#
Effective Dipole Moment Model for Axially Symmetric C_{3v} Molecules: Application to the Precise Study of Absolute Line Strengths of the ν_{6} Fundamental of CH_{3}^{35}Cl

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Materials

## 3. Results and Discussion

#### 3.1. General Information and Assignment of Transitions

Band | Center/cm${}^{-1}$ | ${\mathit{J}}^{\mathbf{max}}$ | ${\mathit{K}}_{\mathit{a}}^{\mathbf{max}}$ | ${\mathit{N}}_{\mathbf{tr}}$ ${}^{\left(\mathbf{a}\right)}$ | ${\mathit{N}}_{\mathit{l}}$ ${}^{\left(\mathbf{b}\right)}$ | ${\mathit{m}}_{1}$ ${}^{\left(\mathbf{c}\right)}$ | ${\mathit{m}}_{2}$ ${}^{\left(\mathbf{c}\right)}$ | ${\mathit{m}}_{3}$${}^{\left(\mathbf{c}\right)}$ |
---|---|---|---|---|---|---|---|---|

${\nu}_{6}$ | 1118.070790 | 68 | 21 | 5124 | 2077 | 88.9 | 8.8 | 2.3 |

${d}_{\mathrm{rms}}$ ${}^{\left(\mathrm{d}\right)}$ | $4.7\times {10}^{-5}$ |

**Table 3.**Small part of the list of transitions assigned to the ${\nu}_{6}$ band of CH${}_{3}{}^{35}$Cl.

J | K | $\Gamma $ | ${\mathit{J}}^{\prime}$ | ${\mathit{K}}^{\prime}$ | ${\Gamma}^{\prime}$ | $\mathit{\nu}$ ${}^{\left(\mathbf{a}\right)}$ | ${\mathit{\delta}}_{\mathit{\nu}}$ ${}^{\left(\mathbf{b}\right)}$ | ${\mathit{S}}_{\mathit{\nu}}^{\mathbf{exp}}$(294.45) ${}^{\left(\mathbf{c}\right)}$ | ${\Delta}_{\mathit{S}}$ ${}^{\left(\mathbf{d}\right)}$ | ${\mathit{S}}_{\mathit{\nu}}^{\mathbf{calc}}$(294.45) ${}^{\left(\mathbf{e}\right)}$ | ${\mathit{\delta}}_{\mathit{S}}$ ${}^{\left(\mathbf{f}\right)}$ | ${\Gamma}_{0}$ ${}^{\left(\mathbf{g}\right)}$ | ${\Delta}_{2}$ ${}^{\left(\mathbf{g}\right)}$ | ${\mathit{\nu}}_{\mathbf{VC}}$ ${}^{\left(\mathbf{g}\right)}$ | R ${}^{\left(\mathbf{h}\right)}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | ||||

5 | 1 | E | 4 | 2 | E | 1010.81799 | 5 | 0.231768 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.5 | 0.2321 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | −0.15 | 2.987 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | −0.224 | −0.422 | 1.3 |

13 | 2 | ${A}_{2}$ | 12 | 3 | ${A}_{1}$ | 1010.83845 | 1 | 0.108382 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 0.2 | 0.5389 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.56 | 3.525 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | −0.150 | −0.449 | 1.7 |

13 | 2 | ${A}_{1}$ | 12 | 3 | ${A}_{2}$ | 1010.83845 | −1 | 0.5389 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | |||||||

38 | 0 | E | 38 | 1 | E | 1010.88494 | 0 | 0.320599 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 3.9 | 0.2939 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 8.32 | 2.815 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

37 | 0 | E | 37 | 1 | E | 1011.00984 | 0 | 0.366556 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.4 | 0.3401 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 7.22 | 3.206 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

36 | 0 | E | 36 | 1 | E | 1011.13137 | −1 | 0.422301 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.6 | 0.3915 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 7.30 | 2.999 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 0.136 | 1.1 | |

30 | 4 | E | 29 | 5 | E | 1011.15162 | 8 | 0.206013 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.0 | 0.1975 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 4.11 | 2.808 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

24 | 3 | E | 25 | 2 | E | 1011.15634 | −2 | 0.499552 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.4 | 0.5107 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | −2.23 | 3.976 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

9 | 1 | ${A}_{2}$ | 10 | 0 | ${A}_{1}$ | 1011.20573 | 1 | 0.157995 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 0.2 | 0.1596 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | −0.99 | 4.175 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

35 | 0 | E | 35 | 1 | E | 1011.24956 | −3 | 0.476969 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.3 | 0.4482 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 6.03 | 2.332 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | −0.750 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | −0.315 | 1.2 |

48 | 6 | E | 47 | 7 | E | 1011.30392 | −59 | 0.8773 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | |||||||

