Rotational constants and centrifugal distortion constants on oblate top molecules

Appendix

A.4 Rotational constants and centrifugal distortion constants on oblate top molecules

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A.4 Rotational constants and centrifugal distortion constants on oblate

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(3) v = 𝐸𝐽+2,𝐾−𝐸𝐽,𝐾 = (4𝐵0− 6𝐷𝐽)(𝐽 +³₂) − 4𝐷𝐽𝐾(𝐽 +³₂)𝐾² for S branch (∆J = 2, ∆K = 0 J,K=0,1,2,3⋯) ⁴²

The rotational constant A and C can be calculated from the value of B and 𝐷_𝐾 can not be obtained in a pure rotational Raman spectrum of benzene in the ground state. The bands from two branches seem to be overlapped in pure rotational Raman spectra. However, bands are not overlapped but there is a distance between two bands due to centrifugal distortion constants.

A.4.3. Relationship between/among three centrifugal distortion constants 𝑫

_𝑱

, 𝑫

_𝑱𝑲

, and 𝑫

_𝑲

Ideally planar molecules with 𝐷_3ℎ and 𝐷_6ℎ symmetry have solid relationships among the three centrifugal distortion constants. These relationships were used to validate the value of obtained centrifugal distortion constants in the planar molecules whose K substructures have not been resolved yet.

(1) 𝐷_𝐽𝐾 = −¹⁵₈ 𝐷_𝐽 = −1.875 𝐷_𝐽, and 𝐷_𝐽 > 0 for all oblate top ³⁸ (2) 2𝐷_𝐽+ 3𝐷_𝐽𝐾+ 4𝐷_𝐾= 0 ³⁹

(3) 𝐷𝐽+ 𝐷𝐽𝐾 + 𝐷𝐾 > 0 ³⁸ (4) 𝐷_𝐾 =²⁹

32 𝐷_𝐽 from (1), (2) ^{38, 39} (5) 𝐷_𝐽+ 𝐷_𝐽𝐾+ 𝐷_𝐾 = ¹

32𝐷_𝐽 > 0 from (1),(2),(3), and (4) ³⁸,³⁹

A.4.4 Quadratic centrifugal distortion constants in planar molecules with

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𝑫

_𝟑𝒉

and 𝑫

_𝟔𝒉

symmetry

(1) 𝐻𝐽, 𝐻𝐽𝐾, 𝐻𝐾𝐽, and 𝐻𝐾 are zero in the 𝐷3ℎ and 𝐷6ℎ symmetry or higher symmetry in oblate top molecules when the rotational constant A=B=2C in the ground state.

Quadratic centrifugal distortion constants (𝐻_𝐽, 𝐻_𝐽𝐾, 𝐻_𝐾𝐽, and 𝐻_𝐾) are determined by internal defects (∆ = 𝐼_𝐶 − 𝐼_𝐵 − 𝐼_𝐴). These constants of the ground state are zero in the assumption of rotational momentum of inertia for planar molecules (𝐼_𝐶 = 𝐼_𝐵 + 𝐼_𝐴 , and A=B=1/2 C). Therefore, the quadratic centrifugal distortion constants are hardly determined in the assumption. They were not calculated to obtain rotational constants of B and two centrifugal distortion constants of 𝐷_𝐽 and 𝐷_𝐽𝐾 in the benzene.

A.4.5 Unresolved K structures of oblate top molecules and additional assumptions to determine the rotational constants and centrifugal distortion constants.

It is hard to accurately determine a value of 𝐷_𝐽𝐾 for a pure rotational Raman spectrum of symmetric oblate top molecules even in rovibronic spectroscopic methods. As the two centrifugal distortion constants are small and have an opposite sign to each other. Therefore, a low-resolution spectrum hardly observes split lines due to 𝐷_𝐽 and 𝐷_𝐽𝐾. Therefore, previous studies on rotational constants and centrifugal distortion constants of benzene needed additional assumptions to calculate centrifugal distortions in the low-resolution spectra. – (1) The higher intensity is corresponding to a transition of (𝐽_𝑓 = 𝑛 , 𝐾_𝑓 = 0) ← (𝐽_𝑓 = 𝑛 , 𝐾_𝑓 = 0). (2) There are no centrifugal distortion constants at all as the molecules are too rigid. (3) There are no 𝐷_𝑗𝑘 and 𝐷_𝐾, but is 𝐻_𝐾 in oblate top molecules. (4)

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Discrepancies between R and S branch can give opportunities to calculate centrifugal distortion constants. (5) There are certain relationships among centrifugal distortions. However, these four assumptions are problematic for oblate top molecules with a high symmetry, and yield discrepancies between literatures.

