_{3}CCH

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This paper deals with review of exploration of resonance in symmetric top molecules in different vibrational excited states, v_{t} = n (n =1, 2, 3, 4). Calculations for CF_{3}CCH shows that resonance take place at
_{10} = 2 and v_{10} = 3 respectively. In order to account for splitting about 3 MHz for the − 2 series in v_{10} = 4 is necessary to introduce the element 〈 J,k, ℓ|f_{24}| J,k + 2, ℓ − 4〉 in fitting program.

_{3}CCH, symmetric top molecules

The vibrational spectrum of CF_{3}CCH was assigned by Berney et al [^{−1}, consequently according to the Boltzman distribution law it has a population of 58% (29%×2 for vibrational degeneracy) of the ground state at dry ice temperature (200 K). Therefore the rotational spectrum in the v_{10} = 1 state is also strong and it is not complicated by interactions with nearby vibrational states. Whereas it is a prolate top, its rotational constants A ≈ 5.7 and B ≈ 2.9 GHz are not large. So, it is a possible candidate for the observation of the A_{1}-A_{2} splitting in the ground state. The
_{3}, v_{4} = 1 [_{3}H and CF_{3}D, v_{6} = 1 [_{3}Cl, v_{6} = 1 [_{3}CF_{3}, v_{12} = 1 [_{3}CCD, the isotopomer of this molecule in the ground, v_{10} = 1 and v_{10} = 2 states have been measured and studied by several authors [

Three identical fluorine atoms (I = 1/2) in CF_{3}CCH have important consequences in the determination of relative intensities of signals. The symmetry operations like exchange of nuclei, have symmetric effect on the total wavefunction for nuclei with integral or zero spin (Bose particles). But for nuclei with half integral spin (Fermi particles) the total wavefunction is antisymmetric. The rotation about the C_{3} axis through 2π/3 or 4π/3 rotation is equal to two exchanges of nuclei. Therefore it is symmetric (A) for Fermi particles as well as Bose particles. A rotation of 2π/3 about top axis will not change ψ_{rot} states with k = 3n (n = integer, including n = 0) since ψ_{rot} ∝ e^{ikψ}. For the symmetric state, the species are A and in antisymmetric state (k ≠ 3n ) the species are E. There are eight different possibilities for combinations of three fluorine atoms with spin 1/2. Two of them can occur when three spins are the same 1/2 or −1/2, these are totally symmetric because they remain unchanged by any permutation of nuclei which are equivalent to a rotation. But this situation can not take place for the remaining 6 arrangements, which are 2A_{1} + 2E. So the 8 spin functions are 4A_{1}+ 2E. The statistical weight depends on the ratio between the spin function of symmetry A and the spin function of symmetry E. Consequently:

For symmetric top molecules like CF_{3}CCH, the ground vibrational state rotational energies are given up to sextic terms by

The spectra in this state are simple, so the different k values (k = 0,1,2,3,4...) for each J transition are assigned easily. This is illustrated in

The centrifugal distortion produces a band head to high frequency at k = 0, with a spread to lower frequency with higher k _{Jk} for CH_{3}Cl, this situation is different

If |k| values are plotted against frequency for this state, the Fortrat diagrams are produced which are shown in _{Jk} parameter. D_{Jk} is 6.27 and −2.517 kHz for CF_{3}CCH and CHCl_{3} respectively [

The sextic centrifugal distortion constants were accurately determined for the first time for the ground state, and result shows that H_{Jk} > |H_{kJ}| and H_{kJ} < 0. For this molecule the sextic splitting constant h_{3} is very small [_{3} for CF_{3}CCH, 2.6×10–11 MHz, gives a splitting about 12 MHz for the highest - J transition (J = 112) [

Because v_{10} state of this molecule has a high population, the rotational spectra in this state are also strong. It is isolated in frequency from other vibrational states, so is not complicated by interactions with nearby states.

