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 σitalic v(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 σitalic v. 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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