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

2.4. Example: Parities of the Rotational Levels in a 1Σ- State

The complete basis set functions $|L\Lambda S\Sigma;\Omega JM\rangle$ for a 1Σ- state can be written as $|0^- 0 0; 0 J M\rangle$. They transform as follows under συ(xz):

eq 2.13 (2.13)

Thus, the rotational levels of even J are of odd parity (-), while those of odd J are of even parity (+).

 

2.5. Example: Parities of the Rotational Levels of 3Σ+ State

The rotational energy levels of a 3Σ state were calculated in sect. 1.10. The three basis set functions used to label the matrix (1.27) transform as follows when the 3Σ state is a 3Σ+ state.

eq 2.14 (2.14)

Consequently the energy levels obtained from the upper 2 × 2 block of (1.27) are of odd parity when J is even and of even parity when J is odd. The energy levels obtained from the lower right-hand corner of (1.27) are of even parity when J is even and of odd parity when J is odd. Since J = N ± 1 for the former wave functions and J = N for the latter, rotational levels of a 3Σ+ state of even N are of even parity; those of odd N are of odd parity.

The wave functions in (2.14) all transform into themselves or into their negatives under the operation συ. This is not true in general of the functions $|\Lambda S\Sigma;\Omega JM\rangle$ of the original basis set. For this reason, the 2Π wave functions used to label the matrix (1.18) cannot be assigned a parity, although appropriate sums and differences of such wave functions could be.

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