#
Improved Theory of the Effective Dipole Moments and Absolute Line Strengths of the XY_{2} Asymmetric Top Molecules in the X^{2}B_{1} Doublet Electronic States

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}(${C}_{2v}$−symmetry) molecule in a doublet electronic state is derived that includes (as special cases) all currently known models of effective dipole moments for such types of molecules, and allows us to take into account the influence of spin–rotation interactions on the effective dipole moment operator that were not considered in the preceding studies. Necessary for the analysis of absolute line strengths, the matrix elements of this dipole moment operator are derived. A comparison with the previous analog models is made and discussed. The efficiency of the obtained results is illustrated, which have been applied to a set of the “forbidden” $\Delta {K}_{a}=2$ transitions of the ${\nu}_{3}$ band of the OClO free radical molecule.

## 1. Introduction

_{2}(${C}_{2v}$-symmetry) molecule and their corresponding matrix elements on the ro-vibrational wave functions have been derived. The general results of that study were successfully used by many authors for the analysis of different XY

_{2}(${C}_{2v}$) molecules, and also for more complicated asymmetric top molecules (not having the possibility to refer here to all these studies, we mention only a few of them—refs. [9,10,11,12,13]—which have been fulfilled by the authors of this paper during recent years).

_{2}one with (${C}_{2v}$-symmetry) ones) can be presented in nature not only in a singlet state but also in multiplet electronic states as well (the NO

_{2}and ClO

_{2}free radical molecules in the ${X}^{2}{B}_{1}$ electronic ground state can be mentioned, for example). The theory and the matrix elements being necessary for calculations of effective dipole moment operators for such molecules differ considerably in some aspects from the basic results of Ref. [1]. The corresponding theory and results for such molecules have been presented in the literature beginning from the eighties of the twentieth century (see, Refs. [14,15,16,17]). However, up to now, not all crucial effects and interactions are completely and correctly taken into account and described. Namely, it is evident that not only pure rotational centrifugal effects but also centrifugal effects that are caused by the spin–rotation interactions should be taken into account. In particular, it is clear (see, e.g., [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33] and our recent studies [34,35,36] where the higher order spin–rotational effects were taken into account) that influences of both the pure rotational and spin–rotational centrifugal effects on the spin–ro-vibrational structure of the discussed type of molecules are comparable in size. Furthemore, related to the absolute strengths of spin–ro-vibrational transitions, up to now, the influence of spin–rotational interactions on the absolute transition strengths have been taken into account only via wave functions, which are eigen functions of the effective Hamiltonian of the considered vibrational band. The dependence of the effective dipole moment operator on the spin–rotational centrifugal effects has never been considered despite the obvious fact that neglect of similar effects in the effective Hamiltonian leads to an increase in the error by several tens of times. In the present study, we intend to fill this gap and derive an effective dipole moment operator of the XY

_{2}(${C}_{2v}$) molecule in a doublet electronic state, taking into account its dependence on the spin–rotational centrifugal effects also. To make the discussion more clear for the reader, we consider the problem of obtaining the effective dipole moment operator and the determination of its matrix elements for the pure rotational problem in Section 2 as a starting point. The main ideas and steps of discussion in Section 2 are then used in Section 3 and Section 4 for the analogous analysis of an effective dipole moment operator and absolute line intensities for the ro-vibrational problem (without spin–rotational interactions; Section 3). This is followed by the discussion of a model that takes into account the presence of spin–rotational interactions in the wave functions, but omits both rotational and spin–rotational effects in the effective dipole moment (Section 4). Finally, Section 5 presents results that are produced by the complete consideration of both pure rotational and spin–rotational centrifugal distortion effects in the wave functions and in the effective dipole moment operator.

