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Article

Rotational Spectra of Symmetric Top Molecules in Ground andDifferent Vibrational Excited States, and Phenomenon of Resonance – Applying in CF3CCH

Department of Chemistry, Faculty of Science, University of Kurdistan, Sanandaj, Iran
Int. J. Mol. Sci. 2007, 8(4), 316-334; https://doi.org/10.3390/i8040316
Submission received: 13 February 2007 / Accepted: 23 March 2007 / Published: 17 April 2007
(This article belongs to the Section Physical Chemistry, Theoretical and Computational Chemistry)

Abstract

:
This paper deals with review of exploration of resonance in symmetric top molecules in different vibrational excited states, vt = n (n =1, 2, 3, 4). Calculations for CF3CCH shows that resonance take place at k = x + ( A - B ) - 2 A ζ A ζ - ( A - B ) and k = x + ( A - B ) - 2 A ζ A ζ - ( A - B ) for v10 = 2 and v10 = 3 respectively. In order to account for splitting about 3 MHz for the − 2 series in v10 = 4 is necessary to introduce the element ⟨ J,k, ℓ|f24| J,k + 2, ℓ − 4⟩ in fitting program.

1. Introduction

The vibrational spectrum of CF3CCH was assigned by Berney et al [1]. The energy level of its lowest lying degenerate vibration is only E = 171 cm−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 v10 = 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 A1-A2 splitting in the ground state. The q t + is rather large and A − B − Aζ is small and it is possible to observe direct ℓ-type resonance transitions in the microwave range [2]. These situation have been previously observed only for PF3, v4 = 1 [3], for CF3H and CF3D, v6 = 1 [4], CF3Cl, v6 = 1 [5] and CH3CF3, v12 = 1 [6]. The rotational spectra of CF3CCD, the isotopomer of this molecule in the ground, v10 = 1 and v10 = 2 states have been measured and studied by several authors [711].
Three identical fluorine atoms (I = 1/2) in CF3CCH 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 C3 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 ψrot states with k = 3n (n = integer, including n = 0) since ψrot ∝ eikψ. 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 2A1 + 2E. So the 8 spin functions are 4A1+ 2E. The statistical weight depends on the ratio between the spin function of symmetry A and the spin function of symmetry E. Consequently:
The intensity of A rotational states = 2 ( intensity of E rotational states )
For symmetric top molecules like CF3CCH, the ground vibrational state rotational energies are given up to sextic terms by equation (1) and by application of selection rules ΔJ = + 1 and Δk = 0, the frequencies are given by equation (2).
E J , k = BJ ( J + 1 ) + ( A - B ) k 2 - D J J 2 ( J + 1 ) 2 - D Jk J ( J + 1 ) k 2 - D k k 4 + H J J 3 ( J + 1 ) 3 + H Jk J 2 ( J + 1 ) 2 k 2 + H kJ J ( J + 1 ) k 4 + H k k 6
ν = 2 B ( J + 1 ) - 4 D J ( J + 1 ) 3 - 2 D Jk ( J + 1 ) k 2 + H J ( J + 1 ) 3 [ ( J + 2 ) 3 - J 3 ] + 4 H JK ( J + 1 ) 3 k 2 + 2 H kJ ( J + 1 ) k 4
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 Figure 1, which shows part of the observed transition J = 24 → 25, and compares it with the calculated spectrum.
The centrifugal distortion produces a band head to high frequency at k = 0, with a spread to lower frequency with higher k Figure 2. Due to negative value of DJk for CH3Cl, this situation is different Figure 3.
If |k| values are plotted against frequency for this state, the Fortrat diagrams are produced which are shown in Figures 2,3. In these diagrams the splitting increase as k increases and this is due to the DJk parameter. DJk is 6.27 and −2.517 kHz for CF3CCH and CHCl3 respectively [2, 12].
The sextic centrifugal distortion constants were accurately determined for the first time for the ground state, and result shows that HJk > |HkJ| and HkJ < 0. For this molecule the sextic splitting constant h3 is very small [2]. Cazzoli et al. [13] have found amount of this value, which is h 3 = 2.3 × 10 - 21 B 4 A - B. The predicted value of h3 for CF3CCH, 2.6×10–11 MHz, gives a splitting about 12 MHz for the highest - J transition (J = 112) [2].

