^{2}University of Tennessee, University of Tennessee Space Institute, Center for Laser Applications, Tullahoma, TN, USA
Keywords: Parity; Molecular spectroscopy; Diatomic molecules;
In this work, the standard framework of parity and angular momentum methods are applied. The effect of parity on the prediction of heteronuclear diatomic molecular spectra is presented. Subsequently, the algorithm for the computation of molecular spectra is described. As a specific example, results for nitric oxide spectra are compared with experimental data.
The variables $\rho ,\text{}\zeta $ and $\text{r}$ are scalars, unaffected by rotations. The physical rotation $\theta $ and the angle of coordinate rotation $\alpha $ are also about the same axis, namely the first intermediate $y$ axis of the full coordinate rotation. The angles $\chi $ and $\gamma $ are both rotations about the ${z}^{\prime}$ axis. Thus, $${\phi}^{\prime}\mathrm{=}\phi \alpha \mathrm{,}\text{\hspace{0.17em}}{\theta}^{\prime}\mathrm{=}\theta \beta \mathrm{,}\text{\hspace{0.17em}}{\chi}^{\prime}\mathrm{=}\chi \gamma \mathrm{.}\text{(4)}$$ In coordinate rotations, one is at liberty to choose $\alpha ,\text{}\beta $ and $\gamma $. If one chooses for the angles $\alpha \mathrm{=}\phi ,\text{}\beta \mathrm{=}\theta $, , all angular dependence of $\langle \rho \mathrm{,}\zeta \mathrm{,}{\chi}^{\prime}\mathrm{,}r{\mathrm{\text{'}}}_{2}\mathrm{,}\dots \mathrm{,}r{\mathrm{\text{'}}}_{N}\mathrm{,}\text{r}\mathrm{,}{\theta}^{\prime}\mathrm{,}{\phi}^{\prime}\text{\hspace{0.05em}}\mathrm{}nvJ\Omega \rangle $ is removed. This yields the Wigner and Witmer [7] diatomic eigenfunction, $$\langle \rho \mathrm{,}\zeta \mathrm{,}\chi \mathrm{,}{r}_{2}\mathrm{,}\dots \mathrm{,}{r}_{n}\mathrm{,}\text{r}\mathrm{,}\theta \mathrm{,}\phi \text{\hspace{0.05em}}\mathrm{}nJM\rangle \mathrm{=}{\displaystyle \sum _{\Omega \mathrm{=}J}^{J}}\langle \rho \mathrm{,}\zeta \mathrm{,}r{\mathrm{\text{'}}}_{2}\mathrm{,}\dots \mathrm{,}r{\mathrm{\text{'}}}_{N}\mathrm{,}\text{r}\text{\hspace{0.05em}}\mathrm{}nv\rangle \text{\hspace{0.05em}}{D}_{M\Omega}^{{J}^{\mathrm{*}}}\mathrm{(}\phi \mathrm{,}\theta \mathrm{,}\chi \mathrm{).}\text{(5)}$$ The values of the quantum numbers, $J$ and $\Omega $ influence the electronicvibrational eigenfunction $\langle \rho \mathrm{,}\zeta \mathrm{,}r{\mathrm{\text{'}}}_{2}\mathrm{,}\dots \mathrm{,}r{\mathrm{\text{'}}}_{N}\mathrm{,}r\text{\hspace{0.05em}}\mathrm{}nv\rangle $, but the electronicvibrational eigenfunction is not an angular momentum state vector.