## 2. Absolute Intensity of an Isolated Line of the XY_{2} (${\mathit{C}}_{\mathbf{2}\mathit{v}}$)
Molecule in a Singlet Electronic State:
Rotational Transitions

**P**(remember that only the first terms in Equation (3) are taken into account in this section) can be expressed (see, e.g., [40]) in the form of three components of the irreducible first rank tensor ${P}_{S}^{\left(1\right)}$ $S=0,\pm 1$:

## 3. Absolute Intensity of an Isolated Line of the XY_{2} (${\mathit{C}}_{\mathbf{2}\mathit{v}}$)
Molecule in a Singlet Electronic State:
Ro-Vibrational Transitions

_{2}(${C}_{2v}$) molecule, the symmetry of vibrational functions $\mid v\rangle $ can be ${A}_{1}$ or ${B}_{1}$, only two types of the “effective dipole moment” operators can be realized, namely ${}^{({v}^{\prime}-{\tilde{v}}^{\prime})}{\mathcal{P}}_{Z}^{A{}_{2}}$ for the parallel bands and ${}^{({v}^{\prime}-{\tilde{v}}^{\prime})}{\mathcal{P}}_{Z}^{B{}_{2}}$ for the perpendicular bands. In both cases, from the general point of view (see, e.g., [8]), the ”effective dipole moment“ operator can be written as:

## 4. Absolute Intensity of an Isolated Line of the XY_{2} (${\mathit{C}}_{\mathbf{2}\mathit{v}}$)
Molecule in Doublet Electronic State: Spin–Rotational Transitions
in the Model That Neglects Spin–Rotational Interactions in the
Effective Dipole Moment Operator

_{2}(${C}_{2v}$ symmetry) molecule in a doublet electronic state. To our knowledge, for a description of absolute strengths of spin–ro-vibrational transitions in such molecules, up to now, the modern chemical physics uses the model that takes into account the presence of spin–rotation interactions in the wave functions of the lower and upper states of the transition considered, but neglects the presence of spin–rotational interactions in the effective dipole moment operator (see, e.g., [14,15,16,17]). It looks rather inconsistent if one takes into account the following arguments: (a) the influence of the rotational centrifugal distortion effects on an effective dipole moment operator is always taken into account for molecules both in singlet and doublet electronic states (see, e.g., above-mentioned Refs. [1,14,15,16,17]); (b) as was discussed above, influences of both the pure rotational and spin–rotational centrifugal distortion effects on the spin–ro-vibrational energies of asymmetric top molecules in doublet electronic states are practically of the same orders of value; (c) in this respect, one can expect that taking into account spin–rotation interactions in an effective dipole moment operator can improve the accuracy of the description of absolute transition strengths in a doublet electronic state molecule by the same order as the pure rotational centrifugal effects improve the description of absolute transition strengths in a singlet electronic state molecule.

## 5. Absolute Intensity of an Isolated Line of the XY_{2} (${\mathit{C}}_{\mathbf{2}\mathit{v}}$)
Molecule in Doublet Electronic State: Spin–Rotational
Transitions: ${\mathbf{P}}_{\mathbf{Z}}$-Operator Depends on Molecular
Vibrations

_{2}(${C}_{2v}$) molecule, the ${R}_{\alpha}$ ($\alpha =x,y,z$) components in the MFS are:

_{2}(${C}_{2v}$ symmetry) molecule (evidently, the same as for the ${R}_{\alpha}$ operators, ${S}_{x}\in {A}_{2}$, ${S}_{y}\in {B}_{1}$, and ${S}_{z}\in {B}_{2}$), it is not difficult to obtain symmetrized combinations of different products of operators ${J}_{\alpha}$ and ${S}_{\beta}$. They are:

#### 5.1. Parallel Ro-Vibrational Bands

#### 5.2. Perpendicular Ro-Vibrational Bands

^{35}ClO

_{2}molecule whose experimental values can be found in Ref. [36] (they are reproduced from [36] in column 3 of Table 5). Column 2 of Table 5 presents theoretically predicted values of the same transition frequencies. These predicted values were obtained as differences between values of corresponding spin–ro-vibrational energies of the (001) upper vibrational state (the latter have been taken from Table 4 of Ref. [36]) and those of the ground vibrational state (in this case, spin–rotational energies of the ground vibrational state have been calculated with the parameters from column 2 of Table 2; [36]).

