Research Article
Open Access
Rotational line strengths for the cyanide B^{2}Σ  X^{2}Σ^{+} (5,4)
band
James O Hornkohl^{1} and Christian G Parigger^{2*}
^{1}Hornkohl Consulting, 344 Turkey Creek Road, Tullahoma, TN, USA
^{2}University of Tennessee, University of Tennessee Space Institute, Center for Laser Applications, Tullahoma, TN, USA
*Corresponding author: Christian Parigger, Associate Professor, University of Tennessee, University of Tennessee Space Institute, Center for
Laser Applications, 411 B.H. Goethert Parkway, Tullahom, TN 373889700, USA, Tel: (931)3937338/509; Email:
@
Received: March 03, 2017; Accepted: April 06, 2017; Published: April 13, 2017
Citation: Christian Parigger, James Hornkoh (2017) Rotational line strengths for the cyanide B2Σ – X 2Σ+ (5,4) band . Int J Mol Theor Phy.(1):16
Rotational line strengths, computed from
eigenvectors of Hund’s case (a) matrix representations of the
upper and lower Hamiltonians using WignerWitmer basis
functions, show a larger than expected influence from the
wellknown perturbation in the (5,4) band. Comparisons with
National Solar Observatory experimental Fourier transform
spectroscopy data reveal nice agreement of measured and
predicted spectra.
Keywords: Diatomic Spectroscopy; Rotational line
strengths; HönlLondon factors; Cyanide spectra violet band
perturbations;
Introduction
The CN violet
$B{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}X{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}$
band system is one of the
most studied band systems. Ram, et al. [1] and Brooke, et al.
[2] reported experimental results and theoretical information,
respectively. Of the many known bands in the violet system, only
the (5,4) band is considered here. This band exhibits a weak,
quantitatively understood perturbation [4] caused by mixing of
the $v\mathrm{=17}$
level of
$A{\text{\hspace{0.05em}}}^{2}\Pi $
with the $v\mathrm{=5}$
level of
$B{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}$
. The particular
perturbation of the CN (5,4) band is evaluated in this work by
isolating the spectral features of this band that is part of the CN
violet system.
Methods
Numerical diagonalizations of upper and lower
Hamiltonians with and without the perturbation are investigated
and compared with available experimental spectra. The
simulations rely on determining rotational strengths without
paritypartitioned Hamiltonians. It is anticipated that the
investigated (5,4) band modifications can be possibly confirmed
with the new PGOPHER program recently released by Western
[3].
For the computation of rotational spectra, the square
of transition moments are numerically computed using the
eigenvectors of upper and lower Hamiltonians. This approach
can also be selected in the new PGOPHER program [3]. For the
diatomic molecule, the results effectively yield the HönlLondon
factors yet we do not utilize tabulated HönlLondon factors that
are available in standard textbooks.
Results
CN (5,4) band spectra
Table 1: Lines in the CN $B{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}X{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}$
(5,4) band near the perturbation.
$\tilde{\nu}$
are the fitted line positions, $S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$
are the rotational line strengths computed in the fitting algorithm.
Without spinorbit mixing, ${S}^{\mathrm{(0)}}\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$
and $\Delta {\tilde{\nu}}^{\mathrm{(0)}}$
are the line strengths and differences of
the fitted line positions, respectively. Spinorbit mixing of
$B{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}$
and $A{\text{\hspace{0.05em}}}^{2}\Pi $
shifts the upper
$e$
parity levels, and significantly reduces the differences, $\Delta {\tilde{\nu}}^{}$
, between measured and
computed line positions.
${J}^{\prime}$