39 | 5 | ${A}_{2}$ | 38 | 6 | ${A}_{1}$ | 1011.36055 | 12 | 0.109227 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.4 | 0.5266 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 3.57 | 2.970 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

39 | 5 | ${A}_{1}$ | 38 | 6 | ${A}_{2}$ | 1011.36055 | 12 | 0.5266 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |||||||

34 | 0 | E | 34 | 1 | E | 1011.36443 | −2 | 0.549711 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.4 | 0.5104 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 7.14 | 2.970 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

31 | 4 | ${A}_{2}$ | 32 | 3 | ${A}_{1}$ | 1011.44531 | −6 | 0.464102 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.6 | 0.2429 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | −4.69 | 3.126 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | −0.159 | 1.1 | |

31 | 4 | ${A}_{1}$ | 32 | 3 | ${A}_{2}$ | 1011.44531 | −6 | 0.2429 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | |||||||

33 | 0 | E | 33 | 1 | E | 1011.47598 | 0 | 0.614718 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.2 | 0.5782 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 5.94 | 2.449 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | −0.129 | −0.317 | 2.0 |

22 | 3 | E | 21 | 4 | E | 1011.50323 | 0 | 0.426391 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.6 | 0.4292 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | −0.67 | 4.391 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

1 | 0 | E | 2 | 1 | E | 1011.53211 | −1 | 0.336528 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.5 | 0.3237 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 3.82 | 4.895 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

32 | 0 | E | 32 | 1 | E | 1011.58418 | −1 | 0.703075 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.4 | 0.6514 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 7.35 | 3.079 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

38 | 5 | E | 39 | 4 | E | 1011.64991 | −9 | 0.916940 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 2.7 | 0.8656 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 5.60 | 2.000 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

14 | 2 | ${A}_{2}$ | 13 | 3 | ${A}_{1}$ | 1011.67808 | 0 | 0.117754 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 0.3 | 0.5676 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 3.60 | 3.914 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

14 | 2 | ${A}_{1}$ | 13 | 3 | ${A}_{2}$ | 1011.67808 | 2 | 0.5676 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | |||||||

6 | 1 | E | 5 | 2 | E | 1011.68498 | 7 | 0.323333 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.5 | 0.3157 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 2.37 | 3.697 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

31 | 0 | E | 31 | 1 | E | 1011.68904 | −2 | 0.781218 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.5 | 0.7299 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 6.57 | 3.697 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

16 | 2 | E | 17 | 1 | E | 1011.71876 | −1 | 0.789033 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.3 | 0.7925 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | −0.44 | 4.412 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | −0.549 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | −0.266 | 1.1 |

45 | 6 | E | 46 | 5 | E | 1011.77417 | 24 | 0.2339 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | |||||||

30 | 0 | E | 30 | 1 | E | 1011.79061 | −1 | 0.854407 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.5 | 0.8133 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 4.81 | 3.123 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 | ||

29 | 0 | E | 29 | 1 | E | 1011.88885 | −1 | 0.938728 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.5 | 0.9012 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 4.00 | 3.487 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.0 |

**Table 4.**Effective dipole moment parameters of the ${\nu}_{6}$ band of CH${}_{3}{}^{35}$Cl (in Debye).

Operator | Parameter | Value |
---|---|---|

${R}_{\sigma}^{E(0,{A}_{1})E}={k}_{\sigma}^{E}$ | ${P}^{E(0,{A}_{1})}\times 10$ | 0.55712(72) |

${R}_{\sigma}^{E(2,1{A}_{1})E}$ | ${P}^{E(2,1{A}_{1})}\times {10}^{3}$ | 0.1466(89) |

${R}_{\sigma}^{{A}_{2}(2,1E)E}$ | ${P}^{{A}_{2}(2,1E)}\times {10}^{3}$ | −0.493(54) |

${R}_{\sigma}^{E(4,3{A}_{1})E}$ | ${P}^{E(4,3{A}_{1})}\times {10}^{5}$ | −0.2249(25) |

${R}_{\sigma}^{{A}_{2}(4,1E)E}$ | ${P}^{{A}_{2}(4,1E)}\times {10}^{5}$ | −0.3378(49) |

${R}_{\sigma}^{{A}_{2}(4,2E)E}$ | ${P}^{{A}_{2}(4,2E)}\times {10}^{5}$ | −1.024(62) |