^{−1}) and corresponding experimental line positions from spectrum I of Ref. [36] (also in cm

^{−1}). The values in column 4 are transmittances of experimental lines. One can see that the “forbidden” transition is strong enough (for a comparison with the “allowed” transitions of the ${\nu}_{3}$ band, see the small fragment of the mentioned experimental spectrum in Figure 1). One can argue that the reason for the appearance of the discussed transitions can be a superposition of the spin–rotational basic functions with $\Delta K=0$ and $\Delta K=\pm 2$ in the effective Hamiltonian eigenfunctions, which are used in the calculation of matrix elements of the effective dipole moment of a molecule. However, the analysis of corresponding wave functions and the estimation of corresponding numerical values show that such influence of superpositions in wave functions is negligible in comparison with the effect of Equation (59).

## 6. Conclusions

_{2}(C

_{2v}-symmetry) molecule in a doublet electronic state by taking into account spin–rotational centrifugal corrections that have never been considered earlier for such kind of problems. Corresponding relations (which are necessary for determination of absolute spin–ro-vibrational transition strengths and which contain all effects known up to now, as special cases) are obtained on the basis of the irreducible tensorial sets theory. The derived results allow us to take into account both the higher-order corrections to the allowed transitions and also to describe weak transitions of the $\Delta K=\pm 2,\pm 3$−types in the parallel bands, of the $\Delta K=\pm 1,\pm 2$−types in the perpendicular bands, and of the $\Delta N=\pm 2$−type in both kinds of spin–ro-vibrational bands. To illustrate the correctness and efficiency of the derived model, we compared the estimated line strengths of a set of the “forbidden” $\Delta {K}_{a}=2$ transitions of the ${\nu}_{3}$ band of the OClO free radical with corresponding experimental data, which confirm the validity of the obtained results.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Small portion of the experimental spectrum of ClO

_{2}compared with Ref. [36]. “Forbidden transitions” $\Delta K=\pm 2$ are marked by dark circles. Experimental conditions: resolution is 0.0015 cm

^{−1}; number of scans is 400; source is a Globar; detector is a MCT313; beam-splitter is made from KBr; optical path length is 0.23 m; aperture is 1.15 mm; temperature is 22 ± 0.3 °C; pressure is 100 Pa; calibration was performed by CO

_{2}and H

_{2}O spectral lines.

**Table 1.**Operators and matrix elements for the perpendicular band $(\Delta k=\pm 1)$ (reproduced from Ref. [1]).

j | ${}^{\mathit{v}}{\mathit{A}}_{\mathit{j}}$ | n | $<\mathit{JK}|{}^{\mathit{v}}{\mathit{A}}_{\mathit{j}}|\mathit{J}+\mathbf{\Delta}\mathit{J}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathit{K}+\mathit{n}\mathbf{\Delta}\mathit{K}>$; $\mathbf{\Delta}\mathit{K}=\pm 1$ | |
---|---|---|---|---|

1 | ${\phi}_{x}$ | 1 | $<JK|{\phi}_{x}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=0,\pm 1$ |

2 | $\{{\phi}_{x},{J}^{2}\}$ | 1 | $[J(J+1)+(J+\Delta J)(J+\Delta J+1)]<JK|{\phi}_{x}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=0,\pm 1$ |

3 | $\{{\phi}_{x},{J}_{z}^{2}\}$ | 1 | $[{K}^{2}+{(K+\Delta K)}^{2}]<JK|{\phi}_{x}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=0,\pm 1$ |

4 | $\{i{\phi}_{y},{J}_{z}\}$ | 1 | $(1+2K\Delta K)<JK|{\phi}_{x}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=0,\pm 1$ |

5 | $\{{\phi}_{z},i{J}_{y}\}$ | 1 | $(1+2K\Delta K)<JK|{\phi}_{x}|J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=0$ |

$(1+2K\Delta K-2m)<JK|{\phi}_{x}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=\pm 1$ | |||

6 | $\{{\phi}_{z},{J}_{x}{J}_{z}+{J}_{z}{J}_{x}\}$ | 1 | ${(1+2K\Delta K)}^{2}<JK|{\phi}_{x}|J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=0$ |