$J$


${p}^{\prime}$

$\tilde{\nu}$

${S}_{{J}^{\prime}J}$

$\Delta \tilde{\nu}$

${S}_{{J}^{\prime}J}^{\mathrm{(0)}}$

$\Delta {\tilde{\nu}}^{\mathrm{(0)}}$

9 ½ 
8 ½ 
${R}_{11}$

$e$

28013.117 
9.474 
0.010 
9.474 
0.337 
9 ½ 
8 ½ 
${R}_{22}$

$+f$

28017.421 
9.474 
0.001 
9.474 
0.059 
10 ½ 
9 ½ 
${R}_{11}$

$+e$

28016.992 
9.199 
0.004 
10.476 
0.600 
10 ½ 
9 ½ 
${R}_{22}$

$f$

28021.651 
11.171 
0.000 
10.476 
0.067 
11 ½ 
10 ½ 
${R}_{11}$

$e$

28020.540 
7.868 
0.041 
11.478 
1.193 
11 ½ 
10 ½ 
${R}_{22}$

$+f$

28025.866 
12.240 
0.006 
11.478 
0.067 
12 ½ 
11 ½ 
${R}_{22}$

$f$

28030.125 
13.288 
0.007 
12.480 
0.072 
12 ½ 
11 ½ 
${R}_{11}$

$+e$

28030.431 
13.812 
0.000 
12.480 
0.000 
13 ½ 
12 ½ 
${R}_{11}$

$e$

28032.081 
17.455 
0.053 
13.481 
1.870 
13 ½ 
12 ½ 
${R}_{22}$

$+f$

28034.428 
14.325 
0.011 
13.481 
0.073 
14 ½ 
13 ½ 
${R}_{11}$

$+e$

28035.672 
17.919 
0.005 
14.483 
1.102 
14 ½ 
13 ½ 
${R}_{22}$

$f$

28038.773 
15.356 
0.013 
14.483 
0.076 
15 ½ 
14 ½ 
${R}_{11}$

$e$

28039.742 
18.442 
0.007 
15.484 
0.807 
15 ½ 
14 ½ 
${R}_{22}$

$+f$

28043.161 
16.383 
0.009 
15.484 
0.084 
16 ½ 
15 ½ 
${R}_{11}$

$+e$

28043.989 
19.132 
0.011 
16.485 
0.655 
16 ½ 
15 ½ 
${R}_{22}$

$f$

28047.590 
17.405 
0.006 
16.485 
0.091 
Figure 1: Synthetic emission spectra. (a) pure upper states ${}^{2}{\Sigma}^{+}$
; (b) upper
states are treated as the sum
${c}_{\text{\hspace{0.05em}}\Sigma}{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}+{c}_{\Pi}{\text{\hspace{0.05em}}}^{2}\Pi $
with ${c}_{\text{\hspace{0.05em}}\Sigma}>>{c}_{\Pi}$
Only R branch lines are shown, including those given in Table 1.
Figure 2: Computed spectra for P and R branches. (a) pure and (b)
perturbed upper states.
Table 1 and Figures 1 and 2 show the results obtained
with and without taking into account the mixing. In Table 1, lines
of the CN (5,4) band are listed that are spectrally close to the
perturbation. Comparsions of line positions,
$\tilde{\nu}$, and the rotational
line strengths, $S({J}^{\prime}\mathrm{,}J\mathrm{)}$ with the corresponding unperturbed
values, ${S}^{\mathrm{(0)}}\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$, are presented. The differences, $\Delta {\tilde{\nu}}^{\mathrm{(0)}}$, for which the
offdiagonal spinorbit coupling constants < AL +> and < BL+> are
set equal to 0, are significantly larger than the usual differences,
$\Delta {\tilde{\nu}}^{}$, of computed and experimentally determined line positions.
The spinorbit mixing of $B{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}$ and $A{\text{\hspace{0.05em}}}^{2}\Pi $ shifts the upper
parity levels. Table 3 also shows that a relatively large fractional
error, e.g., 3.974/17.455 versus 1.870/28032 for ${R}_{11}\mathrm{(}J=\mathrm{12.5)}$ can occur in the computed rotational line strengths,
$S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$ Figure 1 displays the computed spectra of only the branch in
the range of 27980 to 28070 cm1, including the lines listed in
Table 1 with and without spinorbit mixing. The
$v\mathrm{=17,}A\text{\hspace{0.05em}}{\text{\hspace{0.05em}}}^{2}\Pi $ energy eigenvalues lie very near the
$v\mathrm{=5,}B{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}$ eigenvalues,
and this causes a significant effect from the
$A{\text{\hspace{0.05em}}}^{2}\Pi $ level. Figure 2
illustrates computed spectra for P and R branches in the range
of 27905 to 28070 cm^{1} for pure and perturbed upper states. The
perturbed states are affected by the addition of a small amount
of ${}^{2}\Pi $ states to the upper level basis. Clearly, the perturbations
reveal noticeable differences in the appearance of the violet (5,4)
