**1. Introduction**

Identification of diatomic molecular spectra necessitates a clear description of angular momentum (AM) in order to demarcate the various features that comprise optical fingerprints. Quantum mechanics theory (QMT) asserts that not all three components of AM can be measured simultaneously, usually the total AM and one projection of the total AM describe upper and lower states of molecular transitions. The components of AM are formally described by a set of operator equations.

Classical mechanics (CM) description and associated quantization of the asymmetric top [1] suggests occurrence of commutator relations with different signs when computing momenta with respect to the principal axes of inertia. In other words, a laboratory-fixed system shows standard AM commutators, but with respect to the molecule-attached coordinate system, there is a sign change that carries the name "reversed" internal AM [2]. The derivation by Klein in 1929 [1] is based on the correspondence principle that in essence emphasizes that QMT reproduces classical physics in the limit of large quantum numbers. From a CM point of view, reversal of motion occurs when transforming from a lab-fixed to a molecule-attached coordinate system, akin to experience of motion reversal when jumping onto a moving merry-go-around. However, reversal of motion in quantum mechanics (QM) is described by an anti-unitary transformation, requiring sign change and complex conjugation. The reversed internal AM concept [2] and applications actually are communicated and applied in analysis of molecular spectra by Van Vleck in 1951 in his review article on coupling angular momenta, i.e., AM, referring to axes mounted on the molecule, adheres to opposite-sign commutator algebra. This evolved into so-called reversed angular momentum (RAM) concepts for prediction of molecular spectra.

However, orthodox or classic QM abides by strict mathematical rules associated with the theory. Use of RAM techniques is contraindicated, especially since Nöther-type symmetry transformation [3] sustains the standard commutator relations, viz., reversal of motion is an anti-unitary transformation, just like in the Schrödinger wave equation that is invariant with respect to motion-reversal or time-reversal due to anti-unitary operation, as expected. It is important to recognize that a transformation from laboratory-fixed to molecular-attached coordinates within standard QM does not condone anomalous AM operator identities.

This review communicates proofs that the quantum-mechanic AM equations remain the same in a transition from laboratory-fixed to molecular-attached coordinates. Methods that invoke RAM for the prediction of molecular spectra are misleading. Application of standard QM establishes within the concept of line strengths [4] consistent computation of diatomic spectra [5]; examples include hydroxyl, cyanide, and diatomic carbon spectra [6]. First, Oscar Klein's paper [1] is discussed showing his original argumentation. This is followed by presenting proofs consistent with QMT opposing RAM concepts and occurrence of a minus sign in unitary and anti-unitary transformations. The "new" aspect of this review is the emphasis of invoking mathematics consistent with QMT. Subsequently, this review summarizes the approach for prediction of selected diatomic spectra including presentation of computed diatomic spectra of OH and C2 molecules.

### **2. Materials and Methods**

The premise of this article is Oscar Klein's work [1] "Zur Frage der Quantelung des asymmetrischen Kreisels" or "On the question of the quantization of the asymmetric top." This particular work is in German without an available translation; the essential contents are in the Einleitung, viz., the introduction, and on the page following the introduction. Klein's paper reflects the initial argumentation of the RAM method, and essential aspects of this paper are discussed below, up to Equation (6).

The purpose of the 1929 work is, as O. Klein writes, to reduce quantization of the asymmetric top to simple algebra for the components of the angular momentum "... that were developed by Dirac [7] and as well by Born, Heisenberg and Jordan [8]." For a solid body, the main moments of inertia are labeled as *A*, *B*, and *C*, the angular momenta are labeled *P*, *Q*, *R*, and one finds the CM energy of rotation, *E*,

$$E = \frac{1}{2} \left( \frac{P^2}{A} + \frac{Q^2}{B} + \frac{R^2}{C} \right) \tag{1}$$

or perhaps with convenient notation, using for operators ˜*J*1 = *P*, ˜*J*2 = *Q*, ˜*J*3 = *R*, where the tilde-symbol indicates that angular momenta (that would be AM operators in QM) are referred to the main axis of the ellipse of inertia (or in molecules, referred to molecular-fixed coordinates), and for moments of inertia *I*1 = *A*, *I*2 = *B*, *I*3 = *C*,

$$E = \frac{1}{2} \sum\_{k=1}^{k=3} \frac{1}{I\_k} I\_k. \tag{2}$$

Subsequently, O. Klein writes that *P*, *Q*, *R* can be understood to describe matrices satisfying QM equations of motion, with *i* = √ −1 and using the standard *h*¯ for Planck's constant divided by 2 *π*,

$$\frac{dP}{dt} = \frac{i}{\hbar}(EP - PE), \quad \frac{dQ}{dt} = \frac{i}{\hbar}(EQ - QE), \quad \frac{dR}{dt} = \frac{i}{\hbar}(ER - RE). \tag{3}$$

