**3. Results**

### *3.1. Angular Momentum Commutators*

The invariance of standard QMT commutator relations (see Equation (7)) is communicated in this section.

### 3.1.1. Invariance for Unitary Transformations

Application of unitary transformation, viz., transforming from one coordinate system to another, leaves the AM commutator relations invariant [36]. A unitary transformation operator, *U*, acting on an operator O −→ O, with *U*† = *<sup>U</sup>*−1, is defined by

$$\mathcal{O}' = \mathsf{U}\mathcal{O}\mathcal{U}\mathcal{I}^{\dagger} \quad \text{or} \quad \mathcal{O} = \mathsf{U}^{\dagger}\mathcal{O}'\mathcal{U}.\tag{8}$$

The invariance of the AM commutators with respect to a unitary transformation, Equation (8),

$$[f\_{k\prime}f\_l] = i\varepsilon\_{klm}f\_m \qquad \longrightarrow \quad [f'\_{k\prime}f'\_l] = i\varepsilon\_{klm}f'\_{m\prime} \tag{9}$$

can be derived by inserting *Jk* = *U*† *Jk<sup>U</sup>* and *Jl* = *U*† *Jl<sup>U</sup>* in Equation (9) to obtain the intermediate step,

$$\mathcal{U}^{\dagger} f\_k' \mathcal{U} \mathcal{U}^{\dagger} f\_l' \mathcal{U} - \mathcal{U}^{\dagger} f\_l' \mathcal{U} \mathcal{U}^{\dagger} f\_k' \mathcal{U} = \mathcal{U}^{\dagger} f\_k' f\_l' \mathcal{U} - \mathcal{U}^{\dagger} f\_l' f\_k' \mathcal{U} = i \varepsilon\_{klm} \mathcal{U}^{\dagger} f\_m' \mathcal{U}. \tag{10}$$

Multiplying from left with *U* and from right with *U*−<sup>1</sup> yields the transformed identity in Equation (9). In other words, a unitary transformation preserves the quantum-mechanic AM commutators. For example, the Euler rotation matrix is easily demonstrated to be unitary [6]. In other words, there is no anomaly when going from a laboratory-fixed to a molecule-attached coordinate system.

#### 3.1.2. Invariance for Time Reversal or Reversal of Motion

Time reversal or reversal of motion in QMT requires sign changes of the operators and complex conjugation, leaving the QMT commutators invariant,

$$[f\_{k},f\_{l}] = i\varepsilon\_{klm}f\_{m} \quad \longleftrightarrow \quad [(-f\_{k})\_{\prime}(-f\_{l})] = (-i)\varepsilon\_{klm}(-f\_{m}).\tag{11}$$

CM would indicate a reversal of motion when going from a laboratory-fixed to a molecular-fixed coordinate system; however, reversal of motion requires complex conjugation due to the anti-unitary requirement. In other words, the sign is preserved. QMT so-to-speak opposes the hypothesis by O. Klein.

The invariance regarding time reversal or reversal of motion of course also would apply to the abstract form of the time-dependent Schrödinger equation,

$$i\hbar\frac{\partial}{\partial t}\psi = \mathcal{H}\psi \quad \longleftrightarrow \quad (-i)\hbar\frac{\partial}{\partial (-t)}\psi = \mathcal{H}\psi,\tag{12}$$

where *ψ* describes an abstract vector in Hilbert space, and H is a Hamiltonian. Changing time *t* −→ −*t* and applying conjugate complex of *i* preserves the left-hand side of the equation. For example, for a free particle of mass *m* and momentum P, the Hamiltonian is H = P2/2*<sup>m</sup>*, and the form of Schrödinger's equation is preserved.

Equally, the operator equation in the Heisenberg picture, see Equation (4), preserves form under time reversal or reversal of motion,

$$\frac{d\mathcal{O}}{dt} = \frac{i}{\hbar}[\mathcal{H}, \mathcal{O}] + \frac{\partial \mathcal{O}}{\partial t} \quad \longleftrightarrow \quad \frac{d(-\mathcal{O})}{d(-t)} = \frac{(-i)}{\hbar}[\mathcal{H}, (-\mathcal{O})] + \frac{\partial(-\mathcal{O})}{\partial(-t)}.\tag{13}$$

A change of sign for the operators and complex conjugation leaves the equation invariant. The mentioned symmetry can also be associated with usual Nöther symmetries [3].

