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Diatomic Molecular Spectroscopy with Standard and Anomalous Commutators

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Diatomic Molecular Spectroscopy with Standard and Anomalous Commutators © International Science PrDiatomicess, ISSN: Molecular 2229-3159 Spectroscopy with Standard and Anomalous Commutators REVIEW ARTICLE DIATOMIC MOLECULAR SPECTROSCOPY WITH STANDARD AND ANOMALOUS COMMUTATORS 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 of standard angular momentum eigenfunctions. A specific Although Van Vleck's reversed angular momentum element of the rotation matrix cannot represent a standard technique [1] is little used today, the anomalous angular angular momentum state because the rotation matrix momentum commutators [2] upon which it was based element carries two magnetic quantum numbers. A still appear in current texts [3–5]. In this review, we show fundamental property of angular momentum is that the that the standard quantum theory of angular momentum square of the total angular momentum and only one of fully accounts for the rotational states of the diatomic its components are constants of motion. An equation molecule. We find that the commutators which define giving operation of the raising and lowering operators angular momentum are not changed in a transformation on the rotation matrix is derived. Some matrix elements from a laboratory coordinate system to one which rotates of the rotational Hamiltonian are calculated to with the molecule, and the seemingly anomalous behavior demonstrate that standard results can be obtained without of the rotated angular momentum operators J¢ and N¢ is resorting to Van Vleck's argument [1] that the unexpected ± ± ¢ ¢ simply the result of their operating on the complex behavior of J± and N± is the result of the anomalous conjugate of rotation matrix elements. commutators. Specifically, we show below that: A rotation is a unitary transformation and, therefore, ¢ preserves the commutation relationships which define JJ± W = JJJ( +1) - W W ±( 1 W) ± , 1 ... (1) angular momentum. Also, operation of the raising and lowering operators on standard angular momentum states JD¢ J* (abg ) is the same in all coordinate systems connected by proper ±M W rotations. Two approaches are presented to show that + - W WJ* abg = JJD( 1) ( 1) W ( ) proper rotations preserve the angular momentum M , 1 commutators. The frequently applied nonstandard ... (2) behavior of the raising and lowering operators on Notice the molecule-fixed operator J±¢ acts on the state diatomic states is illustrated as a result of the operators whose magnetic quantum number W is referenced to the ¢ W acting on elements of the rotation operator matrix instead molecule-fixed z axis as expected, but that J+ lowers International Review of Atomic and Molecular Physics, 1 (1), January-June 2010 25 Christian G. Parigger and James O. Hornkohl Î when operating on the complex conjugate of the rotation Ji¢ Jj¢ – Jj¢ Ji¢ = –i i¢ j¢ k¢ Jk¢ ... (4) ¢ W J* (abg ) matrix element while J– raises of DM W . Here, the primed index denotes a rotated coordinate. This equation containing the reversed sign of i is known The fully standard formalism of angular momentum as Klein's [2] anomalous commutation formula. is applicable without modification to the theory of diatomic spectra as will be demonstrated here: (i) An Two approaches are debated in this work, namely an exact equation for the total angular momentum states of operator and an algebraic approach, without utilizing the diatomic molecule will be derived. (ii) The Hund's Klein's anomalous commutation formula. Each approach cases (a) and (b) basis sets will be obtained from the begins with the standard commutator formula, Eq. (3). We point out that in principle, Eq. (4) can be utilized in exact equation. (iii) The effects of J , J , J¢ , and J¢ on + – + – building angular momentum theory. However, in analogy J (abg ) DM W and its complex conjugate will be calculated. to distinction between right- and left- handed coordinate (iv) Some case (a) matrix elements of the rotational systems, different signs occur. We only use the standard Hamiltonian will be evaluated to show that the standard sign as indicated in Eq. (3) in computation of a molecular angular momentum formalism gives the accepted results. diatomic spectrum [6], i.e., without resorting to use of Also, Hönl-London factors for all n-photon a « a and Klein's anomaly and Van Vleck's reversed angular b « b transitions obeying the D S = 0 selection rule will momentum method. be calculated. Applications will be illustrated in the study Here, the reversal of the sign in Eq. (3) is briefly of