Hence the possible nonvanishing matrix elements appear to be

 (3.24)

The one remaining point to check involves the matrix element between the two states having Ω = 0. Selection rules require that all matrix elements of µz vanish between 0+ states and 0- states. We are, of course, dealing here with a Σ+ state and a Σ- state, so that this electronic transition is orbitally forbidden. However, it is already known to be spin forbidden, so this orbital forbiddenness is of no great interest. What is of interest is whether or not in the strong spin-orbit coupling limit, corresponding to the nonrotating-molecule basis set |Ω⟩, the ΔΩ = 0 transition is allowed or forbidden. As might be expected, it is the transformation properties of the combined spin and orbital parts of the wave function which determine whether a state having Ω = 0 is a 0+ state or a 0- state. We note that

 (3.25)

Consequently, both the 1Σ+ and the 3Σ- states give rise in the strong spin-orbit coupling limit to 0+ states, and the second matrix element of (3.24) is allowed by symmetry.

Spectroscopists sometimes speak of a doubly forbidden transition. Such a label is useful, if it is employed carefully. The degree of multiple forbiddenness is best defined to be the number of first-order perturbations which must be carried out in succession before a given transition is made allowed. Thus, in the particular case of a 3Σ- - 1Σ+ transition, a single first-order spin-orbit perturbation (satisfying the selection rules ΔS = 0, ±1; ΔΩ = 0) suffices to make the transition allowed (e.g., the mixing of 3Σ- and 1Π), so that this transition is only singly forbidden. On the other hand, a 5Σ- - 1Σ+ transition is made allowed only after two successive first-order spin-orbit perturbations, and it is therefore doubly forbidden.

Taking into account the fact that the first and second terms in (3.23) are identical, the fact that the only nonvanishing matrix elements of the molecule-fixed components of the dipole moment operator in the basis set under consideration are given in (3.24), and the fact that µZ can be expanded as given in (3.12), we can rewrite (3.23) in the form

 (3.26)

For simplicity we define two quantities µ|| and µ

 (3.27)

which can both be made real as follows. Since the two wave functions |0- 1 0⟩ and |0+ 0 0⟩ both have only zero values for the angular momentum projection quantum numbers, their phases can be chosen such that they transform into themselves under the time inversion operation θ (see sect. 2.11)

 (3.28)

Applying the time inversion operation to all quantities in the first equation of (3.27) we obtain

 (3.29)

Clearly, the quantity µ|| is real under these conditions. Applying the time inversion operation in a similar manner to the second equation in (3.27), and using transformation properties for the wave function |0- 1 1&rang obtained from eq (2.38), we find

 (3.30)

Applying the symmetry operation συ to the matrix element on the righthand side of (3.30) allows one to conclude that

 (3.31)

Thus, the quantity µ is also real. (Note that the time inversion operation θ was used together with the reflection operation συ in demonstrating that µ is real. The use of both θ and συ will generally be necessary when the matrix elements under consideration involve functions with nonzero values for angular momentum projection quantum numbers.)

If we now substitute from table 6 and eqs (3.27) in (3.26), we obtain for this matrix element of µZ

 (3.32)

where Ω has been given its value of zero. The intensity is proportional to the square of this quantity summed over M. Thus,

 (3.33)

In a similar fashion.

 (3.34)

These results were recently published by Watson [19]. who corrected the results previously given by Schlapp [20]. An examination of Watson's eq (23) and eq (3.27) above shows that µ0 (his notation) = +µ|| (this notation), but that µ1 (his notation) = -µ (this notation).