band even at low, 2.0 cm^{1}, resolution. Results of modeling
the angular momentum states of the upper $v\mathrm{=5}$ vibrational level as a mixture of ${}^{2}\Sigma $ and ${}^{2}\Pi $ Hund’s case (a)
basis functions, a socalled “deperturbation” or perturbation
analysis, agree well that of Ito, et al. [4] who used the line position
measurements of Engleman [5]. The 100 lines of the more recent
data of Ram, et al. [1] were fitted with a standard deviation of
cm . The standard deviation would be increased from 0.025 to
0.25 cm^{1} without the inclusion of spinorbit mixing of the
$B{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}$ and $A{\text{\hspace{0.05em}}}^{2}\Pi $ basis states.
Changes in the spectra are relatively larger for the
rotational line strengths, $S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$, for the line positions, $\tilde{\nu}$. The
simulation results compare nicely with measured spectra [1]
available from the National Solar Observatory (NSO) at Kitt
Peak [6]. Figure 3 displays the recorded and simulated spectra
for a resolution of 0.03 cm^{1} . The Fourier transform spectrum
920212R0.005 [6] was recorded [1] at a temperature of 20.5
degree Celsius, or at 293.65 Kelvin, and at a spectral resolution of
0.03 cm^{1}. The corresponding R branch spectrum is computed for
a temperature of 300 K and it compares nicely with the recorded
data. The predicted line positions of the R branch match the
vacuum wavenumbers of the experimental spectrum.
Figure 3: Measured and simulated spectra. (a) Segment of the recorded
[1] Fourier transform spectrum 920212R0.005 [6], (b) computed
spectrum for a temperature of 300 K and a spectral resolution of 0.03
cm^{1} . The computed (5,4) R branch is flipped vertically for ease of comparisons.
The influence of ${}^{2}{\Sigma}^{+}+{\text{\hspace{0.05em}}}^{2}\Pi $ mixing on the rotational
line strengths, $S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$, can be recognized because computation
of $S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$ is an integral part of the unique line position fitting
algorithm. Upper and lower Hamiltonian matrices in the Hund’s
case (a) basis are numerically diagonalized, and the spectral line
vacuum wavenumber $\tilde{\nu}$ is the difference between upper and
lower Hamiltonian eigenvalues. To determine which of the many
eigenvalue differences represent allowed spectral lines, the factor
$S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$ is computed from the upper and lower eigenvectors for
each eigenvalue difference. A nonvanishing $S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$
denotes an allowed diatomic spectral line. Parity partitioned effective
Hamiltonians are not used. Parity and branch designation are not
required in the fitting algorithm. Input data to the fitting program
is a table of vacuum wavenumber
$\tilde{\nu}$ versus ${J}^{\prime}$ and $J$. The nonvanishing of the rotational strength is the only selection
rule used. Applications of this rule leads to the establishment
of spectral data bases for diatomic molecular spectroscopy of
selected transitions [7]. Over and above the PGOPHER program
[3], there are other extensive efforts in predicting diatomic
molecular spectra including for instance the socalled Duo
program [8] for diatomic spectroscopy.
WignerWitmer diatomic eigenfunction
The Hund’s case (a) basis functions were derived from
the Wigner and Witmer [9] diatomic eigenfunction,