Rotational frequencies for transitions J → J + 1 in the excited degenerate vibrational state v_{t} = 1, ℓ_{t} = ± 1 of molecules with axial symmetry were calculated by Nielsen [_{3}CCH in the excited state v_{10} = 1. Then the frequencies of lines for transition J → J + 1 were investigated and calculated for v_{t} = 1 and v_{t} = 2 states [

The frequency of transition J → J + 1 for molecules belonging to the point group C_{3v} in singly excited vibrational state v_{t} = 1, is given by the approximate perturbation expression,

where Δν has the value ± q_{t}^{+} (J + 1) if (kℓ − 1) = 0 (ℓ - type doubling) or

Where is the ℓ-type doubling constant, B and A are rotational constants, and ζ is the z-coriolis constant for the vibrational state v_{t}. The degree of ℓ-resonance thus largely depends on the ratio of
_{3v} doubly degenerate states and is predominantly due to the combinations of the splitting of the positive and negative (kℓ-1) series by ℓ- type resonance [which is inversely proportional to (kℓ-1)] and the shift to low frequency due to D_{Jk} [which is proportional to (kℓ-1)^{2}]. Thus one series goes from high frequency at low k to low frequency at high k, while the other is at low frequency both for high and low k, with a head at some intermediate value.

In order to obtain more accuracy than can be obtained from perturbation theory in rotational energies for singly excited vibrational states it is necessary to set up the rotational Hamiltonian as a matrix (H) in equation Hψ = Eψ and diagonalise to obtain the energy (2,10,11,17,18,21,22). This is particularly necessary when[B − A + (Aζ)] is small in the above equation; the diagonalization avoids the errors implicit in perturbation theory or the need to go to higher orders of perturbation.

The Hamiltonian is set up for a symmetric top molecule like CF_{3}CCH [_{1} or A_{2}. If (kℓ- 1) ≠ 3n the species are E. The q_{t}^{+} produces a first order splitting of the (kℓ - 1) = 0, A_{1}A_{2} pair which are the familiar ℓ - doublets, as shown by Grenier-Besson and Amat and others [

The diagonal elements of the Hamiltonian used were:

In addition there are off-diagonal elements given by:

In _{J}, D_{Jk}, D_{k} are quartic, H_{J}, H_{Jk}, H_{kJ} are sextic centrifugal distortion terms respectively. ζ is the Coriolis coupling coefficient between the two components of the degenerate state and η_{J} represents the centrifugal distortion of (Aζ). The general effect of the additional parameters to complicate the spectrum relative to that of a non-degenerate state.

If |kℓ −1| values are plotted against frequency for this state, the Fortrat-diagram is produced which is shown for CH_{3}CCH in

In the present example the rotational Hamiltonian was diagonalised to give exact ℓ-resonance shifts. The spectrum in the millimetre-wave should consist of two widely spaced lines, the ℓ-doublets, whose separation is given by

The vibrational level v_{10} = 2 is located at 342 cm^{−1}. The vibrational levels closest to v_{10} = 2 are v_{5} = 1 and v_{9} = 1 at 536 and 452 cm^{−1}, respectively [_{t} = 2 level is suitable for the analysis of the data. The general features of the direct ℓ-resonance transitions in the v_{t} = 2 level have been discussed by Harder et al. [_{ℓℓ} = (B_{2} − B_{0})/4. Because the main information is obtained from the data of Carpenter et al [_{10} = 4 and show a regular dependence on the vibrational quantum number

These spectra show no obvious regularity from the point of view of frequencies as well as intensities, because there are three series for each J since the vibrational angular momentum quantum number ℓ can have values 0 and ± 2. Therefore it is difficult to identify the quantum numbers k and ℓ associated with each absorption in these spectra. However from the point of view of experimental work, correspondence between the lines of different J value should be observed with respect to their relative position as well as their intensities.

In order to obtain more accuracy in rotational energies for excited vibrational state v_{t} = 2 it is necessary to set up the rotational Hamiltonian as a matrix (H) in eigenvalue equation, Hψ = Eψ, and diagonalise to obtain the energy [_{3}CCH, CF_{3}CCD, CF_{3}CN, and OPF_{3}[_{ℓℓ}ℓ^{2} separates the ℓ = ± 2 and ℓ = 0 blocks and they are coupled by the matrix elements q_{t}^{+} and r_{t}.