2. Rotation Spectrum in Excited State v10 = 1

Because v10 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 vt = 1, ℓt = ± 1 of molecules with axial symmetry were calculated by Nielsen [14]. Although the Nielsen theory was fairly satisfactory for the rotation spectrum of these type of molecules, some of the calculated frequencies by this method were different from the observed frequencies. This formula was extended to the case of higher J value by Gordy et al. [7]. They also interpreted the transitions J = 4 → 5 and J = 8 → 9 of CF3CCH in the excited state v10 = 1. Then the frequencies of lines for transition J → J + 1 were investigated and calculated for vt = 1 and vt = 2 states [15, 16].
The frequency of transition J → J + 1 for molecules belonging to the point group C3v in singly excited vibrational state vt = 1, is given by the approximate perturbation expression, equation (3).
ν = 2 B ( J + 1 ) - 4 D J ( J + 1 ) 3 - 2 D Jk ( J + 1 ) k 2 + 2 η J ( J + 1 ) k + Δ ν
where Δν has the value ± qt+ (J + 1) if (kℓ − 1) = 0 (ℓ - type doubling) or
Δ ν = - ( q t + ) 2 ( J + 1 ) 3 4 ( B - A + A ζ ) ( k - 1 )             if  ( k - 1 ) 0     ( - type resonance )
Where is the ℓ-type doubling constant, B and A are rotational constants, and ζ is the z-coriolis constant for the vibrational state vt. The degree of ℓ-resonance thus largely depends on the ratio of q t + to the resonance denominator (B−A + Aζ). With low to moderate values of this denominator, the spectrum usually found consists of two extreme outer lines for (kℓ-1) = 0 (the ℓ- doublets) and a center group of lines for (kℓ-1) ≠ 0 whose structure depends very strongly on other parameters such as the centrifugal distortion constants. As the strength of the resonance increase, the lines of the center group spread out until the low |kℓ-1| transitions approach the positions of the ℓ-doublets. So for large q t + and small (A − B − Aζ = −458 MHz) value it is possible to observe direct ℓ-resonance transitions in the microwave range. This is a typical pattern for C3v 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 DJk [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 CF3CCH [2, 11]. This rotationvibrational Hamiltonian has two different blocks belong to the different ℓ = + 1 and ℓ = −1 series. k and ℓ are no longer good quantum numbers but (k - ℓ) or (kℓ - 1) may be used to distinguish between the symmetry species; those levels with (kℓ - 1) = 3n, where n is an integer, are of species A1 or A2. If (kℓ- 1) ≠ 3n the species are E. The qt+ produces a first order splitting of the (kℓ - 1) = 0, A1A2 pair which are the familiar ℓ - doublets, as shown by Grenier-Besson and Amat and others [2, 10, 11, 15, 19]. The main difference from the ground state spectra is the splitting of the |k - ℓ| = 0 into two widely separate ℓ - doublets and the splitting, due to ℓ - resonance. Where ℓ is the vibration angular momentum quantum number and ζ is the Coriolis coupling coefficient. Hence k and ℓ are not good quantum numbers, though kℓ - 1 may be used as a satisfactory label.
The diagonal elements of the Hamiltonian used were:
< v t , t , J , k H / h v t , t , J , k > = BJ ( J + 1 ) + ( A - B ) k 2 - 2 A ζ k - D J J 2 ( J + 1 ) 2 - D Jk J ( J + 1 ) k 2 - D k k 4 + H J J 3 ( J + 1 ) + H Jk J 2 ( J + 1 ) 2 k 2 + H kJ J ( J + 1 ) k 4 + H k k 6 + η J J ( J + 1 ) k + η k k 3
In addition there are off-diagonal elements given by:
< v t , , J , k H / h v t , ± 2 , J , k ± 2 > = - 1 4 q t + { ( v t ) ( v t ± + 2 ) [ J ( J + 1 ) - k ( k ± 1 ) ] × [ J ( J + 1 ) - ( k ± 1 ) ( k ± 2 ) ] } 1 / 2 and  < v t , , J , k H / h v t , ± 2 , J , k 1 > = r t [ ( v t + 1 ) 2 - ( ± 1 ) 2 ] 1 / 2 [ J ( J + 1 ) - k ( k 1 ) ] 1 , 2 ( 2 k 1 ) .
In equation 5, the DJ, DJk, Dk are quartic, HJ, HJk, HkJ are sextic centrifugal distortion terms respectively. ζ is the Coriolis coupling coefficient between the two components of the degenerate state and ηJ 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 CH3CCH in Figure 4. This diagram shows the 'ℓ' doublet and two different series ℓ= ± 1. So the spectrum is more complex compared with ground state [2].
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 equation (4), and a central group of transitions with |kℓ −1| > 0, whose frequencies are determined by equation (3). In the latter set the transitions with low values of |kℓ −1| will tend to lie towards the outsides of the group since their ℓ-resonance shifts as predicted by equation (3) will be larges. Two series should thus be observed depending on the sign of kℓ-1. There are two different series in spectrum, making it more difficult to assign than the ground state Figure 4.