$\langle \rho \mathrm{,}\zeta \mathrm{,}\chi \mathrm{,}{r}_{2}\mathrm{,}\dots \mathrm{,}{r}_{N}\mathrm{,}r\mathrm{,}\theta \mathrm{,}\phi \text{\hspace{0.05em}}\mathrm{}nvJM\rangle \mathrm{=}{\displaystyle \sum _{\Omega \mathrm{=}J}^{J}}\langle \rho \mathrm{,}\zeta \mathrm{,}r{\mathrm{\text{'}}}_{2}\mathrm{,}\dots \mathrm{,}r{\mathrm{\text{'}}}_{N}\mathrm{,}r\text{\hspace{0.05em}}\mathrm{}nv\rangle \text{\hspace{0.05em}}{D}_{M\Omega}^{{J}^{\mathrm{*}}}\mathrm{(}\phi \mathrm{,}\theta \mathrm{,}\chi \mathrm{).}\text{(1)}$
The coordinates are: The distance
$\rho $
of one electron (the electron arbitrarily labeled 1 but it could be any one of the
electrons) from the internuclear vector
$r\mathrm{(}r\mathrm{,}\theta \mathrm{,}\phi \mathrm{)}$ the distance $\zeta $
of that electron above or below the plane perpendicular to r,
and passing through the center of mass of the two nuclei (the
coordinate origin), the angle $\chi $ for rotation of that electron
about the internuclear vector r, and the remaining electronic
coordinates ${r}_{2}\mathrm{,}\dots \mathrm{,}{r}_{N}$ in the fixed and $r{\mathrm{\text{'}}}_{2}\mathrm{,}\dots \mathrm{,}r{\mathrm{\text{'}}}_{N}$ in the rotating coordinate system.
The vibrational quantum number
$v$ has been extracted from the quantum numbers collection
$n$ that represents all
required quantum numbers except
$J,\text{}M,\text{}\Omega ,$ and $v$.
The WignerWitmer diatomic eigenfunction has no application in
polyatomic theory, but for the diatomic molecule the exact
separation of the Euler angles is a clear advantage over the Born
Oppenheimer approximation for the diatomic molecule in which
the angle of electronic rotation,
$\theta $ and $\phi $.
Equation (1) can be derived by writing the general equation
for coordinate (passive) rotations
$\alpha ,\text{}\beta \text{and}\gamma $ of the eigenfunction, replacing two
generic coordinate vectors with the diatomic vectors
$r\mathrm{(}r\mathrm{,}\theta \mathrm{,}\phi \mathrm{)}$ and $r\mathrm{\text{'}(}\rho \mathrm{,}\zeta \mathrm{,}\chi \mathrm{)}$
and equating the angles of coordinate rotation to
the angles of physical rotation
$\phi ,\text{}\theta ,\text{and}\phi .$
The general equation for coordinate rotation holds in isotropic space, and therefore
the quantum numbers
$J,\text{}M,\text{and}\Omega $
in the WignerWitmer
eigenfunction include all electronic and nuclear spins. If nuclear
spin were to be included,
$J,\text{}M,\text{and}\Omega $
would be replaced by $F,\text{}{M}_{F}$ and ${\Omega}_{F}$, but hyperfine structure is not resolved in the (5,4) band
data reported by [1], and Eq. (1) is written with the appropriate
spectroscopic quantum numbers.
It is worth noting that the rotation matrix element
${D}_{M\Omega}^{J}\mathrm{(}\phi \mathrm{,}\theta \mathrm{,}\chi \mathrm{)}$
and its complex conjugate
${D}_{M\Omega}^{{J}^{\mathrm{*}}}\mathrm{(}\phi \mathrm{,}\theta \mathrm{,}\chi \mathrm{)}$
do not fully possess the mathematical properties of quantum mechanical
angular momentum. It is well known that a sum of Wigner
Dfunctions is required to build an angular momentum state. The
equation.
$${{J}^{\prime}}_{\pm}\text{\hspace{0.05em}}{D}_{M\Omega}^{{J}^{\mathrm{*}}}\mathrm{(}\phi \mathrm{,}\theta \mathrm{,}\chi \mathrm{)=}\sqrt{J\mathrm{(}J+\mathrm{1)}\Omega \mathrm{(}\Omega \mp \mathrm{1)}}\text{}\text{\hspace{0.05em}}{D}_{M\mathrm{,}\Omega \mp 1}^{{J}^{\mathrm{*}}}\mathrm{(}\phi \mathrm{,}\theta \mathrm{,}\chi \mathrm{)}\text{(2)}$$
is not a phase convention [1012] but a mathematical
result readily obtained from Eq. (1) and
$${{J}^{\prime}}_{\pm}\text{\hspace{0.05em}}\mathrm{}J\Omega \rangle \mathrm{=}\sqrt{J\mathrm{(}J+\mathrm{1)}\Omega \mathrm{(}\Omega \pm \mathrm{1)}}\text{\hspace{0.05em}}\mathrm{}\text{}J\mathrm{,}\Omega \pm 1\rangle \mathrm{,}\text{(3)}$$
in which the prime on the operator
${{J}^{\prime}}_{\pm}$
indicates that it is written in the rotated coordinate system where the appropriate
magnetic quantum number