$(1+2K\Delta K)(1+2K\Delta K-2m)<JK|{\phi}_{x}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=\pm 1$ | |||

7 | $\frac{1}{2}\left(\right)open="["\; close="]">\{{\phi}_{x},{J}_{xy}^{2}\}-\{i{\phi}_{y},i({J}_{x}{J}_{y}+{J}_{y}{J}_{x})\}$ | 1 | $[J(J+1)-K\Delta K-{K}^{2}-1]<JK|{\phi}_{x}|J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=0$ |

$-[m(m-1)-(2m-1)K\Delta K+{K}^{2}+1]<JK|{\phi}_{x}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=\pm 1$ | |||

$\frac{1}{2}\left(\right)open="["\; close="]">\{{\phi}_{x},{J}_{xy}^{2}\}+\{i{\phi}_{y},i({J}_{x}{J}_{y}+{J}_{y}{J}_{x})\}$ | ${\left[(J-K\Delta K-1)(J-K\Delta K-2)(J+K\Delta K+2)(J+K\Delta K+3)\right]}^{1/2}$ | |||

8 | 3 | $<JK|{\phi}_{x}|J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=0$ | |

${\left[(m-K\Delta K-1)(m-K\Delta K-2)(m+K\Delta K+2)(m+K\Delta K+3)\right]}^{1/2}$ | ||||

$<JK|{\phi}_{x}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=\pm 1$ |

**Table 2.**Operators and matrix elements for the parallel band $(\Delta k=0)$ (reproduced from Ref. [1]).

j | ${}^{\mathit{v}}{\mathit{A}}_{\mathit{j}}$ | n | $<\mathit{JK}|{}^{\mathit{v}}{\mathit{A}}_{\mathit{j}}|\mathit{J}+\mathbf{\Delta}\mathit{J}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathit{K}+\mathit{n}\mathbf{\Delta}\mathit{K}>$; $\mathbf{\Delta}\mathit{K}=\pm 1$ | |
---|---|---|---|---|

1 | ${\phi}_{z}$ | 0 | $<JK|{\phi}_{z}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K>$ | $\Delta J=0,\pm 1$ |

2 | $\{{\phi}_{z},{J}^{2}\}$ | 0 | $[J(J+1)+(J+\Delta J)(J+\Delta J+1)]<JK|{\phi}_{z}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K>$ | $\Delta J=0,\pm 1$ |

3 | $\{{\phi}_{z},{J}_{z}^{2}\}$ | 0 | $2{K}^{2}<JK|{\phi}_{z}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K>$ | $\Delta J=0,\pm 1$ |

4 | $\frac{1}{2}\left(\right)open="["\; close="]">\{{\phi}_{x},i{J}_{y}\}-\{i{\phi}_{y},{J}_{x}\}$ | 0 | 0 | $\Delta J=0$ |

0 | $m<JK|{\phi}_{z}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K>$ | $\Delta J=\pm 1$ | ||

5 | $\frac{1}{2}\{{\phi}_{x},{J}_{x}{J}_{z}+{J}_{z}{J}_{x}\}$ | 0 | $[2(J(J+1)-{K}^{2})-1]<JK|{\phi}_{z}|J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K>$ | $\Delta J=0$ |

$-\frac{1}{2}\{i{\phi}_{y},i({J}_{y}{J}_{z}+{J}_{z}{J}_{y})\}$ | $-(1+2{K}^{2})<JK|{\phi}_{z}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K>$ | $\Delta J=\pm 1$ | ||

6 | $\frac{1}{2}\left(\right)open="["\; close="]">\{{\phi}_{x},i{J}_{y}\}+\{i{\phi}_{y},{J}_{x}\}$ | 2 | $\Delta K{\left[(J-K\Delta K-1)(J+K\Delta K+2)\right]}^{1/2}<JK|{\phi}_{x}|J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=0$ |

$\Delta K{\left[(m-K\Delta K-1)(m+K\Delta K+2)\right]}^{1/2}<JK|{\phi}_{x}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=\pm 1$ | |||