3. Rotational Spectra in Excited States, v10 = 2, 3, 4

The vibrational level v10 = 2 is located at 342 cm−1. The vibrational levels closest to v10 = 2 are v5 = 1 and v9 = 1 at 536 and 452 cm−1, respectively [2, 9, 2022]. As the analysis will show, the level can be considered as being isolated and an effective Hamiltonian for a vt = 2 level is suitable for the analysis of the data. The general features of the direct ℓ-resonance transitions in the vt = 2 level have been discussed by Harder et al. [21]. There is a good agreement for the obtained rotational constants B and the ℓ -dependence γℓℓ = (B2 − B0)/4. Because the main information is obtained from the data of Carpenter et al [20]. The fact that the direct ℓ-type resonance transitions provide further information on [21]. Excellent agreement is obtained for the values of g = E v 2 - E v 0 / 4, because the parameter A constrained in its value [20]. The B and other spectroscopic constants determined accurately up to v10 = 4 and show a regular dependence on the vibrational quantum number Figure 5[2, 20, 22, 23].
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 vt = 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 [2022]. The diagonal matrix elements are given by equation (7). This Hamiltonian can set up for a symmetric top molecule such as CF3CCH, CF3CCD, CF3CN, and OPF3[10, 2030]. It has three different vibrational blocks related to ℓ = 2, 0 and −2 Figure 6. The vibrational anharmonicity xℓℓ2 separates the ℓ = ± 2 and ℓ = 0 blocks and they are coupled by the matrix elements qt+ and rt.
In order to explain the form of the spectra we need to consider the energy levels in vt = 2 as given by the Hamiltonian matrix. The diagonal elements are:
< v , , J , k H / h v , , J , k > = ν 0 + x 2 + B [ J ( J + 1 ) - k 2 ] + Ak 2 - 2 A ζ k - D J J 2 ( J + 1 ) 2 - D Jk J ( J + 1 ) k 2 - D k k 4 + η J J ( J + 1 ) k + H J J 3 ( J + 1 ) 3 + H Jk J 2 ( J + 1 ) k 2 + H kJ ( J + 1 ) k 4 + H k k 6 + η JJ J 2 ( J + 1 ) 2 k + η k k 3 η + ..
Where ν0 = pure vibrational frequency which is independent of J, k, ℓ and xℓℓ is the anharmonicity factor in ℓ. The main off-diagonal elements are the ℓ-doubling elements equation (6).