In order to explain the form of the spectra we need to consider the energy levels in v_{t} = 2 as given by the Hamiltonian matrix. The diagonal elements are:

Where ν_{0} = pure vibrational frequency which is independent of J, k, ℓ and x_{ℓℓ} is the anharmonicity factor in ℓ. The main off-diagonal elements are the ℓ-doubling elements

Plotting of
^{2} can help us to analysis of signals in excited states. Which is described below?

From the theoretical point the frequency expression for transition J → J + 1 may be put in following effective form:

where A(J, k, ℓ) and B(J, k, ℓ) are complicated functions of J, k and ℓ.

Therefore:

For analysis and investigation of signals,

If both sides of

and

Hence plotting
^{2} will produce a series of lines; two of them relate to kℓ - 1 = 0 with slope - 2DJ and intercept

The kℓ - 1 ≠ 0 states will produce a series of lines with slope

and intercept B - D_{Jk}k^{2}. Some of them are nearly parallel to each other and some of them show curvature due to strong ℓ resonance. _{J} to the right hand side. This is then an effective B value for the given transition: hence termed BPLOT. Any slope or curvature is then due to the effect of ℓ-resonance.

Plotting of B_{eff} as a function of (J + 1)^{2} will produce a series of lines, each of which belongs to a different (kℓ - 1) value. Correlation of observed lines in different J can then be used to help with the assignment.

In v_{t} = 2 state the Hamiltonian blocks off into factors which for a C_{3v} molecule are of symmetry species A_{1}, A_{2} and E. k and ℓ are no longer good quantum numbers but (k - ℓ) × s remains a good symmetry label. In this label s is sign of ℓ, s = + 1 for ℓ ≥ 0 and s = −1 for ℓ < 0. The spectra are labled by (k-ℓ)^{ℓ} such as 0^{−2}, -1^{−2} and 0^{2} in the spectrum of J =19

Accidental resonances can occur when the ℓ = 0 series are coupled via the q_{t}^{+} term with ℓ = 2 series. This resonance will affect the frequencies of the |J, k, ℓ> →| J + 1, k, ℓ> transitions. The maximum resonance will occur when the energy of state ℓ = 2, k + 2 equals that of ℓ = 0, k i.e.

After omitting terms which are the same from both sides of

If the rotational constants of CF_{3}CCH from _{3}CCH in J = 20 → 21 show this resonance

_{3}CCD,

The Hamiltonian for v_{10} = 3 has four vibrational blocks correspond to the ℓ = +3, +1, −1, −3 _{t}^{+} and r_{t} (abbreviated as q and r, respectively, in

The extension to the v_{10} = 4 is straightforward. The possible values are + 4,+ 2, 0, − 2, and − 4, so that Hamiltonian consists of five blocks and five series are apparent in the spectra. It is worth noting that, to the order of magnitude that was included in the Hamiltonian.

For v_{10} = 2, there are no new parameters in either of the above extension of this Hamiltonian to the cases of v_{10} = 3 and v_{10} = 4.

For the CF_{3}CCH some of the lines belonging to the ℓ = 0, ℓ = 2 and ℓ = − 2 are shown separately in different plots, (

In the ℓ = − 2 series, the lines are nearly parallel to each other, and the distances between these lines become less as |k - ℓ|*s becomes smaller.

The two lines 0^{−2} and 0^{2}, which are expected to have the same frequency but observed to be are separated, the line 0^{−2} is parallel with (J + 1)^{2} axis whereas 0^{2} line is not

The ℓ = 0 series are located between line 9^{0} to line 10^{0} but most of them are on the lower frequency side of the spectrum

The lines 1^{2} to 9^{2} are above the 0^{−2} line whereas the other observed lines related to 10^{2} and greater are hidden under the 0^{−2} lines

Line 10^{0} is above line 0^{−2} and lines 11^{0},12^{0},13^{0},14^{0} cross line 0^{−2} but other observed lines belong to the ℓ = 0 are under the 0^{−2} line

Comparing of Fortrat diagrams of v_{10} = 3 and v_{10} = 4 with corresponding for v_{10} =1 _{10} = 3 [_{10} = 3 are qualitatively similar to those of v_{10} = 1, with ℓ-doublets and strong ℓ-resonance. In the same way the ℓ = 0 and +2 series of v_{10} = 4 show a similar resonance to the corresponding series in v_{10} = 2 while the ℓ = − 2 series is relatively unperturbed.