#### 3.2. Theoretical Background for the Effective Hamiltonian Used

#### 3.2.1. Effective Rotational–Vibrational Hamiltonian

- (1)
- The requirement ${\tilde{H}}_{1n}={\tilde{H}}_{n1}=0(n\ne 1)$ is equivalent to fulfilling the conditions (which follow from Equations (11)–(13), etc):$$\begin{array}{c}\hfill \langle {v}_{1}\mid {h}_{1}\mid {v}_{n}\rangle +({E}_{1}^{0}-{E}_{n}^{0})\langle {v}_{1}\mid i{g}_{1}\mid {v}_{n}\rangle =0,\end{array}$$$$\begin{array}{ccc}& & \langle {v}_{1}\mid {h}_{2}\mid {v}_{n}\rangle +\langle {v}_{1}\mid {h}_{1}i{g}_{1}-i{g}_{1}{h}_{1}\mid {v}_{n}\rangle +({E}_{1}^{0}-{E}_{n}^{0})\langle {v}_{1}\mid i{g}_{2}\mid {v}_{n}\rangle \hfill \\ & & +\frac{1}{2}({E}_{1}^{0}+{E}_{n}^{0})\langle {v}_{1}\mid {\left(i{g}_{1}\right)}^{2}\mid {v}_{n}\rangle -2\langle {v}_{1}\mid i{g}_{1}{H}_{0}i{g}_{1}\mid {v}_{n}\rangle =0,\hfill \end{array}$$
- (2)
- All the above said allows us to present the Hamiltonian $\tilde{H}$ in the following form:$$\begin{array}{ccc}\hfill \tilde{H}& & =\sum _{n,m}\mid {v}_{n}\rangle \left(\langle {v}_{n}\mid \tilde{H}\mid {v}_{m}\rangle \right)\langle {v}_{m}\mid =\sum _{nm}\mid {v}_{n}\rangle {\tilde{H}}_{n,m}\langle {v}_{m}\mid \hfill \\ & & =\mid {v}_{1}\rangle {\tilde{H}}_{11}\langle {v}_{1}\mid +\sum _{n,m\ne 1}\mid {v}_{n}\rangle {\tilde{H}}_{nm}\langle {v}_{m}\mid .\hfill \end{array}$$
- (3)
- As was discussed above, if one is interested in the rotational structure of the only one vibrational state $\mid {v}_{1}\rangle $, then the second term on the right side of Equation (19) is insignificant and can be omitted from the further consideration. As for the first term, it obviously has the form of a function of coordinates of the second “y”-type (for vibrational–rotational problems, they are the Euler angles $\phi ,\theta ,$ and $\chi $; in this case, the dependence of the effective Hamiltonian on the angular variables is manifested in the form of its dependence on the components ${J}_{x},{J}_{y}$, and ${J}_{z}$ of the angular momentum operator J.

- (a)
- Hermitian;
- (b)
- Totally symmetric (transformed in accordance with a symmetric irreducible representation of a molecule symmetry group);
- (c)
- Invariant according to the time reversal operation.

- (1)
- Provide ${a}_{1}/{a}_{2}$ splittings of ro–vibrational energies ${E}_{[J\phantom{\rule{3.33333pt}{0ex}}K,{a}_{1}]}$ and ${E}_{[J\phantom{\rule{3.33333pt}{0ex}}K,{a}_{2}]}$ for different values of the quantum number K (the operators ${C}^{e}$ and ${D}^{e}$ are responsible for the ${a}_{1}/{a}_{2}$ splittings for states with $K=1$, operators ${F}^{e}$ and ${G}^{e}$ are responsible for the ${a}_{1}/{a}_{2}$ splittings for states with $K=2$).
- (2)
- Are responsible (the ${J}_{+}^{2}\pm {J}_{-}^{2}$ operators) for the borrowing of intensities from one $Q\u2014$sub-band to the other.

#### 3.2.2. Symmetrized Ro–Vibrational Functions

- (1)
- The functions $|Jk\rangle $ are also symmetrized functions, and any set of $|Jk\rangle $ functions (for J fixed and $-J\le k\le J$) is transformed in accordance with the irreducible representation ${D}^{\left(J\right)}$ of the SO(3) symmetry group (Ref. [104]).
- (2)
- Because the C${}_{3v}$ symmetry group is a subgroup of the SO(3) group, any irreducible representation ${D}^{\left(J\right)}$ of the SO(3) group is divided into a set of irreducible representations ($\gamma ={a}_{1},{a}_{2},$ and e) of the C${}_{3v}$ group. In this case (in accordance with the general rules of the theory of group, see, e.g., [106]), one can construct superpositions of functions $|Jk\rangle $ which will be transformed already in accordance with irreducible representations of the C${}_{3v}$ group. Such pure rotational (symmetrized in the C${}_{3v}$ group) functions have the form$$\begin{array}{ccc}\hfill \phantom{\rule{-5.69046pt}{0ex}}\sqrt{2}|JK{\gamma}_{r}{\sigma}_{r}\rangle & =& {C}_{JK{\gamma}_{r}{\sigma}_{r}}^{l}\left\{\right|JK\rangle +{(-1)}^{l}{(-1)}^{J+K}|J-K\rangle \}.\hfill \end{array}$$