molecular spectra recorded following laser-induced investigated for an unitary and an anti-unitary optical breakdown. transformation. The Euler rotation matrix is a real, unitary matrix (see Eq. (11) below). The determinate of the Euler 1.1 Review of Standard and Anomalous Angular rotation matrix is +1 meaning that the sign of vectors is Momentum Commutators preserved under rotations. A spatial rotation of coordinates is a proper transformation. The customary starting point for the quantum theory of angular momentum is the commutation formula for Conversely, the inversion or parity operator the cartesian components of the angular momentum constitutes an improper rotation— this transformation operator J, cannot be described exclusively in terms of the Euler angles. However, angular momentum is a pseudo or axial JiJj – JjJi = iijkJk ... (3) vector, preserving the sign of J under improper rotations. where The parity operator is also unitary and Eq. (3) is preserved by the parity operator. Time reversal (time inversion or ì +1i , j , k in cyclic order reversal of motion) changes the sign of J and it complex- ï conjugates the imaginary unit due to time reversal being = í - 1i , j , k not in cyclic order . ijk ï anti-unitary. Thus, Eq. (3) is invariant under time reversal. î 0 if any indices equal As shown in texts (e.g. Messiah [7]), the time reversal operator has been designed to be anti-unitary for the very The above commutator property usually defines the purpose of preserving the sign of i in commutation angular momentum operator. Coordinate transformations formulae. leave the angular momentum operator definition invariant. The conservation law for angular momentum 1.2 Effect of Unitary Transformation on Angular is fundamental. The definition of angular momentum, Momentum Commutators Eq. (3), is, of course, invariant under specifically spatial In a brief review we show that unitary transformations translations and rotations. Furthermore, Eq. (3) is preserve commutation formulae. Consider the operators invariant under coordinate inversion and time reversal. A, B, and C which satisfy the commutation formula Van Vleck's reversed angular momentum method AB – BA = iC, ... (5) starts with Eq. (3) but then utilizes change of sign of i and subject these operators to the unitary transformation for angular momentum when a transformation of U. That is, coordinates to a system attached to a rotating molecule is made, A¢ = U AU † ... (6) 26 International Review of Atomic and Molecular Physics, 1 (1), January-June 2010 Diatomic Molecular Spectroscopy with Standard and Anomalous Commutators and determinants, are equal to the corresponding matrix † abg A = U A¢ U, ... (7) elements of D ( ) labeled mij, i.e., mij is its own cofactor. For example, with similar equations holding for B¢ and C¢ . The operator † J ¢ J ¢ – J ¢ J ¢ U is unitary, i.e., U = U – 1, so x y y x = i [(m12m23 – m13m22) (Jy Jz – Jz Jy) + U † A¢ UU † B¢ U – U † B¢ UU † A¢ U (m13m21 – m11m23) (Jx Jz – Jz Jx) + = U † A¢ B¢ U – U † B¢ A¢ U = i U † C¢ U, ... (8) (m m – m m ) (J J – J J )], ... (13) or 11 22 12 21 x y y x and since A¢ B¢ – B¢ A¢ = iC¢ ... (9) m12m23 – m13m22 = m31, The above textbook discussion (e.g. Davydov [8]) m m – m m = –m , confirms that the angular momentum quantum 13 21 11 23 32 commutators, Eq. (3), are preserved under unitary m11m22 – m12m21 = m33, ... (14) transformations. The Euler rotation matrix, Eq. (11) the right side of the Eq. (12) reduces to iJz¢ . below, is easily demonstrated to be unitary. Comparison with Van Vleck's [1] treatment of Klein's An algebraic approach is used in the following to anomalous formula, given between his Eqs (4) and (5), show that the commutator Eq. (3) remains invariant when shows that we do not require reversal of sign in our proper rotation of coordinates is applied. The laboratory approach. The anomalous sign in Eq. (4) does not reveal referenced J is transformed to the rotated coordinate a novel aspect of the nature of diatomic molecules. system by application of the rotation matrix D (abg ), Noteworthy is that the anomalous commutation formula remains today a time honored tradition in the theory of ¢ abg J = D ( ) J, ... (10) molecular spectra, e.g., see Refs. [2– 5, 14– 22]. Klein's where a , b , and g are the Euler angles and D (abg ) is an anomalous commutators are means by which matrix orthogonal matrix whose determinant is +1, Goldstein[9], elements of various operators in the molecular Hamiltonian, in particular those expressed in terms of D (abg ) = angular momentum raising and lowering operators, are æcosabg-ag cos cos sin sin abg+ag-bg sin cos cos cos sin sin cos ö obtained.
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