7 | $\frac{1}{2}\{{\phi}_{x},{J}_{x}{J}_{z}+{J}_{z}{J}_{x}\}$ | 2 | $2(K+\Delta K){\left[(J-K\Delta K-1)(J+K\Delta K+2)\right]}^{1/2}<JK|{\phi}_{x}|J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=0$ |

$+\frac{1}{2}\{i{\phi}_{y},i({J}_{y}{J}_{z}+{J}_{z}{J}_{y})\}$ | $2(K+\Delta K){\left[(m-K\Delta K-1)(m+K\Delta K+2)\right]}^{1/2}<JK|{\phi}_{x}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=\pm 1$ | ||

8 | $\{{\phi}_{z},{J}_{xy}^{2}\}$ | 2 | $2(K+\Delta K){\left[(J-K\Delta K-1)(J+K\Delta K+2)\right]}^{1/2}<JK|{\phi}_{x}|J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ | $\Delta J=0$ |

$-2\Delta K(m-1-K\Delta K){\left[(m-K\Delta K-1)(m+K\Delta K+2)\right]}^{1/2}$ | $\Delta J=\pm 1$ | |||

$<JK|{\phi}_{x}|J+\Delta J\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}K+\Delta K>$ |

**Table 3.**Nonzero values of the $\tilde{g}(NJ,\tilde{N}\tilde{J})$—coefficients (“relative intensities”) of spin–rotational components of rotational transitions.

$\tilde{\mathit{N}}$ | $\tilde{\mathit{J}}$ | J | $\mathbf{\Delta}\mathit{J}=\tilde{\mathit{J}}-\mathit{J}$ | Value |
---|---|---|---|---|

$\tilde{N}=N-1$ | $\tilde{J}$ = $\tilde{N}+1/2$ = $N-1/2$ | J = $N+1/2$ | $\Delta J=\Delta N$ | $\frac{2N-1}{4N}$ |

$\tilde{J}$ = $\tilde{N}-1/2$ = $N-3/2$ | J = $N-1/2$ | $\Delta J=\Delta N$ | $\frac{2N+1}{4N}$ | |

$\tilde{J}$ = $\tilde{N}+1/2$ = $N-1/2$ | J = $N-1/2$ | $\Delta J\ne \Delta N$ | $\frac{1}{4{N}^{2}}$ | |

$\tilde{J}$ = $\tilde{N}-1/2$ = $N-3/2$ | J = $N+1/2$ | $\Delta J\ne \Delta N$ | 0 | |

$\tilde{N}=N$ | $\tilde{J}$ = $N+1/2$ | J = $N+1/2$ | $\Delta J=\Delta N$ | $\frac{N(2N+3)}{4{(N+1)}^{2}}$ |

$\tilde{J}$ = $N-1/2$ | J = $N-1/2$ | $\Delta J=\Delta N$ | $\frac{(N+1)(2N-1)}{4{N}^{2}}$ | |

$\tilde{J}$ = $N-1/2$ | J = $N+1/2$ | $\Delta J\ne \Delta N$ | $\frac{1}{4N(N+1)}$ | |

$\tilde{J}$ = $N+1/2$ | J = $N-1/2$ | $\Delta J\ne \Delta N$ | $\frac{1}{4N(N+1)}$ | |

$\tilde{N}=N+1$ | $\tilde{J}$ = $\tilde{N}+1/2$ = $N+3/2$ | J = $N+1/2$ | $\Delta J=\Delta N$ | $\frac{2N+1}{4(N+1)}$ |

$\tilde{J}$ = $\tilde{N}-1/2$ = $N+1/2$ | J = $N-1/2$ | $\Delta J=\Delta N$ | $\frac{2N+3}{4(N+1)}$ | |

$\tilde{J}$ = $\tilde{N}-1/2$ = $N+1/2$ | J = $N+1/2$ | $\Delta J\ne \Delta N$ | $\frac{1}{4{(N+1)}^{2}}$ | |

$\tilde{J}$ = $\tilde{N}+1/2$ = $N+3/2$ | J = $N-1/2$ | $\Delta J\ne \Delta N$ | 0 |

**Table 4.**Possible combinations of indexes for nonzero values of matrix elements; Equations (48) and (52) ${}^{\left(a\right)}$.

