NIST logo

Bookmark and Share

NBS Monograph 115 - Version History

The Calculation of Rotational Energy Levels
and Rotational Line Intensities in Diatomic Molecules

Jon T. Hougen
Gloria Wiersma, HTML conversion

Procedures are described, in this pedagogical monograph, for making quantum mechanical calculations of rotational energy levels and rotational line intensities in diatomic molecules. The procedures are illustrated by sample calculations. A familiarity with the material of this report should enable a practicing electronic spectroscopist to carry out, though in a rather mechanical way, his own theoretical calculations for molecules under experimental investigation. The material of this report is aimed at electronic spectroscopists who have had the equivalent of one semester of graduate level quantum mechanics.

Key words: Diatomic molecules; Hund's coupling cases; rotational levels; rotational line intensities; theoretical calculations.

Table of Contents - Cover Page 1970 document

1. Calculation of rotational energy levels
1.1. Hund's coupling cases (a), (b), (c), and (d)
1.2. General approach to the calculations
1.3. Nonrotating-molecule Hamiltonian
1.4. Nonrotating-molecule basis set
1.5. Nonrotating-molecule matrix elements
1.6. Rotating-molecule Hamiltonian
1.7. Rotating-molecule basis set
1.8. Rotating-molecule matrix elements
1.9. Example: The Hill and Van Vleck expression for 2Π states
1.10. Example: The Schlapp expression for 3Σ States
2. Symmetry properties of the rotational energy levels
2.1. Geometric symmetry operations
2.2. Permutation-inversion symmetry operations
2.3. The symmetry operation σv
2.4. Example: Parities of the rotational levels in a 1Σ- state
2.5. Example: Parities of the rotational levels in a 3Σ+ state
2.6. The symmetry operation i
2.7. The symmetry operation C2
2.8. Example: Symmetry properties of the rotational levels in a 1Πu state
2.9. Relations between matrix elements
2.10. Example: $L_perp^2 equiv L^2 - L_z^2$
2.11. The time inversion operation Θ
3. Calculation of rotational line intensities
3.1. Laboratory-fixed components of the electric dipole moment operator µ
3.2. Molecule-fixed components of µ
3.3. The direction cosine matrix α
3.4. Example: Hönl-London intensity expressions for a 1Π -1Σ+ transition
3.5. Example: Rotational intensity distribution in a 3Σ- - 1Σ+ transition
3.6. Intensity calculations when closed-form expressions cannot be obtained.
Example: Rotational intensity distribution in a 4Δ - 6Σ+ transition
4. Perturbations
4.1. General remarks
4.2. Homogeneous and heterogeneous perturbations
4.3. Example: 1Π - 1Σ+ heterogeneous perturbation
4.4. Example: 3Δ - 1Π homogeneous perturbation
4.5. Van Vleck transformations
4.6. Example: Λ-type doubling in a 1Π state
4.7. Centrifugal distortion corrections to rotational energy levels
5. References