### *3.2. Diatomic Wave Function*

For diatomic molecules, symmetry properties allow one to invoke simplifications when evaluating the laboratory wave-function in terms of rotated coordinates [5]. For internuclear geometry, the spherical polar coordinates are *r*, *φ*, and *θ*, and one (arbitrary) electron is described by cylindrical coordinates *ρ*, *χ*, *ζ*. For coordinate rotation, one uses

Euler angles *α*, *β*, *γ*, and without loss of generality, one can choose *α* = *φ*, *β* = *θ*, *χ* = *γ* [5]. The result is the Wigner–Witmer eigenfunction (WWE) for diatomic molecules [37,38],

$$
\langle \rho\_{\prime} \tilde{\zeta}\_{\prime}, \chi\_{\prime} \mathbf{r}\_{2}, \dots, \mathbf{r}\_{N\prime} r\_{\prime} \theta\_{\prime} \phi \mid n v \rangle M \rangle = \sum\_{\Omega = - \parallel}^{\parallel} \langle \rho\_{\prime} \tilde{\zeta}\_{\prime}, \mathbf{r}\_{2}^{\prime}, \dots, \mathbf{r}\_{N\prime}^{\prime} r \mid n v \rangle \, D\_{M\Omega}^{\prime \ast} (\phi\_{\prime} \theta\_{\prime} \chi) . \tag{14}$$

The usual total AM quantum numbers are *J* and *M*, and the electronic–vibrational eigenfunction is explicitly written by extracting *v* from the collection of quantum numbers, *n*. The WWE exactly separates *φ*, *θ*, *χ*. The quantum numbers *J*, *M*, Ω refer to the total AM. The sum over Ω in Equation (14) originates from the usual abstract transformation,

$$
\langle JM \rangle = \sum\_{\Omega = -J}^{J} |J\Omega\rangle \langle J\Omega | JM \rangle , \tag{15}
$$

where Ω is the magnetic quantum number along the rotated, or new, *z*-axis. The sum in Equation (15) ensures that the quantum numbers for total AM are *J* and *M*. In Hund's case *a* [39], Ω describes the projection of the total AM, within L-S coupling. Hund's case *a* eigenfunctions form a basis; therefore, from a computational point of view, these eigenfunctions form a complete (sufficient) set. In various approximate descriptions and for specific diatomic molecules, it may be desirable to use other Hund cases.

From the rotation operator R(*<sup>α</sup>*, *β*, *<sup>γ</sup>*), with the Euler angles *α*, *β*, *γ*, one finds for D-matrix elements,

$$D\_{M\Omega}^{J^\*} (\mathfrak{a}, \mathfrak{z}, \gamma) = \langle IM | \mathcal{R} (\mathfrak{a}, \mathfrak{z}, \gamma) | l\Omega \rangle^\*. \tag{16}$$

D-matrices are the usual mathematical tool for transformation from one basis to another, but the D-matrix cannot represent an eigenfunction due to presence of two magnetic quantum numbers *M* and Ω, so the sum over Ω is needed in the transformed coordinates.

Diatomic spectra composed of line positions and line strengths are based on WWE [5] instead of eigenfunctions used for the Born-Oppenheimer approximation. Extensive experimental studies confirm agreemen<sup>t</sup> of computed spectra with measured emission spectra from laser-induced optical plasma [5].

### *3.3. Selected Diatomic Spectra*

Typical spectra of some diatomic molecules of general interest are presented. Figure 1 illustrates OH molecular spectra for different spectral resolutions. Figures 2 and 3 show computed C2 Swan spectra for the vibrational sequences Δ*ν* = −1, +1. The OH spectra, Figure 1, are a superposition of 0-0 (band head near 306 nm), 1-1 (band head near 312 nm), and 2-2 (band head near 318 nm) vibrational transition along with rotational contributions. Four C2 vibrational peaks, Figures 2 and 3, are clearly discernible. Rotational contributions for the selected spectral resolution, Δ*λ*, appear to have beats (especially Figure 2) that, however, are purely coincidental.

The details for the computation, line strength data for *C*2 Swan bands, and programs are published [40]. Computation of diatomic spectra utilizes high-resolution data for determination of molecular constants of selected molecular transitions from an upper to a lower energy level. Numerical solution of the Schrödinger equation for potentials yield r-centroids and transition-factors associated with vibrational transitions, viz. Franck– Condon factors. Calculated rotational factors are interpreted as selection rules because these factors are zero for forbidden transitions, viz. Hönl–London factors. Hönl–London factors in traditional molecular spectroscopy involve selection rules that may require use of anomalous commutators and use of two magnetic quantum numbers *M* and Ω for a given total angular momentum *J*. Anomalous selection rules and two quantum numbers for angular momentum *J* appear to be associated with approximations. The published line strength data [40,41] are derived consistent with standard quantum mechanics, in other words, without anomalous commutators and without states that have two magnetic quantum numbers associated with angular momentum.

