Study of Millimeter-Wave Rotational Spectra of Trioxane in Ground , v ( A ) , v 1 ( E ) = 1 and v 2 ( E ) = 1 States

The millimeter-wave rotational spectra of the ground and excited vibrational states v(A), v1 (E) =1 and v2 (E ) =1 of the oblate symmetric top molecule, (CH2O)3, have been analyzed again. The B0 = 5273.25747MHz, D J = 1.334547 kHz, D Jk = -2.0206 kHz, H J (-1.01 mHz), H JK (-3.80 mHz), and HKJ (4.1 mHz) have been determined for ground state. For non degenerate excited state, vA(1), the B = 5260.227723 MHz and D J and D JK were determined 1.27171 kHz and -1.8789 kHz respectively. The 1=±1 series have been assigned in two different excited states v1 (E) =1 and v2 (E) =1.Most of the parameters were determined with higher accuracy compare with before. For the v2 (E) =1 state the Cζ=-1940.54(11) MHz and q J = 0.0753 (97) kHz were determined for the first time.


Introduction
1,3,5-Trioxane, (CH 2 O) 3, is a cyclic trimer of formaldehyde, is an oblate symmetric top molecule belonging to the C 3v group.Its vibrational spectrum exhibits 7 type A fundamental absorption bands and 10 type E fundamental bands.Several authors have studied the rotational spectra of the ground state and some excited states [1][2][3][4][5][6][7][8] .In their study observed numerous excited states corresponding to type E and type A vibrational modes.The centrifugal distortion constants are very small and that no structure characteristic of the excited states could be resolved.A more study of the rotational spectrum of this molecule requires the measurement of high J transitions, which are located in the millimeter region and have greater intensity.The microwave spectrum of trioxane was first studied by Oka et al 1 , who determined the rotational constants for the ground state, for low-lying excited states, and for 13 C isotopic species.They measured also the dipole moment [2][3][4][5][6][7][8] , µ=2.07(4)D.Cox et al 3 .analyzed also the degenerate excited states v 1 (E) = 1 and v 2 (E) = 1 by direct diagonalization of the Hamiltonian matrix.
If rotational transitions of two degenerate states v 1 (E) = 1 and v 2 (E) = 1 are analyzed using the perturbation formula of Grenier-Besson and Amat 9 , the significant discrepancies between calculated and observed frequencies are obtained for these states, particularly for high J, and Kl -1 transitions of the v 2 (E) = 1 state.The perturbation treatment is only expected to be valid 3,9 if the following inequalities hold: Errors arise through the use of the perturbation treatment due to l-type resonance associated with the <1, KH VR 1 ± 2, K ± 2> matrix elements, where H VR is the vibrationrotation Hamiltonian.The presence of l-type resonance precludes r-type resonance effects, associated with the <1,K|H VR |1±2, K±1>elements, arising from inequality . If the only off-diagonal elements considered are (2,2) elements, the energy matrix can be factored conveniently into a number of tridiagonal submatrices thus facilitating computation (Figure 1).
The aim of this study is determination of rotational parameters for mixing the low and high J values in ground and v(A) , v 1 (E) = 1 and v 2 (E) = 1 states, which are more accurate and reliable.

Spectra in the ground and nondegenerate states
The rovibrational Hamiltonian for a non-degenerate vibrational state is essentially similar to that of the ground state.For oblate top species: If centrifugal distortion is taken into account, the energy levels change and so, Where D J , D JK , D K are quartic centrifugal distortion constants, and H J , H JK , H KJ , are sextic centrifugal distortion terms, these are small correction terms for non-rigidity.The selection rules are 0 ∆K 1, ∆J = ± = .The 0 ∆K = selection rule means that the C (A in prolate symmetric top molecules) and D K terms play no part in the spectrum, but the D JK term removes the superposition of the different K components in the spectrum.The transition frequencies for 1) (J J + → can be expressed as ν = 2B(J + 1) -4D J (J + 1) 3 -2D Jk (J + 1)k 2 + H J (J + 1) 3 [(J + 2) 3 -J 3 ] + 4H Jk (J + 1) 3 k 2 + 2H kJ (J + 1)k 4 +… (3)   The frequencies of Refs 2,8 were mixed in order to improve the accuracy and reliability of the rotational and other centrifugal distortion constants.Then a weighted least-squares method 10 was used to fit the mixed frequencies for low and high J values to the parameters of Eq (3) in which the weights were taken to be where σ in MHz is estimated uncertainty in an observation for each unblended line.
The fit to the transitions are given in Tables 1, 2 and obtained parameters are given in Table 5.Some of the lines have an observed error of 0.02 to 0.2 MHz to allow for overlapping or broadening.These tables show that the mixed frequencies of low and high J values of trioxane were fitted very well and correlation coefficients of parameters are reasonable.The structural parameters of this compound as shown in Table 5 were obtained with higher accuracy and compared with previous work.The frequencies of Refs 2,8 were mixed together in order to obtain improved constants.The rotational and quartic constants were obtained with good agreement with previous work.It should be noticed that the distortion constants obtained are different from those of the ground state.The H J , H Jk , H kJ centrifugal distortions constants are not obtained by these available data.The results of refinements are listed in Tables 3, 4 and derived parameters are given in Table 5.