4. Plot of Beff

Plotting of ν 2 ( J + 1 ) against (J + 1)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:
ν = ( J + 1 ) A ( J , k , ) + ( J + 1 ) 3 B ( J , k , )
where A(J, k, ℓ) and B(J, k, ℓ) are complicated functions of J, k and ℓ.
Therefore:
ν ( J + 1 ) = A ( J , k , ) + ( J + 1 ) 2 B ( J , k , )
For analysis and investigation of signals, equation (7) can be changed after some abbreviation and alteration to the following form:
ν = 2 B ( J + 1 ) - 4 D J ( J + 1 ) 3 - 2 D Jk ( J + 1 ) k 2 ± ( q t + ) ( J + 1 )             for k - 1 = 0 or            - ( q t + ) 2 ( J + 1 ) 3 4 ( B - A + A ζ ) ( k - 1 )             for k - 1 0
If both sides of Eq (13) are divided by 2(J + 1), then:
ν 2 ( J + 1 ) = B - D Jk k 2 ± 1 2 q t + - 2 D J ( J + 1 ) 2             for k - 1 = 0
and
ν 2 ( J + 1 ) = B - D Jk k 2 - 2 D J ( J + 1 ) 2 - ( q t + ) 2 ( J + 1 ) 2 8 ( B - A + A ζ ) ( k - 1 ) for  ( k - 1 ) 0
Hence plotting ν 2 ( J + 1 ) as a function of (J + 1)2 will produce a series of lines; two of them relate to kℓ - 1 = 0 with slope - 2DJ and intercept B - D Jk k 2 ± 1 2 q t +.
The kℓ - 1 ≠ 0 states will produce a series of lines with slope
- 2 D J - ( q t + ) 2 8 ( B - A + A ζ ) ( k - 1 )
and intercept B - DJkk2. Some of them are nearly parallel to each other and some of them show curvature due to strong ℓ resonance. Equation (10) can be further simplified by taking the term in DJ 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.
ν = 2 B eff ( J + 1 ) - 4 D J ( J + 1 ) 3
            B eff = ν 2 ( J + 1 ) + 2 D J ( J + 1 ) 2
Plotting of Beff 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 vt = 2 state the Hamiltonian blocks off into factors which for a C3v molecule are of symmetry species A1, A2 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) such as 0−2, -1−2 and 02 in the spectrum of J =19 Figure 7.
Accidental resonances can occur when the ℓ = 0 series are coupled via the qt+ 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.
E ( J , k , = 0 > ) = E ( J , k + 2 , = 2 > )
After omitting terms which are the same from both sides of equation (8) this gives:
( A - B ) k 2 = 4 x + ( A - B ) ( k + 2 ) 2 - 4 A ζ ( k + 2 )
Hence            k = x + ( A - B ) - 2 A ζ A ζ - ( A - B )
If the rotational constants of CF3CCH from Table 1 are put in equation (18) then k is found to be 9.7. Experimentally the resonance is indeed observed between k = 9 and k = 10 for this molecule and is independent of J, as expected from equation (18). The calculated spectrum of CF3CCH in J = 20 → 21 show this resonance Figure 8. Like this resonance take places in different k values for different molecules which depends on their parameters Table 2.
Figure 9 shows the energy levels as ℓ = 2, 0 and −2 stacks. The lines between states in adjacent stacks represent the ℓ-type interaction; for example k = 4 in ℓ = − 2 is linked to k = 6, ℓ = 0 and this also interacts with k = (6 + 2) = 8, ℓ = + 2. The interaction results in the energy levels being pushed apart; for example k = 4 is shifted up and k = 6 is shifted down. The extent of this shift depends on how nearly equal are the energy levels. When they are close, a large shift is obtained. The maximum shift will occur when the energies are almost equal, see equation (18) and is found in the region of k ≈ 9. This phenomenon can occur at k ≈ 16 value for CF3CCD, Table 2.
The Hamiltonian for v10 = 3 has four vibrational blocks correspond to the ℓ = +3, +1, −1, −3 Figure 10. The ℓ = ± 3 and ℓ = ± 1 blocks are separated in vibrational energy by the anharmonic term. There are, however, couplings between +3 and +1, +1, −1 and −1 and −3, due to the off-diagonal matrix elements qt+ and rt (abbreviated as q and r, respectively, in Figures. 6, 10). The labelling of the eigenvalues of this Hamiltonian by (k-ℓ) s where s is 1 for ℓ = + 3 and +1, and −1 for ℓ = −3 and − 1; the symmetry of a given transition can then be easily deduced from the fact that
A 1 A 2 symmetry  ( k - ) s = 3 n , n = 0 , 1 , 2 , E symmetry  ( k - ) s 3 n , n = 0 , 1 , 2 , . v , , J , k >
The extension to the v10 = 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 v10 = 2, there are no new parameters in either of the above extension of this Hamiltonian to the cases of v10 = 3 and v10 = 4.