In the case of v_{10} = 2 [_{10} = 3, but in this case, because the vibrational separation due to the x_{ℓℓ} term is so much larger, this resonance only starts to become obvious at high k [

Omitting all centrifugal distortion and higher terms and terms which are the same for both states this gives:

If appropriate values input in

The ℓ = 0 and ℓ = ± 2 series have separation in v_{10} = 4 which are close found in the v_{10} = 2 vibrational state. Consequently, this resonance arises in the same position found for v_{10} = 2, namely between k = 9 and 10. The ℓ = ± 4 series are further away in the energy due to the x_{ℓℓ} ℓ^{2} term. In the v_{10} = 4 spectrum _{10} = 2 and v_{10} = 3 values. In order to account for this, it is necessary to introduce the element 〈 J,k, ℓ|f_{24}| J,k + 2, ℓ − 4〉. This provides a first-order coupling between the otherwise degenerate energies of the k = +1, ℓ = −2, and k = −1, ℓ = + 2 states which comprise the −3 pair of levels and leads to a splitting of this transition.

Part of the J = 24 → 25 spectrum of CF_{3}CCH in ground state. The top trace (A) is the observed spectrum, the lower (B) is a computer simulation.

Fortrat diagram of CF_{3}CCH in ground state J = 21 → 22.

Fortrat diagram of CHCl_{3} in ground state J = 21 → 22.

Fortrat diagram of CF_{3}CCH in v_{10} = 1 state. J = 20 → 21.

Plot of B against v_{t} for CF_{3}CCH.

The Hamiltonian matrix for v_{10} = 2 state, J = 2.

Part of the J = 19 → 20 spectrum of CF_{3}CCH in v_{10} = 2 state. The top trace (A) is the observed spectrum, the lower (B) is a computer simulation.

Calculated Spectrum of CF_{3}CCH in v_{10} = 2 State. J = 20 → 21.

Diagram of energy levels for CF_{3}CCH in v_{10} = 2 State.

The Hamiltonian matrix for v_{10} = 3, J = 2.

Plot of B_{eff} against (J + 1)^{2} for v_{10} = 2, ℓ = + 0 of CF_{3}CCH.

Plot of B_{eff} against (J + 1)^{2} for v_{10} = 2, ℓ = 2 of CF_{3}CCH.

Blot of B_{eff} against (J + 1)^{2} for v_{10} = 2, ℓ = −2 of CF_{3}CCH.

Plot of B_{eff} against (J + 1)^{2} for selected values of (k-ℓ)s of CF_{3}CCH.

The splitting (k - ℓ)s = −3 for −2 series state in part of the J = 20 → 21 spectrum of CF_{3}CCH in the v_{10} = 4. The top trace is the observed spectrum, the lower is a computer simulation.

Comparison of the Parameters for the Excited States v_{10} = n (n = 0, 1, 2, 3, 4) of CF3CCH.

Parameter | v_{10} = 0 [ |
v_{10} = 1 [ |
v_{10} = 2 [ |
v_{10} = 3 [ |
v_{10} = 4 [ |
---|---|---|---|---|---|

A/MHz | 5718.8436 |
5718.8436 |
5718.8436 |
5718.8436 |
5718.8436 |

B/MHz | 2877.9535(78) | 2883.4613(1) | 2888.9678(3) | 2894.3884(75) | 2899.7481(14) |