$\Omega $
Hund’s basis function
The Hund’s case (a) basis function based upon the
WignerWitmer diatomic eigenfunction is
$$\begin{array}{l}\mathrm{}a\rangle \mathrm{=}\langle \rho \mathrm{,}\zeta \mathrm{,}\chi \mathrm{,}r{\mathrm{\text{'}}}_{2}\mathrm{,}\dots \mathrm{,}r{\mathrm{\text{'}}}_{\ufffd}\mathrm{,}r\mathrm{,}\theta \mathrm{,}\phi \text{\hspace{0.05em}}\mathrm{}nvJMS\Lambda \Sigma \Omega \rangle \mathrm{=}\\ \sqrt{\frac{2J+1}{8{\pi}^{2}}}\langle \rho \mathrm{,}\zeta \mathrm{,}r{\mathrm{\text{'}}}_{2}\mathrm{,}\dots \mathrm{,}r{\mathrm{\text{'}}}_{\ufffd}\mathrm{,}r\text{\hspace{0.05em}}\mathrm{}nv\rangle \text{\hspace{0.05em}}\mathrm{}S\Sigma \rangle \text{\hspace{0.05em}}{D}_{M\Omega}^{{J}^{\mathrm{*}}}\mathrm{(}\phi \mathrm{,}\theta \mathrm{,}\chi \mathrm{).}\text{(4)}\end{array}$$
As noted above, a sum of basis functions is required
to build an eigenstate of angular momentum. The basis function
would also not be an eigenstate of the parity operator. The case
(a) matrix elements,
${p}_{ij}^{\mathrm{(}a\mathrm{)}}$, of the parity operator P,
$${p}_{ij}^{\mathrm{(}a\mathrm{)}}\mathrm{=}{p}_{\text{\hspace{0.05em}}\Sigma}{\mathrm{(}\mathrm{)}}^{J}\text{}\delta \mathrm{(}{J}_{i}{J}_{j}\mathrm{)}\text{\hspace{0.05em}}\delta \mathrm{(}{\Omega}_{i}\mathrm{,}{\Omega}_{j}\mathrm{)}\text{\hspace{0.05em}}\delta \mathrm{(}{\Lambda}_{i}\mathrm{,}{\Lambda}_{i}\mathrm{)}\text{\hspace{0.05em}}\delta \mathrm{(}{n}_{i}{n}_{j}\mathrm{),}\text{(5)}$$
show that a single
$\mathrm{}a\rangle $
basis function is not an eigenstate of parity.
The procedure called parity symmetrization adds
$\mathrm{}JM\Omega \rangle $
and
$\mathrm{}JM\mathrm{,}\Omega \rangle $
basis functions thereby destroying the second magnetic
quantum number
$\Omega $
and yielding a function which at least
possesses the minimal mathematical properties of an eigenstate
of angular momentum, parity, and the other members of the
complete set of commuting operators. The general procedure
would be to continue adding basis functions to the upper and
lower bases until eigenvalue differences between the upper and
lower Hamiltonians accurately predict measured line positions.
The upper Hamiltonian matrix for the (5,4) band
Electronic spin S interactions with electronic orbital
momentum L and nuclear orbital momentum R produce both
diagonal and offdiagonal matrix elements in the Hund’s case (a)
representation of the Hamiltonian. The offdiagonal elements
connect different basis states. For example, both of the mentioned
spin orbit interactions connect
${}^{2}{\Sigma}^{+}$
and ${}^{2}\Pi $.
Because van
Vleck transformed Hamiltonians are not used, the appropriate
parameters for the strength of these interactions are < AL+> and
< BL+>.
The presented work relies on Hamiltonians that are not
paritypartitioned. Table 2 lists the molecular parameters. The
values for the A2Π state were determined utilizing the Nelder
Mead minimization algorithm using values given by Brooke, et al.
[2] as trial values. Error estimates were not computed, and the
values of Brooke, et al. [2] were only very slightly changed.
In Table 2, parameters not followed by a number
in parenthesis were held fixed or an error estimate was not
computed. The value in parenthesis is the standard deviation of
the fitted value.
Table 2:Molecular parameters used in this work. A value in parenthesis indicates the standard deviation in the fitted value.
Tables 3 and 4 show the Hamiltonian matrices for levels modeled as a mixture of,
${}^{2}{\Sigma}^{+}$
and ${}^{2}\Pi $
basis states, without and with spinorbit interactions, respectively.
Table 3:Hamiltonian matrix without spinorbit coupling. The bottom row contains the energy eigenvalues.