#### 3.2.3. Hamiltonian Matrix Elements

#### 3.3. Ro–Vibrational Analysis and Parameters of the Effective Hamiltonian

#### 3.4. Line Strengths: Experimental Intensities of Ro–Vibrational Lines of the ${\nu}_{6}$ Band

#### 3.5. Line Strength Analysis: Improvement of the Model and Calculation Scheme

#### 3.5.1. Effective Dipole Moment Operator for the $E$-Type Band

#### 3.5.2. Irreducible Rotational Operators of the SO(3) and ${C}_{3v}$ Symmetry First-Order Operators

- (1)
- The rotational operators, Equations (56)–(58), satisfy the commutation relations; see, e.g., [95]:$$\begin{array}{c}\hfill {({J}_{\alpha},{J}_{\beta})}_{-}={J}_{\alpha}{J}_{\beta}-{J}_{\beta}{J}_{\alpha}=-i\sum _{\gamma}{\u03f5}_{\alpha \beta \gamma}{J}_{\gamma},\end{array}$$
- (2)
- They also satisfy the important transformation conditions [107]:$$\begin{array}{c}\hfill ({J}_{x}\mp i{J}_{y})\mid Jk\rangle ={\left\{J(J+1)-k(k\pm 1)\right\}}^{1/2}\mid Jk\pm 1\rangle \end{array}$$$$\begin{array}{c}\hfill {J}_{z}\mid Jk\rangle =k\mid Jk\rangle ;\end{array}$$
- (3)
- As a consequence, three operators ${R}_{m}^{1\left(1\right)}$ ($m=0,\pm 1$) can be constructed:$$\begin{array}{c}\hfill {R}_{0}^{1\left(1\right)}={J}_{z},\end{array}$$$$\begin{array}{c}\hfill {R}_{\pm 1}^{1\left(1\right)}=\mp \frac{1}{\sqrt{2}}({J}_{x}\mp i{J}_{y}),\end{array}$$$$\begin{array}{c}\hfill \langle \tilde{J}\tilde{k}\mid {R}_{m}^{1\left(1\right)}\mid Jk\rangle =\frac{1}{{(2\tilde{J}+1)}^{1/2}}{C}_{Jk,1m}^{\tilde{J}\tilde{k}}<\tilde{J}\Vert {R}^{1\left(1\right)}\Vert J>\end{array}$$$$\begin{array}{c}\hfill \langle J\Vert {R}^{1\left(1\right)}\Vert J\rangle ={\left\{(2J+1)J(J+1)\right\}}^{1/2}.\end{array}$$Following the scheme of connection of irreducible tensorial operators of the $SO\left(3\right)$ and ${C}_{3v}$ symmetry groups (see, e.g., [78]), one can obtain three first-order irreducible rotational operators that are symmetrized in accordance with irreducible representations of the ${C}_{3v}$ point symmetry group:$$\begin{array}{c}\hfill {R}^{1\left({A}_{2}\right)}\equiv {J}_{z}={R}_{0}^{1\left(1\right)}\in {A}_{2},\end{array}$$$$\begin{array}{c}\hfill {R}_{1}^{1\left(E\right)}\equiv {J}_{y}=-\frac{i}{\sqrt{2}}({R}_{-1}^{1\left(1\right)}+{R}_{1}^{1\left(1\right)})\in {E}_{1},\end{array}$$$$\begin{array}{c}\hfill {R}_{2}^{1\left(E\right)}\equiv {J}_{x}=\frac{1}{\sqrt{2}}({R}_{-1}^{1\left(1\right)}-{R}_{1}^{1\left(1\right)})\in {E}_{2}.\end{array}$$Taking into account Equations (60)–(65), one can obtain$$\begin{array}{c}\hfill \langle \tilde{J}\tilde{k}\mid {R}^{1\left({A}_{2}\right)}\mid Jk\rangle \equiv \langle \tilde{J}\tilde{k}\mid {J}_{z}\mid Jk\rangle ={\delta}_{J\tilde{J}}{\delta}_{k\tilde{k}}k,\end{array}$$$$\begin{array}{ccc}& & \langle \tilde{J}k\pm 1\mid {R}_{2}^{1\left(E\right)}\mid Jk\rangle \equiv \langle \tilde{J}k\pm 1\mid {J}_{x}\mid Jk\rangle =\mp \langle \tilde{J}k\pm 1\mid i{R}_{1}^{1\left(E\right)}\mid Jk\rangle \hfill \\ & & \equiv \mp \langle \tilde{J}k\pm 1\mid i{J}_{y}\mid Jk\rangle ={\delta}_{J\tilde{J}}{\left\{(J\pm k+1)(J\mp k)\right\}}^{1/2}\hfill \end{array}$$

#### 3.5.3. The ${\lambda}_{m}^{\left(1\right)}$ and ${\lambda}_{\sigma}^{(\Gamma )}$ Irreducible Direction Cosines Operators