Aζ/MHz | - | 3293.5541(71) | 3293.064(9) | 3289.833(162) | 3285.999(334) |

q_{t}^{+}/MHz |
- | 3.6206334(72) | 3.6206334 |
3.6154(4) | 3.6169(8) |

r_{t}^{+}/MHz |
- | 0.0 |
0.203(24) | 0.165(12) | 0.197(9) |

D_{J}/kHz |
0.2684784(190) | 0.275890(33) | 0.2817(2) | 0.2888(8) | 0.2954(16) |

D_{Jk}/kHz |
6.27696(20) | 6.239328(46) | 6.2073(15) | 6.1628(8) | 6.1351(17) |

D_{k}/kHz |
−5.23 |
−5.23 |
−5.23 |
−5.23 |
−5.23 |

η_{J}/kHz |
- | 25.50129(89) | 25.025(13) | 24.977(6) | 24.847(8) |

η_{k}/kHz |
- | −19.1916(42) | −19.1916 |
−15.38(71) | −28.86(177) |

D^{ℓ}_{J}/Hz |
- | - | −0.3950(62) | −0.395 |
−0.395 |

D^{ℓ}_{Jk}/Hz |
- | - | −0.967(280) | −0.967 |
−0.967 |

H_{J}/mHz |
0.03195(110) | 0.0348(33) | 0.0 |
0.0 |
0.0 |

H_{Jk}/mHz |
18.0092(170) | −17.824(43) | 15.15(96) | 15.15 |
15.15 |

H_{kJ}/mHz |
−11.5756(170) | −12.19(66) | 0.0 |
0.0 |
0.0 |

H_{k}/mHz |
- | 0.0 |
0.0 |
0.0 |
0.0 |

q_{J}/Hz |
- | 4.29834(36) | 0.0 |
0.0 |
0.0 |

γ_{ℓℓ}/kHz |
- | - | −27.189(86) | −27.030(36) | −28.193(32) |

x_{ℓℓ}/MHz |
- | - | 8260.443(85) | 8135.83(2.1) | 8056.1(3.1) |

η_{JJ}/Hz |
- | 0.0 |
−72.2(67) | −72.2 |
−72.2 |

f_{24}/kHz |
- | - | 0.0 |
0.0 |
−1.039(23) |

Constrained at this value.

Resonance phenomena and k value for some different symmetric top molecules.

compound | CF_{3}CCH [ |
CF_{3}CCD [ |
CF_{3}CN [ |
CH_{3}CN [ |
CD_{3}CN[ |
---|---|---|---|---|---|

Resonance take place at | k = 9 →10 | k =16 →17 | k = 21→ 22 | k = 3 → 4 | k = 3 → 4 |

_{3}CCH, CF

_{3}CCD, and CF

_{3}CCCF

_{3}

_{10}= 1 Vibrational States and Observation of Direct ℓ-Type Resonance Transitions

_{3}in the v

_{4}= 1 state

_{3}H and CF

_{3}D in the v

_{6}= 1 state: Direct ℓ-type resonance transition and J = 3 ← 2

_{3}CCH

_{3}CCD in the excited vibrational state v

_{10}= 2

_{3}CCD in the Ground and Excited Vibrational States v

_{10}= 1, Bull

_{3}splitting

_{t}= 1

_{3v}Symmetry in an Excited State v

_{t}= 2

_{3}CN in the centimetre, millimetre and submillimetre wave ranges, Observation of direct ℓ-type resonance transitions

_{3v}symmetric top molecules, with application to CH

_{3}C

_{15}N

_{3}CCH in the Excited Vibrational State v

_{10}= 2

_{3}CCH in the vibrational state v

_{10}= 2

_{3}CCH in the Excited Vibrational States v

_{10}= 3 and v

_{10}= 4

_{3}CN in the excited vibrational state v

_{8}= 2

_{3}CN in the Excited Vibrational States v

_{8}= 3 and v

_{8}= 4

_{3}in the vibrationall excited state v

_{6}= 2

_{3}C

_{14}N and CH

_{3}C

_{15}N in the 2v8 State

_{3}C

_{14}N and CH

_{3}C

_{15}N in a Doubly Excited Degenerate Vibrational State

_{3}CN and CD

_{3}CN. Analysis of the v

_{8}= 1 and v

_{8}= 2 Excited Vibrational States

_{3}CCH in the vibrational state v

_{10}= 3