**Figure 1.** Computed spectrum of the *A*2Σ → *X*2Π uv band of OH, T = 4 k K, (top) spectral resolutions of Δ*λ* = 0.32 nm (Δ*ν*˜ = 32 cm<sup>−</sup>1) and (bottom) idealized resolution for the stick spectrum Δ*λ* = 0.002 nm (Δ*ν*˜ = 0.2 cm<sup>−</sup>1) of the Δ*ν* = 0 sequence (adapted from [6]).

**Figure 2.** C2 Swan *<sup>d</sup>*3Π*g* → *<sup>a</sup>*3Π*u* band Δ*ν* = −1 sequence, T = 8 kK, Δ*λ* = 0.13 nm (Δ*ν*˜ = 6 cm<sup>−</sup>1) (adapted from [6]).

**Figure 3.** C2 Swan *<sup>d</sup>*3Π*g* → *<sup>a</sup>*3Π*u* band Δ*ν* = +1 sequence, T = 8 kK, Δ*λ* = 0.18 nm (Δ*ν*˜ = 6 cm<sup>−</sup>1) (adapted from [6]).

The published program package [40] also includes a worked high-temperature cyanide example, the Boltzmann equilibrium spectrum program (BESP) for computation of equilibrium spectra, and the Nelder–Mead temperature (NMT) routine that utilizes a non-linear fitting algorithm. The OH line strength data have been made available recently [41].

Various reported studies of plasma spectra, including astrophysics plasma, and of molecular laser-induced breakdown spectroscopy (LIBS) [40–43] illustrate nice comparisons of recorded and of computed diatomic spectra. In LIBS, plasma generated by focusing coherent radiation is analyzed primarily in visible/optical or in near-uv to near-ir regions. After initiation of optical breakdown with typically 10 nanoseconds, 100 mJ laser pulses focused in standard ambient temperature and pressure (SATP) air or in gas mixtures [42], molecule formation including, for example, OH in air, C2 in carbon monoxide, and CN in 1:1 molar N2:CO2 mixture, leads to recombination radiation that is typically measured using time-resolved optical emission laser spectroscopy. When using a metallic target, other diatomic molecules can be investigated, e.g., TiO or AlO, and molecular spectra can be computed from line strength data [40].

### **4. Discussion and Conclusions**

Angular momentum operators are well defined in quantum mechanics theory, including the fact that there is an inherent limit in measurement of its components. Another way of formulating this could be: There are only two quantum numbers needed for description of angular momentum, usually the total angular momentum and its projection onto a quantization axis. The use of the correspondence principle to ensure compatibility with classical mechanics equations of motion brings about an ad hoc hypothesis of a negative sign for the commutators, as originally communicated by Oscar Klein in 1929. Subsequent application of reversed angular momentum coupling continues to find support in analytic description of molecules that also includes modeling of quantum mechanic vector-operators as vectors.

However, quantum mechanics theory already ensures how to mathematically describe angular momentum, not supporting heuristic conclusions involving reversed angular momentum concepts, nor occurrence of more than two quantum numbers for the total angular momentum of diatomic molecules. This review emphasizes that there is no mathematical justification of reversed angular momentum algebra, and it also discusses applications in diatomic molecular spectroscopy. Consistent application of standard quantum mechanics theory is preferred, including avoidance of a priori use of separating electronic, vibrational, rotational wave functions. Subsequent to implementation of diatomic molecular symmetries, line strengths for selected diatomic molecules that contain effects of spin splitting and lambda-doubling as function of wavelength are in agreemen<sup>t</sup> with results from optical emission spectroscopy. The computed and fitted diatomic spectra nicely match within reasonable error bars, but without invoking heuristic selection rules that may be affected by initial approximations or by spurious use of reversal of angular momentum.

**Funding:** This research received no specific grant-number external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data for Figures 1–3 can be computed using provided programs and published line strengths for C2 and OH in Refs. [39,40], respectively.

**Acknowledgments:** The author acknowledges support in part from the State of Tennessee funded Center for Laser Applications at the University of Tennessee Space Institute.

**Conflicts of Interest:** The author declares no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.