- (1)
- The direction cosines, which are obtained in such a way, satisfy the commutation relations$$\begin{array}{c}\hfill {({J}_{\alpha},{k}_{Z\beta})}_{-}=-i\sum _{\gamma}{\u03f5}_{\alpha \beta \gamma}{k}_{Z\gamma},\end{array}$$
- (2)
- Their combinations$$\begin{array}{c}\hfill {\lambda}_{0}^{1}={k}_{Zz},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{\pm 1}^{1}=\mp \frac{1}{\sqrt{2}}({k}_{Zx}\mp \mathrm{i}{k}_{Zy})\end{array}$$
- (3)
- As a consequence, for the nonzero matrix elements $<\tilde{J}\tilde{k}\mid {\lambda}_{m}^{\left(1\right)}\mid Jk>$, the following relations (analogous to Equation (64)) are valid:$$\begin{array}{c}\hfill \langle \tilde{J}\tilde{k}\mid {\lambda}_{m}^{\left(1\right)}\mid Jk\rangle =\frac{1}{{(2J+1)}^{1/2}}{C}_{Jk,1m}^{\tilde{J}\tilde{k}}\langle \tilde{J}\Vert {\lambda}^{\left(1\right)}\Vert J\rangle \end{array}$$$$\begin{array}{c}\hfill \langle J\Vert {\lambda}^{\left(1\right)}\Vert J\rangle =(2J+1),\end{array}$$$$\begin{array}{c}\hfill \langle J\Vert {\lambda}^{\left(1\right)}\Vert J+1\rangle =-{\left\{(2J+1)(2J+3)\right\}}^{1/2},\end{array}$$$$\begin{array}{c}\hfill \langle J\Vert {\lambda}^{\left(1\right)}\Vert J-1\rangle ={\left\{(2J+1)(2J-1)\right\}}^{1/2};\end{array}$$
- (4)
- Analogously to Equations (66)–(68), it is possible to show that three irreducible operators ${\lambda}_{\sigma}^{(\Gamma )}$ of the ${C}_{3v}$ point symmetry group are$$\begin{array}{c}\hfill {\lambda}^{\left({A}_{2}\right)}\equiv {k}^{\left({A}_{2}\right)}={k}_{Zz},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{1}^{\left(E\right)}\equiv {k}_{1}^{\left(E\right)}={k}_{Zy},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{2}^{\left(E\right)}\equiv {k}_{2}^{\left(E\right)}={k}_{Zx},\end{array}$$$\mid Jk>$ are$$\begin{array}{c}\hfill \langle Jk\mid {k}_{Zz}\mid Jk\rangle =k{\left\{\frac{(2J+1)}{J(J+1)}\right\}}^{1/2},\end{array}$$$$\begin{array}{ccc}& & \langle Jk\mid {k}_{Zx}\mid Jk\pm 1\rangle =\pm \langle Jk\mid i{k}_{Zy}\mid Jk\pm 1\rangle \hfill \\ & & =\frac{1}{2}{\left\{\frac{(2J+1)(J\mp k)(J\pm k+1)}{J(J+1)}\right\}}^{1/2};\hfill \end{array}$$$$\begin{array}{c}\hfill \langle Jk\mid {k}_{Zz}\mid J+1k\rangle ={\left\{\frac{(J+k+1)(J-k+1)}{(J+1)}\right\}}^{1/2},\end{array}$$$$\begin{array}{ccc}& & \langle Jk\mid {k}_{Zx}\mid J+1k\pm 1\rangle =\pm \langle Jk\mid i{k}_{Zy}\mid J+1k\pm 1\rangle \hfill \\ & & =\mp \frac{1}{2}{\left\{\frac{(J\pm k+1)(J\pm k+2)}{(J+1)}\right\}}^{1/2};\hfill \end{array}$$$$\begin{array}{c}\hfill \langle Jk\mid {k}_{Zz}\mid J-1k\rangle ={\left\{\frac{(J+k)(J-k)}{J}\right\}}^{1/2},\end{array}$$$$\begin{array}{ccc}& & \langle Jk\mid {k}_{Zx}\mid J-1k\pm 1\rangle =\pm \langle Jk\mid i{k}_{Zy}\mid J-1k\pm 1\rangle \hfill \\ & & =\pm \frac{1}{2}{\left\{\frac{(J\mp k)(J\mp k-1)}{J}\right\}}^{1/2}.\hfill \end{array}$$

#### 3.5.4. Effective Dipole Moment Operator: The Main Part

#### 3.5.5. Effective Dipole Moment Operator: First Order Corrections

#### 3.5.6. Effective Dipole Moment Operator: Second-Order Corrections

#### 3.5.7. Effective Dipole Moment Operator: Higher-Order Corrections

#### 3.6. Line Strengths Analysis: Determination of Effective Dipole Moment Parameters and Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Small part of the high-resolution spectrum I in the region of the $Q$—branch of CH${}_{3}{}^{35}$Cl. The ${}^{P}{Q}_{4}\left(J\right)$ transitions are marked by dark circles. Dark triangles denote other transitions (not belonging to the ${}^{P}{Q}_{4}\left(J\right)$ set). Unmarked lines belong probably to the CH${}_{3}{}^{37}$Cl ${\nu}_{6}$ band transitions.