Spectum in the v 1 (E) = 1 state
The v 1 (E) and v 2 (E) degenerate modes are at 307 cm -1 and 472 cm -1 .These two vibrations come from the COC and OCO bending vibration.These have populations at 298 K relative to the ground state of (≈22.7%) and (≈10%) respectively.The theory for the rotational spectrum of an E degenerate vibrational state of a C 3v molecule is well known 9 .Essentially the two components of the vibration each give rise to a series of K levels which are coupled to each other by l-resonance.This results in two series of transitions which are best labeled by the pseudo-quantum number Kl -1.The l-resonance splitting is greatest for levels with low values of this label.Perturbation theory yields the approximate frequency expression where + t q is the l-type doubling constant, B and C are rotational constants, and t ζ is the z-coriolis constant for the vibrational state v t .The degree of l-resonance thus largely depends on the ratio of + t q to the resonance denominator . 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 whose structure depends very strongly on other parameters such as the centrifugal distortion constants.As the strength of the resonance increases, the lines of the center group spread out until the low 1 Kl − transitions approach the positions of the l-doublets.
In order to obtain more accuracy in rotational energies for singly excited vibrational states than can be obtained from perturbation theory, it is necessary to set up a rotational Hamiltonian as a matrix (H) in equation Hψ = Eψ and diagonalise to obtain the energy [11][12][13][14][15][16][17][18] .The Hamiltonian was set up for a symmetric top molecule like (CH 2 O) 3 .This rotationvibrational Hamiltonian has two different blocks that belong to the different l= +1 and l = -1 series Figure 1.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 A 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 9 .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.In addition, there are off-diagonal terms that arise from the transformation.The major one of these gives rise to the l doubling: and hence, the lines can be assigned.
The 273 frequencies of Refs 2,3-8 for this state were mixed together in order to improve the accuracy of the parameters in this state, then a weighted least-squares method 10 was used to fit these frequencies to Eq (5), in which the weights were taken to be w = 1/(observed error) 2 = 1/(0.1) 2 , where 0.1 in MHz is estimated uncertainty in an observation for each unblended line.Some of the lines have uncertainty between 0.2 to 0.5 MHz due to overlapping or blending lines.The fit to the transitions are given in Tables 6,7 and the constants obtained are given in Table 10.

Spectrum in v (E)= 1 state
This state shows the characteristic intensity pattern of doubly degenerate state in its k structure as v 1 (E) = 1 state.This state has enough population (≈10%) relative to ground state to observe the signals.
The frequencies of Refs 2,3,8 were mixed together in order to obtain improved constants.The results of refinements are listed in Tables 8, 9 and derived parameters are given in Table 10.The H J , H Jk , H kJ , H k centrifugal distortions constants are not obtained with these available data.

Results and Discussion
Ignoring sextic term in Eq(3) for K = 0 the frequency expression becomes ν = 2B(J + 1) -4D J (J + 1) 3 Eq (8) has enough accuracy for practical works.From a series of lines, D J can be evaluated and therefore B is obtained very accurately.Plotting 1) (J ν + against (J + 1) 2 gives a straight line with slope -4D J and intercept 2B Figure 2.