5. Results and Discussion

For the CF3CCH some of the lines belonging to the ℓ = 0, ℓ = 2 and ℓ = − 2 are shown separately in different plots, (Figures 11, 12, 13).
  • 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 02, 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 02 line is not Figure 14.
  • The ℓ = 0 series are located between line 90 to line 100 but most of them are on the lower frequency side of the spectrum Figure 11.
  • The lines 12 to 92 are above the 0−2 line whereas the other observed lines related to 102 and greater are hidden under the 0−2 lines Figures. 11,13.
  • Line 100 is above line 0−2 and lines 110,120,130,140 cross line 0−2 but other observed lines belong to the ℓ = 0 are under the 0−2 line Figures. 11,13.
Comparing of Fortrat diagrams of v10 = 3 and v10 = 4 with corresponding for v10 =1 Figure. 4 and v10 = 3 [22] is instructive. The ℓ ± series of v10 = 3 are qualitatively similar to those of v10 = 1, with ℓ-doublets and strong ℓ-resonance. In the same way the ℓ = 0 and +2 series of v10 = 4 show a similar resonance to the corresponding series in v10 = 2 while the ℓ = − 2 series is relatively unperturbed.
In the case of v10 = 2 [20], as k increase, the ℓ = 0 energies come into accidental resonance with those of the positive (k - ℓ) s series. These two series of energies are coupled via the q t +. ℓ-resonance matrix element, giving an avoided crossing. Such a resonance is also present for v10 = 3, but in this case, because the vibrational separation due to the xℓℓ term is so much larger, this resonance only starts to become obvious at high k [22]. The maximum resonance can occur when
E J , k + 2 , = 3 = E J , k , = 1
Omitting all centrifugal distortion and higher terms and terms which are the same for both states this gives: equation (20).
k = x + ( A - B ) - 2 A ζ A ζ - ( A - B )
If appropriate values input in equation 20 then solving this equation yields a value of k between 19 and 20. There is a slight kink in this diagram which is caused by the accidental degeneracy of the |J, k, ℓ = −1〉 and |J,k + 4, ℓ = +3〉 states which are not directly connected by a matrix element but are coupled in second order by the interaction of both of them with |J, k = + 2, ℓ = +1〉. Applying the equation 21 yields a value for the k = 8.4 which is indeed where the perturbation is observed.
k = x + 2 ( A - B ) - 3 A ζ A ζ + ( A - B )
The ℓ = 0 and ℓ = ± 2 series have separation in v10 = 4 which are close found in the v10 = 2 vibrational state. Consequently, this resonance arises in the same position found for v10 = 2, namely between k = 9 and 10. The ℓ = ± 4 series are further away in the energy due to the xℓℓ2 term. In the v10 = 4 spectrum Figure. 15, the transition with (k - ℓ) s = −3 for the −2 series is split by about 3 MHz in the J = 20 spectrum. This splitting is not reproduced by the parameters obtained by extrapolation of the v10 = 2 and v10 = 3 values. In order to account for this, it is necessary to introduce the element 〈 J,k, ℓ|f24| 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.
Figure 1. Part of the J = 24 → 25 spectrum of CF3CCH in ground state. The top trace (A) is the observed spectrum, the lower (B) is a computer simulation.
Figure 1. Part of the J = 24 → 25 spectrum of CF3CCH in ground state. The top trace (A) is the observed spectrum, the lower (B) is a computer simulation.
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Figure 2. Fortrat diagram of CF3CCH in ground state J = 21 → 22.
Figure 2. Fortrat diagram of CF3CCH in ground state J = 21 → 22.
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Figure 3. Fortrat diagram of CHCl3 in ground state J = 21 → 22.
Figure 3. Fortrat diagram of CHCl3 in ground state J = 21 → 22.
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Figure 4. Fortrat diagram of CF3CCH in v10 = 1 state. J = 20 → 21.
Figure 4. Fortrat diagram of CF3CCH in v10 = 1 state. J = 20 → 21.
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Figure 5. Plot of B against vt for CF3CCH.
Figure 5. Plot of B against vt for CF3CCH.
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Figure 6. The Hamiltonian matrix for v10 = 2 state, J = 2.
Figure 6. The Hamiltonian matrix for v10 = 2 state, J = 2.
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Figure 7. Part of the J = 19 → 20 spectrum of CF3CCH in v10 = 2 state. The top trace (A) is the observed spectrum, the lower (B) is a computer simulation.