**Figure 3.**Part of the high-resolution spectrum I in the region of the $R\u2014$branch of CH${}_{3}{}^{35}$Cl. Two sets of transitions (${}^{R}{R}_{9}\left(J\right)$ marked by dark circles, and ${}^{R}{R}_{8}\left(J\right)$ marked by dark triangles) are shown. Some sets of the ${}^{R}{Q}_{K}\left(J\right)$ clusters are also seen.

**Figure 4.**Detail of the infrared spectrum I of the ${\nu}_{6}$ band of CH${}_{3}{}^{35}$Cl showing sets of ${}^{P}{P}_{9}\left(J\right)$ (dark circles) and ${}^{P}{P}_{6}\left(J\right)$ (dark triangles) transitions. Three $Q$-type clusters are also indicated.

**Figure 5.**Residuals ${E}_{i}^{\mathrm{obs}}-{E}_{i}^{\mathrm{calc}}$ of effective Hamiltonian fit calculations of the ${\nu}_{6}\left(E\right)$ band of CH${}_{3}{}^{35}$Cl dependent on upper state quantum number J.

Spectrum | Region /cm${}^{-1}$ | Resolution /cm${}^{-1}$ | No. of Scans | Source | Detector | Beam- Splitter | Aperture /mm | Opt. Path- Length/m | Temp. /K | Pressure /Pa | Calibr. Gas |
---|---|---|---|---|---|---|---|---|---|---|---|

I | 500–1700 | 0.0025 | 1000 | Globar | MCT | KBr | 1.5 | 4 | $(294.45\pm 0.3)$ K | 50 | H${}_{2}$O, CO${}_{2}$ |

II | 500–5000 | 0.003 | 1860 | Globar | MCT | KBr | 1.0 | 24 | $(294.45\pm 0.3)$ K | 300 | H${}_{2}$O, CO${}_{2}$ |

J | K | ${\mathit{\gamma}}_{\mathit{r}}{\mathit{\sigma}}_{\mathit{r}}$ | l | Value |
---|---|---|---|---|

$\mathrm{Even}$ | 0 | ${a}_{1}$ | 0 | ${\left(-i\right)}^{J}\sqrt{2}$ |

$\mathrm{Odd}$ | 0 | ${a}_{2}$ | 1 | ${\left(-i\right)}^{J-1}\sqrt{2}$ |

$\mathrm{Any}$ | 3, 6, 9, … | ${a}_{1}$ | 0 | ${(-1)}^{J}$ |

${a}_{2}$ | 1 | ${(-1)}^{J+1}$ | ||

1, 4, 7, … | ${e}_{1}$ | 0 | ${(-1)}^{J}$ | |

${e}_{2}$ | 1 | ${(-1)}^{J+1}$ | ||

2, 5, 8, … | ${e}_{1}$ | 0 | ${(-1)}^{J}$ | |

${e}_{2}$ | 1 | ${(-1)}^{J-1}$ |

${\mathit{\gamma}}_{1}{\mathit{\sigma}}_{1}$ | ${\mathit{\gamma}}_{2}{\mathit{\sigma}}_{2}$ | ${\mathit{\gamma}}_{3}{\mathit{\sigma}}_{3}$ | Value | ${\mathit{\gamma}}_{1}{\mathit{\sigma}}_{1}$ | ${\mathit{\gamma}}_{2}{\mathit{\sigma}}_{2}$ | ${\mathit{\gamma}}_{3}{\mathit{\sigma}}_{3}$ | Value |
---|---|---|---|---|---|---|---|

${A}_{1}$ | ${A}_{1}$ | ${A}_{1}$ | 1 | ${A}_{2}$ | ${E}_{1}$ | ${E}_{2}$ | $1/\sqrt{2}$ |

${A}_{1}$ | ${A}_{2}$ | ${A}_{2}$ | 1 | ${E}_{1}$ | ${E}_{1}$ | ${E}_{1}$ | $-1/2$ |

${A}_{1}$ | ${E}_{1}$ | ${E}_{1}$ | $1/\sqrt{2}$ | ${E}_{1}$ | ${E}_{2}$ | ${E}_{2}$ | $1/2$ |

${A}_{1}$ | ${E}_{2}$ | ${E}_{2}$ | $1/\sqrt{2}$ |

**Table 7.**Coefficients ${A}_{JKm\gamma \sigma}^{{\gamma}_{r}{\sigma}_{r}}$ and ${B}_{JKm\gamma \sigma}^{{\gamma}_{r}{\sigma}_{r}}$ of symmetrized ro–vibrational functions.