The researchers did set the highest intensity have a K=0 transition in a given quantum number J ¹¹. It is only valid in prolate top molecules but inappropriate for oblate top molecules due to the negative centrifugal distortion term. In the symmetric top, there are two models. One is a prolate top, and the other is oblate top. However, properties of prolate tops and oblate top are totally different from each other. If the K components have not been resolved in the rotational spectra, intensity analyses could be helpful for prolate top molecules but not for oblate top molecules. ⁴³ For the prolate top. The intensity of lines of a given J falls when K number increases. Even if K splitting is not observed in a low-resolution spectrum, the position of the maximum intensity is corresponding to K=0 in the given J value. In the other hands, Boltzmann factors and rotational line intensities with a given J values can be high or low. The energy of E(J,K) does not always increase because the terms of 𝐷_𝐽𝐾

(−4𝐷_𝐽𝐾(𝐽 + 1)𝐾² 𝑓𝑜𝑟 𝑅 𝑏𝑟𝑎𝑛𝑐ℎ, −4𝐷𝐽𝐾(𝐽 + 3/2)𝐾² 𝑓𝑜𝑟 𝑆 𝑏𝑟𝑎𝑛𝑐ℎ) cancel out the terms of 𝐷𝐽

(−6𝐷_𝐽(𝐽 + 1) 𝑓𝑜𝑟 𝑅 𝑏𝑟𝑎𝑛𝑐ℎ − 6𝐷_𝐽(𝐽 +³

2) for S branch) degeneracy also affects an intensity of each individual transitions in a certain number of K. Therefore, it is hard to predict exact J, K transitions in the oblate top.

(2) Two centrifugal distortions of 𝐷_𝐽𝐾, and 𝐷_𝐾 have been ignored. Some researchers set the centrifugal distortions to be zero in the fitting as a resolution of spectra could no resolve them. They attempted to calculated both 𝐷_𝐽𝐾 and 𝐷_𝐾 in the oblate top molecules but they gained different sets of rotational constant and centrifugal distortion constants in individual S and R branch. According to Cloppenburg, the unresolved K-splitting made displacements of the S and R branches. The

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displacements cause difficulties when we determine the rotational constant centrifugal distortion constants 𝐷_𝐽 and 𝐷_𝐽𝐾. However, the difficulties can be theoretically solved with properly estimated values of 𝐷_𝐽𝐾. ⁴² However, When we neglect effects of 𝐷_𝐽𝐾 and the K-structures of the given J transitions, the assumption leads to an error in the values of the B and 𝐷_𝐽. ³³

(3) The quadratic terms of centrifugal distortion constants, 𝐻_𝐽, 𝐻_𝐽𝐾, 𝐻_𝐾𝐽 , and 𝐻_𝐾 hardly

contribute to an intensity and a position of bands in pure rotational Raman spectra in the ground state.

For pure rotational Raman spectra in the cold temperature, there are a few bands, and it is not enough to resolve 𝐷_𝐽𝐾. One of quadratic term can be plugged into energy equations in order to explain discrepancy between S and R branch in a low-resolution spectrum due to possible effects of line shifts from unresolved K structure. They did set the constants 𝐷_𝐽𝐾, 𝐻_𝐽𝐾, and 𝐻_𝐾𝐽 equal to zero as the K- structure of the Raman transition of s-tetrazine was not resolved and bands seem to be symmetric enough to ignore them. ⁵¹ However, according to Aliev ³⁸ and Dowling ³⁹, the term of 𝐷_𝐽𝐾 can not be ignored in the pure rotational Raman spectra for planar molecules with 𝐷_3ℎ and 𝐷_6ℎ molecules.

(4) The differences between R and S branch have been discussed in not only pure rotational Raman spectra but also rovibronic spectra and rotationally resolved spectra for molecular structure

investigations. ^{51 52}

(5) There are certain relationships among three distortion constants in section. Relationship between three centrifugal distortion constants 𝐷_𝐽, 𝐷_𝐽𝐾, and 𝐷_𝐾. These theoretical relationships were used to check determined constants in the experimental spectra and compare them with distortion constants of literature. The relationships between the centrifugal distortion constants would manage to validate their values of obtained 𝐷_𝐽 and 𝐷_𝐽𝐾 in low resolution spectra with the unresolved K structure, but not assign the transition lines. Hence, the relationships do not give by themselves a single exact value

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of rotational constant, and centrifugal distortion constants in the low-resolution spectrum.

Unresolved K structures irritate researchers when they investigate molecular structures with a planar structure and their harmonicity with a low-resolution rotational spectra or rotationally resolved rovibronic spectra. Owing to unresolved K structures in the oblate top, a lot of assumptions and calculation methods have been applied. Even if there were reasonable assumptions in assignment of bands and lines of rotational spectra or rotationally resolved spectra for benzene, these different assumptions merely showed numerical discrepancies of the rotational constant and centrifugal distortion constants. ³³