Consider the algorithm for computation of the wavelengths and intensities in the spectrum of a molecule from the first principles of quantum mechanics. The upper ${H}^{\prime}$ and lower $H$ Hamiltonian matrices are computed and numerically diagonalized by unitary matrices, ${U}^{\prime}$ and $U$. The upper ${{F}^{\prime}}_{{n}^{\prime}{v}^{\prime}{J}^{\prime}}$ and lower ${F}_{nvJ}$ terms are the eigenvalues of the Hamiltonians, $${{F}^{\prime}}_{{n}^{\prime}{v}^{\prime}{J}^{\prime}}\mathrm{=}{{U}^{\prime}}^{\u2020}{H}^{\prime}\text{\hspace{0.05em}}{U}^{\prime}\text{(9)}$$ $${F}_{nvJ}\mathrm{=}{U}^{\u2020}H\text{\hspace{0.05em}}U\mathrm{,}\text{(10)}$$ and the vacuum wavenumbers, $\tilde{\nu}$, of the predicted spectral lines, $$\tilde{\nu}\mathrm{=}{{F}^{\prime}}_{{n}^{\prime}{v}^{\prime}{J}^{\prime}}{F}_{nvJ}\mathrm{,}\text{(11)}$$ are term differences. Of the very large number of computed term differences only those for which the Condon and Shortley [8] line strength does not vanish are spectral lines. The line strength, $S\mathrm{(}nvJ\mathrm{,}{n}^{\prime}{v}^{\prime}{J}^{\prime}\mathrm{)},$, is the sum over all $M$ and ${M}^{\prime}$ of the irreducible tensor ${T}_{k}^{\mathrm{(}q\mathrm{)}}$ expectation values, $\langle nvJM\text{\hspace{0.05em}}\mathrm{}{T}_{k}^{\mathrm{(}q\mathrm{)}}\mathrm{}{n}^{\prime}{v}^{\prime}{J}^{\prime}{M}^{\prime}\rangle .$ The exact separation of the total angular momentum in the Wigner Witmer diatomic eigenfunction results in a diatomic line strength composed of two parts, $$S\mathrm{(}nvJ\mathrm{,}{n}^{\prime}{v}^{\prime}{J}^{\prime}\mathrm{)=}S\mathrm{(}nv\mathrm{,}{n}^{\prime}{v}^{\prime}\mathrm{)}\text{\hspace{0.05em}}S\mathrm{(}J\mathrm{,}{J}^{\prime}\mathrm{),}\text{(12)}$$ the electronicvibrational strength, $S\mathrm{(}nv\mathrm{,}{n}^{\prime}{v}^{\prime}\mathrm{)},$ and the unitless rotational line strength or HönlLondon factor, $S\mathrm{(}J\mathrm{,}{J}^{\prime}\mathrm{)}$. The BornOppenheimer approximation separates the electronicvibrational strength into electronic and vibrational parts. In the Hund’s case (a) basis built from the WignerWitmer eigenfunction, the third Euler angle, $\chi \mathrm{=}\gamma $, appears in the Wigner $D$ function, $$\mathrm{}a\rangle \mathrm{=}nvJM\Omega S\Sigma \rangle \mathrm{=}\sqrt{\frac{2J+1}{8{\pi}^{2}}}\langle \rho \mathrm{,}\zeta \mathrm{,}r{\mathrm{\text{'}}}_{2}\mathrm{,}\dots \mathrm{,}r{\mathrm{\text{'}}}_{N}\mathrm{,}\text{r}\text{\hspace{0.05em}}\mathrm{}nv\rangle \text{\hspace{0.05em}}\mathrm{}S\Sigma \rangle \text{\hspace{0.05em}}{D}_{M\Omega}^{{J}^{\mathrm{*}}}\mathrm{(}\phi \mathrm{,}\theta \mathrm{,}\chi \mathrm{).}\text{(13)}$$ The algorithm for computation of the vacuum wavenumbers, $\tilde{\nu}$, and line strengths, $S\mathrm{(}nvJ\mathrm{,}{n}^{\prime}{v}^{\prime}{J}^{\prime}\mathrm{)},$ of diatomic spectral lines is a straightforward application of quantum mechanics, but except for the very simplest molecules is also very far removed from the realm of the possible. However, with two very stringent caveats, the algorithm can be implemented for the diatomic molecule. The first caveat is that the vacuum wavenumbers, $\tilde{\nu}$, for many spectral lines in many bands of a band system must have been experimentally measured with high accuracy such as that provided by Fourier transform spectroscopy. Secondly, using semiempirical molecular constants one must be able to build upper and lower Hamiltonian matrices whose eigenvalue differences accurately predict the measured vacuum wavenumbers. A fitting process is required [1]. One assumes trial values for the molecular constants, computes the spectral lines positions, $\tilde{\nu}$, and from the differences between $\tilde{\nu}{\tilde{\nu}}_{\text{e}xp}$ finds the corrections to the molecular parameters. The difference between computed and measured line positions will typically equal the measurement error margins.
The Hund’s case (a) basis is mathematically complete. A sum of basis functions, $\mathrm{}a\rangle $, can be quantitatively very accurate. The parity operator, P, commutes with the Hamiltonian. Thus, the orthogonal matrix that diagonalizes the case (a) representation of the Hamiltonian will also diagonalize the case (a) representation of P.