$v$

5 
5 
17 
17 
17 
17 



$\Lambda $

0 
0 
1 
1 
1 
1 



$\Sigma $

0.5 
0.5 
0.5 
0.5 
0.5 
0.5 
$v$

$\Lambda $

$\Sigma $

$\Omega $

0.5 
0.5 
1.5 
0.5 
0.5 
1.5 
5 
0 
0.5 
0.5 
36351.6409 
25.6707 
0 
0 
0 
0 
5 
0 
0.5 
0.5 
25.6707 
36351.6409 
0 
0 
0 
0 
17 
1 
0.5 
1.5 
0 
0 
36257.6340 
19.5866 
0 
0 
17 
1 
0.5 
0.5 
0 
2.8639 
19.5866 
36310.9646 
0 
0 
17 
1 
0.5 
0.5 
0 
2.3274 
0 
0 
36310.9646 
19.5866 
17 
1 
0.5 
1.5 
0 
2.8566 
0 
0 
19.5866 
36257.6340 


${E}_{nvJ}$

36377.3116 
36325.9702 
36251.2135 
36317.3851 
36317.3851 
36251.2135 
Table 4:Hamiltonian matrix including spinorbit coupling, but otherwise using the same layout as in Table 3. A reduction by one order of magnitude in the standard deviation of the spectral line fitting can be accomplished when including the perturbations.



