Rotational energy level notation

The vibration-rotation energy levels of a diatomic molecule are given in traditional notation (Herzberg, 1950) by the expression... 

Diatomic molecules provide a simple introduction to the relation between force constants in the potential energy function, and the observed vibration-rotation spectrum. The essential theory was worked out by Dunham20 as long ago as 1932 however, Dunham used a different notation to that presented here, which is chosen to parallel the notation for polyatomic molecules used in later sections. He also developed the theory to a higher order than that presented here. For a diatomic molecule the energy levels are observed empirically to be well represented by a convergent power-series expansion in the vibrational quantum number v and the rotational quantum number J, the term... 

Bunker has recently introduced a different labeling of the inversion states according to the number of nodes t inv of the inversion function i//,- (p). Thus, the 0 label corresponds to v-, v = 0, 0 to 1, I" " to 2 etc. (Fig. 3). The notation of Bunker allows one to label the energy levels by their symmetry and to determine the vibration and rotation selection rules in a very straightforward way We feel, however, that for high inversion barriers and especially for the inversion states below the inversion barrier it is more natural to use the old labeling (but we may be too conservative in this respect). 

The energy matrix of this interaction is an infinite matrix but we have found that for the calculation of the 2v2 and vn energy levels it is sufficient to work with a 7x7 matrix for each value of the rotational quantum number/. In the notation I k), the off-diagonal matrix elements of connect the following... 

Figure 2.1.b. Schematic diagram of the energy levels and vibration-rotation transitions (after Herzberg, 1950). The lower lines (v/ = 0) correspond to the fundamental vibrational state with different rotational states (J// = 0,1, 2,3,4). The first vibrationally excited state (v/ = 1) with related rotational states (J/ = 0 to 4) are shown by the upper lines. The transitions corresponding to the P and R branches are indicated. Transition AJ = 0 is not allowed in the case of the idealized rigid rotor, harmonic oscillator for a linear molecule, but can be observed (Q-branch) in other cases. The notations are from Herzberg (1950). 

Rotation-Vibration-Electronic Energy Levels and Standard Notation... 

As the species looked at in chemical reactions are mostly molecules, the two electronic levels depicted in Figure 7.2 spht into sub-levels, according to the molecular vibrational and rotational energy quanta. The vibrational levels are customarily numbered with the quantum number v,(i = 0, 1, 2,...). The notation for the rotational levels is more complex and depends as well on the size of the molecule, but typically one associates the rotation with the quantum number Ji i = 0, 1, 2,...). In order to distinguish between states, double primes are used to mark the (lower) ground-state levels and single-primed quantum numbers mark the excited (upper) state levels. The main processes observed in molecule-laser photon interactions are shown in Figure 7.3. 

The full-rotational group compatibility tables show how a free-ion J level is broken up into crystal-field levels when the ion is placed in a crystalline environment with a distinct point symmetry. The irreducible representations (irreps) are labelled according to the notations of Koster et al. (1963). The tables are given up to J = 8 for even-electron systems and up to J = 17/2 for odd-electron systems. The double groups are marked by an asterisk. Although higher J values may occur for divalent lanthanide ions, they are of less importance for the study of the energy levels in the ultraviolet, visible and near-infrared parts of the spectra. 

In our discussion which concerns chiefly diatomic or linear molecules we will often apply the same notation (J, or J, J") for the total angular momentum and for the rotational quantum number because of the decisive contribution of rotational motion. Some remarks on accounting for nuclear spin can be found in Sections 1.4 and 5.3. The calculations of the rotational level energies and of the transition intensities have been... 

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