J | K | $\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathit{m}\mathit{\gamma}\mathit{\sigma}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$ | $\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mathit{\gamma}}_{\mathit{r}}{\mathit{\sigma}}_{\mathit{r}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$ | ${\mathit{A}}_{\mathit{J}\mathit{K}\mathit{m}\mathit{\gamma}\mathit{\sigma}}^{{\mathit{\gamma}}_{\mathit{r}}\mathit{\sigma}}$ | ${\mathit{B}}_{\mathit{J}\mathit{K}\mathit{m}\mathit{\gamma}\mathit{\sigma}}^{{\mathit{\gamma}}_{\mathit{r}}\mathit{\sigma}}$ |
---|---|---|---|---|---|

$any$ | $k\ne 0,3,6,9,\dots $ | ${a}_{1}$ | ${e}_{1}$ | 1 | |

${e}_{2}$ | 1 | ||||

${a}_{2}$ | ${e}_{1}$ | $-1$ | |||

${e}_{2}$ | 1 | ||||

${e}_{1}$ | ${e}_{1}$ | $-1$ | |||

${e}_{2}$ | 1 | ||||

${e}_{2}$ | ${e}_{1}$ | 1 | |||

$any$ | k = 3, 6, 9,… | $1{e}_{1}$ | ${a}_{1}$ | 1 | |

${a}_{2}$ | 1 | ||||

$1{e}_{2}$ | ${a}_{1}$ | 1 | |||

${a}_{2}$ | $-1$ | ||||

$2{e}_{1}$ | ${a}_{1}$ | 1 | |||

${a}_{2}$ | $-1$ | ||||

$2{e}_{2}$ | ${a}_{1}$ | 1 | |||

${a}_{2}$ | 1 | ||||

$even$ | k = 0 | ${e}_{1}$ | ${a}_{1}$ | $\sqrt{2}$ | |

${e}_{2}$ | ${a}_{1}$ | $\sqrt{2}$ | |||

$odd$ | k = 0 | ${e}_{1}$ | ${a}_{2}$ | $\sqrt{2}$ | |

${e}_{2}$ | ${a}_{2}$ | $-\sqrt{2}$ |

**Table 8.**Spectroscopic parameters of the $({v}_{6}=1)$ state of CH${}_{3}{}^{35}$Cl (in cm${}^{-1}$) ${}^{\left(\mathrm{a}\right)}$.

Parameter | Value |
---|---|

E | 1018.0707900(43) |

B | 0.4417686446(97) |

C | 5.23070591(17) |

${D}_{J}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | 0.6049990(56) |

${D}_{JK}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}$ | 0.678610(25) |

${D}_{K}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$ | 0.85663560(15) |

${H}_{J}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{12}$ | −0.33657(93) |

${H}_{JK}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{10}$ | 0.14360(66) |

${H}_{KJ}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{9}$ | 0.1873(22) |

${H}_{K}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{8}$ | 0.13300(49) |

${L}_{JK}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{13}$ | −0.1389(42) |

${L}_{KKJ}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{12}$ | −0.2198(49) |

$2C\zeta $ | 2.6202558(10) |

${\eta}_{J}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$ | −0.156270(14) |

${\eta}_{K}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$ | −0.112648(17) |

${\eta}_{JJ}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{10}$ | 0.8931(92) |

${\eta}_{JK}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{8}$ | −0.2986(22) |

${\eta}_{KK}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{7}$ | 0.12944(56) |

${\eta}_{JKK}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{11}$ | −0.8241(79) |

$\gamma \phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$ | −0.1205968(96) |

${\gamma}_{J}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{9}$ | 0.5859(14) |

${\gamma}_{K}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ | −0.4166(42) |

$\kappa \phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{9}$ | 0.145(15) |

${\kappa}_{J}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{12}$ | −0.2327(92) |

${\kappa}_{JJ}\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{16}$ | 0.415(20) |

Spectrum | CH${}_{3}$${}^{35}$Cl | CH${}_{3}$${}^{37}$Cl | H${}_{2}$O | CO${}_{2}$ |
---|---|---|---|---|

I | 74.21 | 23.73 | 2.05 ± 0.22 | 0.013 ± 0.001 |

**Table 10.**Effective dipole moment parameters of the E-type band of axially symmetric (${C}_{3v}$) molecule.

Operator, ${\mathit{R}}_{\mathit{\sigma}}^{\Gamma (\mathrm{\Omega},\mathit{n}\tilde{\Gamma})\mathit{E}}$ | Parameter, ${\mathit{P}}^{\Gamma (\mathrm{\Omega},\mathit{n}\tilde{\Gamma})}$ | ${\mathit{C}}^{\Gamma (\mathrm{\Omega},\mathit{n}\tilde{\Gamma})}$ Coefficient, Equation (102) | |
---|---|---|---|