$v$

5 
5 
17 
17 
17 
17 



$\Lambda $

0 
0 
1 
1 
1 
1 



$\Sigma $

0.5 
0.5 
0.5 
0.5 
0.5 
0.5 
$v$

$\Lambda $

$\Sigma $

$\Omega $

0.5 
0.5 
1.5 
0.5 
0.5 
1.5 
5 
0 
0.5 
0.5 
36351.6409 
25.6707 
2.8566 
2.3274 
2.8639 
0 
5 
0 
0.5 
0.5 
25.6707 
36351.6409 
0 
2.8639 
2.3274 
2.8566 
17 
1 
0.5 
1.5 
2.8566 
0 
36257.6340 
19.5866 
0 
0 
17 
1 
0.5 
0.5 
2.3274 
2.8639 
19.5866 
36310.9646 
0 
0 
17 
1 
0.5 
0.5 
2.8639 
2.3274 
0 
0 
36310.9646 
19.5866 
17 
1 
0.5 
1.5 
0 
2.8566 
0 
0 
19.5866 
36257.6340 


${E}_{nvJ}$

36377.3957 
36327.7869 
36250.9625 
36317.3525 
36315.8194 
36251.1620 
In Table 3, the Hamiltonian was computed for
$\langle AL+\rangle \mathrm{=}\langle BL+\rangle \mathrm{=0}$
in other words the offdiagonal spinorbit coupling has been
removed. Consequently, the 2×2 matrices along the main
diagonal are independent, and could be individually diagonalized.
Using matrices like these to model upper states of the CN violet
(5,4) band, the 100 experimental spectral lines reported by
Ram et al. [1] were fitted with standard deviation of 0.25 cm^{1}.
Standard Hund’s case (a) matrix elements [10, 12] were used. In
Table 4, offdiagonal spinorbit coupling mixes the Hund’s case
(a) basis states, and the standard deviation of the spectral line fit
mentioned in Table 3 is reduced by a factor of 10 to 0.025
cm^{1}. The spinorbit coupling constants
$\langle AL+\rangle \mathrm{=4.25(0.03)}$
and $\langle BL+\rangle \mathrm{=0.205(0.001)}$
listed in Table 2 were used to determine the
Hamiltonian in Table 4. This single 6×6 matrix describing
${}^{2}\Pi {}^{2}{\Sigma}^{+}$
mixing can be compared with the two 3×3 parity partitioned
matrices of Brown and Carrington [13].
A diatomic line position fitting algorithm
A basic tool for the diatomic spectroscopist is a
computer program that accepts a table of experimentally
measured vacuum wave numbers
${\tilde{\nu}}_{\text{e}xp}$
versus ${J}^{\prime}$
and $J$,
and outputs a set of molecular parameters with which one can
reproduce the
${\tilde{\nu}}_{\text{e}xp}$
with a standard deviation comparable to
the estimated experimental error. In practice, an experimental
line list frequently shows gaps, viz. spectral lines are missing.
Following a successful fitting process, one can use the molecular
parameters to predict all lines. A computed line list is especially
useful when it includes the Condon and Shortley [14] line strength
from which the Einstein coefficients and oscillator strength
[15, 16] and the HITRAN line strength [17] can be calculated. A
feature of the line fitting program described below is its use of
nonzero rotational strengths (see Eq. (9) below) to mark which
of the many computed differences between upper and lower
term values represents the vacuum wavenumber of an allowed
spectral line. Consequently, the fitting process creates a complete
line list including rotational factors. Parity plays no part in the
fitting process, but the same orthogonal matrix that diagonalizes
the case (a) Hamiltonian matrix will also diagonalize the case
(a) parity matrix whose elements are given in Equation (5). The
$p\mathrm{=}\pm 1$
parity eigenvalue becomes a computed quantity, and the
e/f parity designation is established from the parity eigenvalue
using the accepted convention Brown et al. [18].
Trial values of upper and lower state molecular
parameters, typically taken from previous works [2] for the band
system in question, are used to compute upper H’ and lower H
Hamiltonian matrices in the case (a) basis given by Eq. (4) for
specific values of
${J}^{\prime}$
and $J$
The upper and lower Hamiltonians
are numerically diagonalized,
$${T}^{\prime}\mathrm{=}\tilde{{U}^{\prime}}\text{\hspace{0.05em}}{H}^{\prime}\text{\hspace{0.05em}}{U}^{\prime}\text{(6)}$$
$$T\mathrm{=}\tilde{U}\text{\hspace{0.05em}}H\text{\hspace{0.05em}}U\text{(7)}$$
giving the upper
${T}^{\prime}$
and $T$
term values. The vacuum wavenumber
$\tilde{\nu}$
is determined,
$${\tilde{\nu}}_{ij}\mathrm{=}{{T}^{\prime}}_{i}{T}_{j}\mathrm{,}\text{(8)}$$
and the rotational strength is evaluated,
$${S}_{ij}\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)=(2}J+\mathrm{1)}{\left{\displaystyle \sum _{n}}{\displaystyle \sum _{m}}{\tilde{{U}^{\prime}}}_{in}\langle J\Omega \mathrm{;}q\mathrm{,}{\Omega}^{\prime}\Omega \text{\hspace{0.05em}}\mathrm{}{J}^{\prime}{\Omega}^{\prime}\rangle \text{\hspace{0.05em}}{U}_{mj}\text{\hspace{0.05em}}\delta \mathrm{(}{{\Sigma}^{\prime}}_{n}{\Sigma}_{m}\mathrm{)}\right}^{2}\mathrm{.}\text{(9)}$$
The degree of the tensor operator
$q$, responsible for the transitions amounts to
$q\mathrm{=1}$
for electric dipole transitions. For a nonzero rotational factors,
$S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$, the vacuum wavenumber is
added to a table of computed line positions to be compared with
the experimental list
${\tilde{\nu}}_{\text{e}xp}$
versus ${J}^{\prime}$
and $J$. The ClebschGordan coefficient,
$\langle J\Omega \mathrm{;}q\mathrm{,}{\Omega}^{\prime}\Omega \text{\hspace{0.05em}}\mathrm{}{J}^{\prime}{\Omega}^{\prime}\rangle $,
is the same one appearing in the pure case (a)  case (a) formulae for
$S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$.
For a specific values of
${J}^{\prime}$
and $J$
one constructs tables for
${\tilde{\nu}}_{\text{e}xp}$
and computed
${\tilde{\nu}}_{ij}$. The differences
$\Delta {\tilde{\nu}}_{ij}$
$$\Delta {\tilde{\nu}}_{ij}\mathrm{=}{\tilde{\nu}}_{ij}{\tilde{\nu}}_{\text{e}xp}\mathrm{,}\text{(10)}$$
are computed where each
${\tilde{\nu}}_{ij}$
is the one that most closely equals
one of the
${\tilde{\nu}}_{\text{e}xp}$.
Once values of ${\tilde{\nu}}_{ij}$ and ${\tilde{\nu}}_{\text{e}xp}$
are matched, each
is marked unavailable until a new list of
${\tilde{\nu}}_{ij}$
is computed. The
indicated computations are performed for all values of