Figure 7. Part of the J = 19 → 20 spectrum of CF3CCH in v10 = 2 state. The top trace (A) is the observed spectrum, the lower (B) is a computer simulation.
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Figure 8. Calculated Spectrum of CF3CCH in v10 = 2 State. J = 20 → 21.
Figure 8. Calculated Spectrum of CF3CCH in v10 = 2 State. J = 20 → 21.
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Figure 9. Diagram of energy levels for CF3CCH in v10 = 2 State.
Figure 9. Diagram of energy levels for CF3CCH in v10 = 2 State.
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Figure 10. The Hamiltonian matrix for v10 = 3, J = 2.
Figure 10. The Hamiltonian matrix for v10 = 3, J = 2.
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Figure 11. Plot of Beff against (J + 1)2 for v10 = 2, ℓ = + 0 of CF3CCH.
Figure 11. Plot of Beff against (J + 1)2 for v10 = 2, ℓ = + 0 of CF3CCH.
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Figure 12. Plot of Beff against (J + 1)2 for v10 = 2, ℓ = 2 of CF3CCH.
Figure 12. Plot of Beff against (J + 1)2 for v10 = 2, ℓ = 2 of CF3CCH.
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Figure 13. Blot of Beff against (J + 1)2 for v10 = 2, ℓ = −2 of CF3CCH.
Figure 13. Blot of Beff against (J + 1)2 for v10 = 2, ℓ = −2 of CF3CCH.
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Figure 14. Plot of Beff against (J + 1)2 for selected values of (k-ℓ)s of CF3CCH.
Figure 14. Plot of Beff against (J + 1)2 for selected values of (k-ℓ)s of CF3CCH.
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Figure 15. The splitting (k - ℓ)s = −3 for −2 series state in part of the J = 20 → 21 spectrum of CF3CCH in the v10 = 4. The top trace is the observed spectrum, the lower is a computer simulation.
Figure 15. The splitting (k - ℓ)s = −3 for −2 series state in part of the J = 20 → 21 spectrum of CF3CCH in the v10 = 4. The top trace is the observed spectrum, the lower is a computer simulation.
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Table 1. Comparison of the Parameters for the Excited States v10 = n (n = 0, 1, 2, 3, 4) of CF3CCH.
Table 1. Comparison of the Parameters for the Excited States v10 = n (n = 0, 1, 2, 3, 4) of CF3CCH.
Parameterv10 = 0 [2]v10 = 1 [2]v10 = 2 [20]v10 = 3 [22]v10 = 4 [22]
A/MHz5718.8436*5718.8436*5718.8436*5718.8436*5718.8436*
B/MHz2877.9535(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)
qt+/MHz-3.6206334(72)3.6206334*3.6154(4)3.6169(8)
rt+/MHz-0.0*0.203(24)0.165(12)0.197(9)
DJ/kHz0.2684784(190)0.275890(33)0.2817(2)0.2888(8)0.2954(16)
DJk/kHz6.27696(20)6.239328(46)6.2073(15)6.1628(8)6.1351(17)
Dk/kHz−5.23*−5.23*−5.23*−5.23*−5.23*
ηJ/kHz-25.50129(89)25.025(13)24.977(6)24.847(8)
ηk/kHz-−19.1916(42)−19.1916*−15.38(71)−28.86(177)
DJ/Hz--−0.3950(62)−0.395*−0.395*
DJk/Hz--−0.967(280)−0.967*−0.967*
HJ/mHz0.03195(110)0.0348(33)0.0*0.0*0.0*
HJk/mHz18.0092(170)−17.824(43)15.15(96)15.15*15.15*
HkJ/mHz−11.5756(170)−12.19(66)0.0*0.0*0.0*
Hk/mHz-0.0*0.0*0.0*0.0*
qJ/Hz-4.29834(36)0.0*0.0*0.0*
γℓℓ/kHz--−27.189(86)−27.030(36)−28.193(32)
xℓℓ/MHz--8260.443(85)8135.83(2.1)8056.1(3.1)
ηJJ/Hz-0.0*−72.2(67)−72.2*−72.2*
f24/kHz--0.0*0.0*−1.039(23)
*Constrained at this value.
Table 2. Resonance phenomena and k value for some different symmetric top molecules.
Table 2. Resonance phenomena and k value for some different symmetric top molecules.
compoundCF3CCH [20]CF3CCD [10]CF3CN [23]CH3CN [29]CD3CN[29]
Resonance take place atk = 9 →10k =16 →17k = 21→ 22k = 3 → 4k = 3 → 4

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Motamedi, M. Rotational Spectra of Symmetric Top Molecules in Ground andDifferent Vibrational Excited States, and Phenomenon of Resonance – Applying in CF3CCH. Int. J. Mol. Sci. 2007, 8, 316-334. https://doi.org/10.3390/i8040316

AMA Style

Motamedi M. Rotational Spectra of Symmetric Top Molecules in Ground andDifferent Vibrational Excited States, and Phenomenon of Resonance – Applying in CF3CCH. International Journal of Molecular Sciences. 2007; 8(4):316-334. https://doi.org/10.3390/i8040316

Chicago/Turabian Style

Motamedi, Masoud. 2007. "Rotational Spectra of Symmetric Top Molecules in Ground andDifferent Vibrational Excited States, and Phenomenon of Resonance – Applying in CF3CCH" International Journal of Molecular Sciences 8, no. 4: 316-334. https://doi.org/10.3390/i8040316

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