$J$ 

$p$ 
${F}_{{J}^{\prime}}$ 
${F}_{J}$ 
$\tilde{\nu}$ 
$S\mathrm{(}J\mathrm{,}{J}^{\prime}\mathrm{)}$ 
$\tilde{\nu}{\tilde{\nu}}_{\text{e}xp}$ 



(cm^{1}) 
(cm^{1}) 
(cm^{1}) 

(cm^{1}) 
24.5 
${P}_{21}$ 
$f$ 
49111.456 
982.866 
48128.590 
0.813 
0.021 
24.5 
${P}_{21}$ 
$+e$ 
49111.501 
982.588 
48128.914 
0.812 
0.078 
24.5 
${P}_{11}$ 
$f$ 
48952.618 
822.538 
48130.080 
21.749 
0.014 
24.5 
${P}_{11}$ 
$+e$ 
48952.673 
822.282 
48130.390 
21.805 
0.020 
24.5 
${P}_{22}$ 
$+f$ 
48876.118 
744.504 
48131.615 
18.832 
0.007 
24.5 
${P}_{22}$ 
$e$ 
48876.146 
744.505 
48131.641 
18.786 

24.5 
${P}_{12}$ 
$+f$ 
48747.194 
614.306 
48132.888 
0.515 
0.013 
24.5 
${P}_{12}$ 
$e$ 
48747.244 
614.307 
48132.936 
0.527 
0.012 
24.5 
${R}_{12}$ 
$f$ 
48952.618 
814.699 
48137.919 
0.881 
0.034 
24.5 
${R}_{12}$ 
$+e$ 
48952.673 
814.701 
48137.971 
0.880 
0.077 
24.5 
${R}_{11}$ 
$+f$ 
49210.930 
1068.021 
48142.908 
25.424 
0.034 
24.5 
${R}_{11}$ 
$e$ 
49210.985 
1067.731 
48143.254 
25.317 
0.018 
A single selection rule handles all types of diatomic spectra. If the HönlLondon factor, $S\mathrm{(}J\mathrm{,}{J}^{\prime}\mathrm{)}$ is nonvanishing, then the transition is allowed. Parity plays no part in the fitting process which determines the molecular parameters, but the parity eigenvalues are computed from the finalized values of the molecular parameters. The presented algorithm can be used to predict molecular spectra for the purpose of fitting measured data [9].
 Zare RN, Schmeltekopf AL, Harrop WJ, Albritton DL. A direct approach for the reduction of diatomic spectra to molecular constants for the construction of RKR potentials. J Mol Spectrosc. 1973;46(1):37–66.
 Brown JM, Hougen JT, Huber KP, Johns JWC, Kopp I, LefebvreBrion H, et al. The labeling of parity doublet levels in linear molecules. J Mol Spectrosc. 1975;55(13):500–503.
 Hougen JT. The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules. National Institute of Standards and Technology: Boulder, CO. 2001.
 Røeggen I. The inversion eigenvalues of nonσ states of diatomic molecules, expressed in terms of quantum numbers. Theor Chim Acta. 1971;21(4):398–409. doi:10.1007/BF00528562
 Judd B. Angular Momentum Theory for Diatomic Molecules. Academic Press: New York, NY. 1975.
 Larsson M. Phase Conventions for Rotating Diatomic Molecules. Phys Scr. 1981;23(5A):835–836.
 Wigner E, Witmer EE. On the structure of the diatomic molecular spectra according to quantum mechanics. Z Phys. 1928;51(11):859–886. Hettema H, editor. On the structure of the spectra of twoatomic molecules according to quantum mechanics. Quantum Chemistry: Classic Scientific Papers. World Scientific: Singapore. 2000;287311. doi: 10.1142/9789812795762_0018
 Condon EU, Shortley GH. The Theory of Atomic Spectra.Cambridge Univ. Press, Cambridge, UK. 1953.
 Parigger CG, Woods AC, Surmick DM, Gautam G, Witte MJ, Hornkohl JO. Computation of diatomic molecular spectra for selected transitions of aluminum monoxide, cyanide, diatomic carbon, and titanium monoxide. Spectrochim Acta Part B At Spectrosc. 2015;107:132–138.
 Faris GW, Cosby PC. Observation of NO B 2Π(v=3)←X 2Π(v=0) absorptions with 1+1 multiphoton ionization: Precision line position measurements and parity assignment of the B 2Π state. J Chem Phys. 1992;97(10):7073–7086.