${J}^{\prime}$
and $J$
in the experimental line list, and corrections to the trial
values of the molecular parameters are subsequently determined
from the resulting
$\Delta {\tilde{\nu}}_{ij}$.
The entire process is iterated until the
parameter corrections become negligibly small. As this fitting
process successfully concludes, one obtains a set of molecular
parameters that predict measured line positions,
${\tilde{\nu}}_{\text{e}xp}$,
with a
standard deviations that equal the experimental estimates within
the accuracy of the
${\tilde{\nu}}_{\text{e}xp}$.
Discussion
The influence on intensities in the (5,4) band of the
CN violet system caused by the weak spinorbit mixing, Figures
1 and 2 is significantly larger than initially anticipated. This
can be noticed because computation of the rotational strengths
is an integral part of our line position fitting program. The
eigenvectors that diagonalize the Hamiltonian to yield fitted line
position
$\tilde{\nu}$
also yield $S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$.
In established diatomic molecular
practice, HönlLondon factors are determined independently of
line positions. Analytical approximations utilize the parameter
$Y\mathrm{=}A\mathrm{/}B$
to account for the influence of spinorbit interaction on
$S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$.
Kovács [19] gives many examples, Li, et al. [20] give a
more recent application. These analytical approximations can
accurately account for intermediate spinorbit coupling which
smooth transitions between case (a) and case (b) with increasing
${J}^{\prime}$
and $J$, but show limited sensitivity to abrupt changes in
$S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$
near perturbations such as those seen the CN (5,4) band.
Conclusions
The WignerWitmer diatomic eigenfunction makes it
possible to form an exact, mathematical connection between
computation of
$\tilde{\nu}$
and $S\mathrm{(}{J}^{\prime}\mathrm{,}J\mathrm{)}$
in a single algorithm. The
concept of the nonvanishing rotational strengths as the
omnipotent selection rule initially conceived as a simplifying
convenience in a computer algorithm is now seen to be more
valuable, as evidenced in this work’s analysis of the CN (5,4)
band perturbations by isolating a specific branch. Future work
is planned for comparisons of the CN (10,10) band spectra that
include perturbation and that show promising agreements with
experiments and PGOPHER predictions.
Acknowledgments
One of us (CGP) acknowledges support in part by the
Center for Laser and greatly thanks for the outstanding dedication
of late James O. Hornkohl.
 Ram RS, Davis SP, Wallace L, Englman R, Appadoo DRT, Bernath PF. Fourier transform emission spectroscopy of the $B{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}X{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}$
system of CN. J Mol Spectrosc. 2006;237(2):225–231.
 Brooke JSA, Ram RS, Western CM, Li G, Schwenke DW, Bernath PF. Einstein A Coefficients and Oscillator Strengths for the $A\text{\hspace{0.05em}}{\text{\hspace{0.05em}}}^{2}\Pi X{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}$
(Red) and $B{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}X\text{\hspace{0.05em}}{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}$
(Violet) Systems and Rovibrational Transitions in the $X\text{\hspace{0.05em}}{\text{\hspace{0.05em}}}^{2}{\Sigma}^{+}$
State of CN ApJS.2014;210(2):115. doi:10.1088/00670049/210/2/23
 Western CM. PGOPHER: A program for simulating rotational, vibrational and electronic spectra. J Quant Spectrosc Rad Transfer. 2017;186(SI):221–242.
 Ito H, Fukuda Y, Ozaki Y, Kondow T, Kuchitsu K. Analysis of perturnation between the v = 5 and v = 17 levels of the CN radical. J Molec Spectrosc. 1987;121(1):84–90.
 Engleman R. The v = 0 and +1 sequence bands of the CN violet system observed during the flash photolysis of BrCN. J Mol Spectrosc. 1974;49(1):106–116.
 National Solar Observatory (NSO) at Kitt Peak, McMathPierce Fourier Transform Spectrometer (FTS) data. [cited 2016 Dec 20]; Available from: ftp://vso.nso.edu/FTS cdrom/FTS30/920212R0.005
 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.
 Yurchenko SN, Lodi L, Tennyson J, Stolyarov AV. Duo: a general program for calculating spectra of diatomic molecules. Comput Phys Commun. 2016;202:262–275.
 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. On the structure of the spectra of twoatomic molecules according to quantum mechanics. Quantum Chemistry: Classic Scientific Papers. World Scientific: Singapore; 2000;287–311.
 Zare RN, Schmeltekopf AL, Harrop DL, 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, Howard BJ. Approach to the anomalous commutator relations of rotational angular momentum in molecules. Mol Phys. 1976;31(5):1517–1525.
 LefebvreBrion H, Field RW. The Spectra and Dynamics of Diatomic Molecules. Elsevier: Amsterdam, 2004.
 Brown JM, Carrington A. Rotational Spectroscopy of Diatomic Molecules. Cambridge Univ Press: Cambridge; 2003;516–517.
 Condon EU, Shortley GH. The Theory of Atomic Spectra. Cambridge Univ Press: Cambridge; 1964.
 Hilborn RC. Einstein coefficients, cross sections, f values, dipole moments, and all that. Am J Phys. 1982;50(11):982–986.
 Thorne AP. Spectrophysics, 2nd ed. Chapman and Hall. New York. 1988.
 Rothman LS, Rinsland CP, Goldman A, Massie ST, Edwards DP, Flaud JM, et al. The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 Edition. J Quant Spectrosc Rad Transfer. 1998;60(5):665–710.
 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.
 Kovács I. Rotational Structure in the Spectra of Diatomic Molecules. Elsevier: New York. 1969.
 Li G, Harrison JJ, Ram RS, Western CM, Bernath PF. Einstein A coefficients and absolute line intensities for the E 2Π – X 2Σ+ transition of CaH. J Quant Spectrosc Rad Transfer. 2012;113(1):67–74.