In terms of operators, using the Hamilton operator H instead of *E* and writing the equation in the Heisenberg-picture for an abstract observable (operator), O, without explicit time-dependence of the observable, i.e., *∂*O *∂t* = 0, and using the commutator [H, O] = HO − OH, gives

$$\frac{d\mathcal{O}}{dt} = \frac{i}{\hbar}[\mathcal{H}, \mathcal{O}] + \frac{\partial \mathcal{O}}{\partial t}.\tag{4}$$

The hypothesis of O. Klein comprises the requirement of utilizing Equation (3) in Equation (1). Consequently, O. Klein assumes commutator relations for *P*, *Q*, *R*,

$$i\hbar P = RQ - \mathbb{Q}R, \quad i\hbar \mathbb{Q} = PR - RP, \quad i\hbar R = \mathbb{Q}P - PQ,\tag{5}$$

or using abbreviated nomenclature and the Levi-Civita symbol, with *εklm* = 1 for even permutations, and *εklm* = −1 for odd ones, otherwise *εklm* = 0 for identical indices, *k*, *l*, *m* = 1, 2, 3,

$$[f\_{k\prime}f\_l] = -i\hbar \varepsilon\_{klm} f\_m. \tag{6}$$

With the commutator relations in Equation (5), the correspondence principle leads to the equations of motion, and as O. Klein writes, "... as we overlook occurrence of the action-quant ...," viz. overlook *h*¯. Further, O. Klein remarks that Equation (5) differs only by the sign of *i* from the well-known quantum-mechanical commutators for a laboratory-fixed system. In summary, O. Klein's work concludes that a minus sign is required for consistency with classical mechanics and a result of the application of the correspondence principle.

Clearly, writing Equation (5) in the compact form of Equation (6) highlights the minus sign that differs from the standard equations of AM operators *Jk*, *k* = 1, 2, 3,

$$\left[f\_{k\prime}f\_{l}\right] = \ \imath\hbar\varepsilon\_{klm}f\_{m}.\tag{7}$$

The minus sign in Equation (6) is labeled "anomalous" by some authors, e.g., J. Van Vleck [2], but there is no justification for the anomalous minus sign to occur within QMT. Usually, one considers right-hand systems, so Equation (7) is termed as the standard quantum-mechanic AM operator identity. Sustenance of RAM concepts may appear convenient, even calling the negative sign an "anomaly" but without QMT support. In scientific approach and in spite of the initial success in explaining spectra within various approximations, one usually avoids starting with an "anomaly" and/or inaccurate presuppositions that are readily falsified [9]. However, several textbooks and works continue support of RAM in the theory of molecular spectra [10–22], in spite of obvious falsification by QMT. This work emphasizes that there is no need to resort to RAM "cook book" [22] methods.

The methods in this work utilize standard QMT [23,24] and standard mathematical methods [25], showing that there is no sign change of the standard commutator relations when transforming from a laboratory-fixed to a molecule-attached coordinate system. Consistent application of standard AM algebra in the establishment of computed spectra yield nice agreemen<sup>t</sup> with laboratory experimental results [5] and agreemen<sup>t</sup> in analysis of astrophysical C2 Swan data from the white dwarf Procyon B [5], including agreemen<sup>t</sup> in comparisons with computed spectra that are obtained with other molecular fitting programs such as PGOPHER [26].

Methods for measurement of optical emission signals from diatomic molecules are comprised of standard molecular spectroscopy experimental arrangements such as in laserinduced plasma or breakdown spectroscopy [27–34], encountered as well in stellar plasma physics or astrophysics to name other areas of interest. Particular interests in astrophysics include "cool" stars, brown dwarfs, and extra-solar planets, and the associated need for accurate theoretical models for ab initio calculations of diatomic molecular spectra, nicely reviewed recently [35].