Figure 2.
Diagram of v/(J + 1) against (J + 1) 2 for trioxane for ground state, from J = 0 to J = 19.Abbreviation of equation Eq(3) gives Eq (9).(J + 1)*2 ν = 2B(J + 1) -4D J (J + 1) 3 -2D Jk (J + 1)K 2 The constant D JK is obtained from the slopes of By this method it is possible to determine the rotational constant and other two quartic constants D J and D JK which are very important to predict these values for fitting the data of ground and other excited states.
Since 12 C and 16 O are Bose particles with spin zero, the symmetry of the spin-wave function of trioxane is determined solely from the hydrogen atoms.Table 5 shows that rotational constants, obtained quartic parameters for ground and v(A) =1 states are in good agreement with the previous results and have more accuracy compare with them.For the ground state, it is possible to determine the tree sextic centrifugal distortion constants.The H JK and H J are well determined, but H KJ is obtained with lower accuracy.
For the v(A) state, all of the sextic parameters were constrained to zero since there inclusion in the fit did not improve the sum of weighted squares of errors and resulted in a values not significantly different from zero and not well determined by the data.Probably these parameters are important in analysis of high J rotational spectrum of this particular molecule.
The oblate symmetric top (CH 2 O) 3 shows a strong 0 ∆K 1, ∆J = + = ground state spectrum.The spectra in ground and v(A) = 1 states are simple, so the different K values (K = 0,1,2,3,4...) for each J transition are assigned easily.The centrifugal distortion produces a band head to high frequency at K = 0, with a spread to lower frequency with higher K.
If |K| values are plotted against frequency for ground and v(A) = 1states, the Fortrat diagram is produced which is shown in (Figures 4, 5).In these diagrams the splitting increase as K increases and this is due to the D JK parameter.In other words the term -2D Jk (J + 1)K 2 has the effect of separating the (J + 1) components of each (J + 1) -J transition.Table 10 shows that our D J parameter for v 1 (E) state is obtained with positive value but its value in Ref 8 is negative.The parameters MHz (11) 1940.54 -Cζ = and q J = 0.0753 (97) kHZ were determined for the v 2 (E) = 1 state for the first time.
For the v 1 (E) = 1 and v 2 (E) = 1 states the important parameters in determining form of the spectra are the l-type doubling constant q v and the value of Aζ.The parameter η J , which is to be regarded as a type of centrifugal distortion constant, has a negative value.As can be seen from Eq (5) this results in the positive series being displaced to lower frequency and the negative series to higher frequency.If |K-l| values are plotted against frequency for v 1 (E) =   Fortrat diagram for (CH 2 O) 3 in v 2 (E) = 1 state for J = 9.If all the sextic constants were included in the fit, these were not only strongly correlated but also had standard deviations which were of about the same absolute magnitudes as the quantities themselves.This means that actual values are not significant in this state and constrained at the zero value.
Investigations in Ref 2 and Tables 5 and 10 shows that D JK kHz) 0.02 1.87 ( ± − for the v(A) state in trioxane was significantly lower in magnitude than the ground state value kHz) 0.01 2.03 ( ± − . From the present analysis D JK for the v 2 (E) = 1 state kHz) 2.080(54) (− is significantly greater than the ground state while the lowest E state v 1 (E) = 1 is essentially the same as the ground state.These changes in D JK probably arise through a Coriolis interaction between the v 2 (E) = 1 and v 1 (A) states.
If B v values of (CH 2 O) 3 are plotted against vibrational quantum number v t , the straight line is obtained Figure 6.Extrapolating of this line gives roughly B 4 = 5253.5MHz.It seems these different vibrational states show that these levels are free of interaction with any other vibrations and act as isolated levels.

Figure 1 .
Figure 1.The Hamiltonian matrix for v t 1 state, J = 3.The diagonal matrix elements are given by:

Figure 5 .
Figure 5. Fortrat diagram for calculated frequencies of (CH 2 O) 3 in v(A) state for J = 12Table10shows that our D J parameter for v 1 (E) state is obtained with positive value but its value in Ref8 is negative.The parameters MHz (11) 1940.54 -Cζ = and q J = 0.0753 (97) kHZ were determined for the v 2 (E) = 1 state for the first time.For the v 1 (E) = 1 and v 2 (E) = 1 states the important parameters in determining form of the spectra are the l-type doubling constant q v and the value of Aζ.The parameter η J , which is to be regarded as a type of centrifugal distortion constant, has a negative value.As can be seen from Eq(5) this results in the positive series being displaced to lower frequency and the

1 and v 2 (
E) = 1 states, the Fortrat diagrams are produced which are shown in Figures 6 & 7.

Figure 7 .
Figure 7. Fortrat diagram for (CH 2 O) 3 in v 2 (E) = 1 state for J = 9.If all the sextic constants were included in the fit, these were not only strongly correlated but also had standard deviations which were of about the same absolute magnitudes as the quantities themselves.This means that actual values are not significant in this state and constrained at the zero value.Investigations in Ref2 and Tables5 and 10shows that D JK kHz) 0.02 1.87 ( ± − for the v(A) state in trioxane was significantly lower in magnitude than the ground state value kHz) 0.01 2.03 ( ± −.From the present analysis D JK for the v 2 (E) = 1 state kHz) 2.080(54) (− is significantly greater than the ground state while the lowest E state v 1 (E) = 1 is essentially the same as the ground state.These changes in D JK probably arise through a Coriolis interaction between the v 2 (E) = 1 and v 1 (A) states.If B v values of (CH 2 O) 3 are plotted against vibrational quantum number v t , the straight line is obtained Figure6.Extrapolating of this line gives roughly B 4 = 5253.5MHz.It seems these different vibrational states show that these levels are free of interaction with any other vibrations and act as isolated levels.

Figure 8 .
Figure 8. Variation of B value with vibrational quantum number v t for trioxane in different excited states of v 1 (E) = state.

Table 1 .
Results of refinement of observed frequencies for (CH 2 O) 3 in ground state.
The state indicated by v A (E) corresponds to the lowest excited non degenerate vibrational state, with an energy 524 cm -1 .The population of this state at 298 K has enough population (≈8%) relative to ground state to observe the signals.This state shows the characteristic intensity pattern of non-degenerate state in its K structure.

Table 2 .
Correlation coefficients of obtained parameters for (CH 2 O) 3 in ground state.

Table 3 .
Results of refinement of observed frequencies for (CH 2 O) 3 in v(A) state.

Table 4 .
Correlation coefficients of obtained parameters for (CH 2 O) 3 in v(A) state.

Table 5 .
Results of refinement of observed frequencies for (CH 2 O) 3 in ground and v(A) = 1

Table 6 .
Results of Refinement of Observed Frequencies for (CH 2 O) 3 in v 1 (E) = 1 State.

Table 8 .
Results of refinement of observed frequencies for (CH 2 O) 3 in v 2 (E) = 1 state.