$\mathbf{\Delta}\mathit{N}=\tilde{\mathit{N}}-\mathit{N}$ | J | $\mathbf{\Delta}\mathit{J}=\tilde{\mathit{J}}-\mathit{J}$ | L | M |
---|---|---|---|---|

0 | $N\pm 1/2$ | 0 | N | N |

$N-1/2$ | 1 | $N+1$ | $N-1$ | |

$N+1/2$ | −1 | $N-1$ | $N+1$ | |

1 | $N+1/2$ | 0 | N | $N+1$ |

$N\pm 1/2$ | 1 | $N+1$ | N | |

−1 | $N-1/2$ | 0 | N | $N-1$ |

$N\pm 1/2$ | −1 | $N-1$ | N | |

2 | $N+1/2$ | 1 | $N+1$ | $N+1$ |

−2 | $N-1/2$ | -1 | $N-1$ | $N-1$ |

**Table 5.**Illustration of the “forbidden” $\Delta K=2$ transitions in the ${\nu}_{3}$ band of

^{35}ClO

_{2}.

Transition | ${\mathit{\nu}}_{\mathbf{calc}.}$ | ${\mathit{\nu}}_{\mathbf{exp}.}$ | Transmitt. | ||
---|---|---|---|---|---|

$\left(\right)$ | − | $\left(\right)$ | in cm^{−1} | in cm^{−1} | in % |

1 | 2 | 3 | 4 | ||

$\left(\right)$ | − | $\left(\right)$ | 1106.7267 | 1106.7262 | 92 |

$\left(\right)$ | − | $\left(\right)$ | 1106.7824 | 1106.7828 | 92 |

$\left(\right)$ | − | $\left(\right)$ | 1107.7978 | 1107.7976 | 95 |

$\left(\right)$ | − | $\left(\right)$ | 1107.8319 | 1107.8315 | 89 |

$\left(\right)$ | − | $\left(\right)$ | 1108.6547 | 1108.6546 | 86 |

$\left(\right)$ | − | $\left(\right)$ | 1108.6882 | covered | 62 |

$\left(\right)$ | − | $\left(\right)$ | 1109.1992 | 1109.2000 | 80 |

$\left(\right)$ | − | $\left(\right)$ | 1109.2575 | covered | 61 |

$\left(\right)$ | − | $\left(\right)$ | 1109.3568 | 1109.3563 | 85 |

$\left(\right)$ | − | $\left(\right)$ | 1109.4050 | 1109.4050 | 80 |

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Ulenikov, O.; Bekhtereva, E.; Gromova, O.; Kakaulin, A.; Sydow, C.; Bauerecker, S.
Improved Theory of the Effective Dipole Moments and Absolute Line Strengths of the XY_{2} Asymmetric Top Molecules in the *X*^{2}*B*_{1} Doublet Electronic States. *Int. J. Mol. Sci.* **2023**, *24*, 12734.
https://doi.org/10.3390/ijms241612734

**AMA Style**

Ulenikov O, Bekhtereva E, Gromova O, Kakaulin A, Sydow C, Bauerecker S.
Improved Theory of the Effective Dipole Moments and Absolute Line Strengths of the XY_{2} Asymmetric Top Molecules in the *X*^{2}*B*_{1} Doublet Electronic States. *International Journal of Molecular Sciences*. 2023; 24(16):12734.
https://doi.org/10.3390/ijms241612734

**Chicago/Turabian Style**

Ulenikov, Oleg, Elena Bekhtereva, Olga Gromova, Aleksei Kakaulin, Christian Sydow, and Sigurd Bauerecker.
2023. "Improved Theory of the Effective Dipole Moments and Absolute Line Strengths of the XY_{2} Asymmetric Top Molecules in the *X*^{2}*B*_{1} Doublet Electronic States" *International Journal of Molecular Sciences* 24, no. 16: 12734.
https://doi.org/10.3390/ijms241612734