Diatomic Molecular Spectroscopy with Standard and Anomalous Commutators International Review of Atomic and Molecular Physics, 1 (1), January-June 2010 25 D I ATOM I C M OLECU LAR SPECTROSCOPY W I TH STAN DARD AN D AN OM ALOU S COM M U TATORS CHRISTIAN G. PARIGGER AND JAMES O. HORNKOHL The University of Tennessee Space Institute, Tullahoma, Tennessee 37388, U.S.A. Abstract: In this review, we address computation of diatomic molecular spectra. An overview of the theory is discussed based on symmetries of the diatomic molecule. The standard quantum theory of angular momentum fully accounts for the rotational states of the diatomic molecule. Details are elaborated in view of standard versus anomalous commutators for generation of a synthetic spectrum. Specific example spectra are presented for selected diatomic molecules in view of diagnostic applications in laser-induced optical breakdown spectroscopy. Keywords: Molecular Spectra (PACS 33.20), Intensities and Shapes of Molecular Spectral Lines and Bands (PACS 33.70), Plasma Diagnostics (PACS 52.70), Laser-Induced Optical Breakdown. 1. INTRODUCTION Although Van Vleck's reversed angular momentum technique [1] is little used today, the anomalous angular momentum commutators [2] upon which it was based still appear in current texts [3– 5]. In this review, we show that the standard quantum theory of angular momentum fully accounts for the rotational states of the diatomic molecule. We find that the commutators which define angular momentum are not changed in a transformation from a laboratory coordinate system to one which rotates with the molecule, and the seemingly anomalous behavior of the rotated angular momentum operators J ′ ± and N ′ ± is simply the result of their operating on the complex conjugate of rotation matrix elements. A rotation is a unitary transformation and, therefore, preserves the commutation relationships which define angular momentum. Also, operation of the raising and lowering operators on standard angular momentum states is the same in all coordinate systems connected by proper rotations. Two approaches are presented to show that proper rotations preserve the angular momentum commutators. The frequently applied nonstandard behavior of the raising and lowering operators on diatomic states is illustrated as a result of the operators acting on elements of the rotation operator matrix instead of standard angular momentum eigenfunctions. A specific element of the rotation matrix cannot represent a standard angular momentum state because the rotation matrix element carries two magnetic quantum numbers. A fundamental property of angular momentum is that the square of the total angular momentum and only one of its components are constants of motion. An equation giving operation of the raising and lowering operators on the rotation matrix is derived. Some matrix elements of the rotational Hamiltonian are calculated to demonstrate that standard results can be obtained without resorting to Van Vleck's argument [1] that the unexpected behavior of J ′ ± and N ′ ± is the result of the anomalous commutators. Specifically, we show below that: J J ′ ± Ω = ( 29 ( 29 1 1 , 1 J J J + -Ω Ω± Ω± ... (1) ( 29 * J M J D ′ ± Ω αβγ = ( 29 ( 29 ( 29 * , 1 1 1 J M J J D Ω + -Ω Ω αβγ   ... (2) Notice the molecule-fixed operator J′ ± acts on the state whose magnetic quantum number Ω is referenced to the molecule-fixed z′ axis as expected, but that J + lowers Ω REVIEW ARTICLE © International Science Press, ISSN: 2229-3159