1 | ${R}_{\sigma}^{E(0,{A}_{1})E}={k}_{\sigma}^{E}$ | ${P}^{E(0,{A}_{1})}$ | 1 |

2 | ${R}_{\sigma}^{E(2,1{A}_{1})E}$ | ${P}^{E(2,1{A}_{1})}$ | $\frac{1}{2}\left\{J(J+1)+\tilde{J}(\tilde{J}+1)\right\}$ |

3 | ${R}_{\sigma}^{E(2,2{A}_{1})E}$ | ${P}^{E(2,2{A}_{1})}$ | $\frac{1}{2}\left({k}^{2}+{\tilde{k}}^{2}\right)$ |

4 | ${R}_{\sigma}^{E(2,2E)E}$ | ${P}^{E(2,2E)}$ | $-\frac{1}{2}\left\{J(J+1)-k\tilde{k}-1\right\}$, $\Delta J=0,\Delta k=\pm 1$ |

$\frac{1}{2}\left\{(J-\Delta J\Delta kk)(\tilde{J}-\Delta J\Delta k\tilde{k}+1)+1\right\}$, $\Delta J=\pm 1,\Delta k=\pm 1$ | |||

5 | ${R}_{\sigma}^{{A}_{2}(2,1E)E}$ | ${P}^{{A}_{2}(2,1E)}$ | $-\frac{\Delta k}{2}\left(k+\tilde{k}\right)$ |

6 | ${R}_{\sigma}^{E(4,1{A}_{1})E}$ | ${P}^{E(4,1{A}_{1})}$ | $\frac{1}{2}\left\{{J}^{2}{(J+1)}^{2}+{\tilde{J}}^{2}{(\tilde{J}+1)}^{2}\right\}$ |

7 | ${R}_{\sigma}^{E(4,2{A}_{1})E}$ | ${P}^{E(4,2{A}_{1})}$ | $\frac{1}{2}\left\{J(J+1){k}^{2}+\tilde{J}(\tilde{J}+1){\tilde{k}}^{2}\right\}$ |

8 | ${R}_{\sigma}^{E(4,3{A}_{1})E}$ | ${P}^{E(4,3{A}_{1})}$ | $\frac{1}{2}\left\{{k}^{4}+{\tilde{k}}^{4}\right\}$ |

9 | ${R}_{\sigma}^{{A}_{2}(4,1E)E}$ ${}^{\left(\mathrm{a}\right)}$ | ${P}^{{A}_{2}(4,1E)}$ | $-\frac{\Delta k}{2}\left(k+\tilde{k}\right)\left\{J(J+1)+\tilde{J}(\tilde{J}+1)\right\}$ |

10 | ${R}_{\sigma}^{{A}_{2}(4,2E)E}$ ${}^{\left(\mathrm{b}\right)}$ | ${P}^{{A}_{2}(4,2E)}$ | $-\frac{\Delta k}{2}(k+\tilde{k})({k}^{2}+{\tilde{k}}^{2})$ |

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Ulenikov, O.; Bekhtereva, E.; Gromova, O.; Fomchenko, A.; Morzhikova, Y.; Sidko, S.; Sydow, C.; Bauerecker, S.
Effective Dipole Moment Model for Axially Symmetric *C*_{3v} Molecules: Application to the Precise Study of Absolute Line Strengths of the *ν*_{6} Fundamental of CH_{3}^{35}Cl. *Int. J. Mol. Sci.* **2023**, *24*, 12122.
https://doi.org/10.3390/ijms241512122

**AMA Style**

Ulenikov O, Bekhtereva E, Gromova O, Fomchenko A, Morzhikova Y, Sidko S, Sydow C, Bauerecker S.
Effective Dipole Moment Model for Axially Symmetric *C*_{3v} Molecules: Application to the Precise Study of Absolute Line Strengths of the *ν*_{6} Fundamental of CH_{3}^{35}Cl. *International Journal of Molecular Sciences*. 2023; 24(15):12122.
https://doi.org/10.3390/ijms241512122

**Chicago/Turabian Style**

Ulenikov, Oleg, Elena Bekhtereva, Olga Gromova, Anna Fomchenko, Yulia Morzhikova, Sergei Sidko, Christian Sydow, and Sigurd Bauerecker.
2023. "Effective Dipole Moment Model for Axially Symmetric *C*_{3v} Molecules: Application to the Precise Study of Absolute Line Strengths of the *ν*_{6} Fundamental of CH_{3}^{35}Cl" *International Journal of Molecular Sciences* 24, no. 15: 12122.
https://doi.org/10.3390/ijms241512122