Study of Millimeter-Wave Rotational Spectra of Phosphorus Trifluoride in Ground , v 2 = 1 and v 4 = 1 States

The millimeter-wave rotational spectra of the ground and excited vibrational states v2 =1 and v4 =1 of the symmetric top molecule, PF3, have been analyzed again. The B0 = 7819.9907(13) MHz, DJ = 7.84984(41) kHz, DJk = -11.7644 (11) kHz, HJ = 15.678 (36) mHz, HJk = -66.46 (12) mHz and HkJ = 87.42 (15) mHz have been determined for ground state. The 1=±1 series have been assigned and the rotational parameters including B4 =7823.09212(41) MHz, (q


Introduction
Phosphorus trifluoride (PF 3 ) has one threefold axis of symmetry (C 3 ) and three vertical planes of symmetry (σ v ).This molecule, therefore, belongs to the spectroscopic group C 3v .Selection rules for a molecule of this configuration show that there should be two totally symmetric fundamentals (Type A 1 ) and two doubly degenerate fundamentals (Type E).
Several authors have studied the rotational spectra of the ground state and some excited states [1][2][3][4][5][6][7][8][9][10] .The lowest doubly degenerate vibrational level, v 4 =1, lies at 344 cm -1 (Deg deform).The next highest frequency, v 2 , is found at 487 cm -1 (Sym deform) 11 .There is no doubt about the assignment of the series of lines which lie to high frequency of the ground state rotational transitions.Owing to the large dipole moment and the large thermal population, the spectra are intense.The aim of this study is determination of rotational parameters for mixing the low and high J values in ground and v 2 =1 and v 4 =1 states, which are more accurate and reliable.

Spectra in the ground and v 2 =1 states
The rovibrational Hamiltonian for a non-degenerate vibrational state is essentially similar to that of the ground state.The transition frequencies for J→(J+1) 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 +…. ( We have mixed the 181 frequencies of Refs 5 and 7 in order to improve the accuracy of the rotational and other centrifugal distortion constants.Then a weighted least-squares method 12 was used to fit the mixed frequencies for low and high J values to the parameters of equation 1, in which the weights were taken to be w = 1/(observed error) 2 = 1/(0.2) 2 , where 0.2 in MHz is estimated uncertainty in an observation for each unblended line.
The fit to the transitions are given in Table 1 and the constants obtained are given in Table 5.Some of the lines have an observed error of 0.3 to 0.9 MHz to allow for overlapping or broadening.These Tables show that the mixed frequencies of low and high J values of PF 3 were fitted very well and correlation coefficients of parameters are reasonable (Table 2).The structural parameters of this compound as shown in Table 5 were obtained with higher accuracy and compared with previous work.

Spectrum in the State v 2 = 1
This state shows the characteristic intensity pattern of non-degenerate state in its k structure.The population of the four fundamental vibrations of PF 3 are given in the Table 3.This Table shows the v 2 =1 state has enough population (9.5%) relative to ground state to observe the signals.The v 2 =1 state was assigned by Hirota and Morino 1 to the intense vibrational satellite to low frequency of the ground state.Their assignment was based on intensity arguments and the absence of l-doublets.The frequencies of Refs 5 and 7 were mixed together in order to obtain improved constants.The quartic constants were obtained with good agreement with previous work.It should be noticed that the distortion constants obtained are not very different from those of the ground state.H J , H Jk , H kJ , H k centrifugal distortions were obtained with new values and higher accuracy.For PF 3 , H Jk <0 while both H J and H kJ >0.This behavior is typical of an oblate top.The results of refinements are listed in Table 4 and obtained parameters are shown in Table 5.

Spectrum in the state v 4 = 1
As Table 3 shows, this state has enough population relative to the ground state to show strong signals in its spectrum.The theory for the rotational spectrum of an E degenerate vibrational state of a C 3v molecule is well known 13 .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 1) where + t q is the l-type doubling constant, B and C are rotational constants, and ζ t 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 (B-C+Cζ t ).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 fork1-1≠0 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 (k1-1) 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 14- 22 .The Hamiltonian was set up for a symmetric top molecule like PF 3 .This rotationvibrational Hamiltonian has two different blocks that 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 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 ldoublets, as shown by Grenier-Besson and Amat 13 .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.The diagonal matrix elements are given by: <v t , l t , J, k | H/h | v t , l t , J, k> = BJ(J + 1) + (C -B)k 2 -2Cζkl -D J J 2 (J + 1) 2 -D Jk J(J + 1)k 2 -D k k 4 + η J J(J + 1)kl + η k k 3 l + 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 In addition, there are off-diagonal terms that arise from the transformation.The major one of these gives rise to the l doubling: <v t , l, J, k (5) and hence, the lines can be assigned.
The frequencies of Refs 4 and 6 were mixed together in order to improve the accuracy of the parameters in this state, then a weighted least-squares method 12 was used to fit these frequencies to equation 3, 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 0.5MHz 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 8.

Results and Discussion
The oblate symmetric top PF 3 shows a strong ∆J=+1, ∆K=0 ground state spectrum.The v 2 =1 rotational spectrum is also quite strong, lines are about 10 times weaker than the ground state.Due to the fluorine nuclear spin statistic for both the spectra, K = 3n lines are twice those for K≠3n.The spectra in ground and v 2 =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 2 =1states, the Fortrat diagram is produced which is shown in Figure 1.In this diagram 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.).J The first term has ∆K=±3 matrix elements and the second one ∆K=±6 matrix elements.ε and ' 3 h are quartic and sextic centrifugal distortion constants respectively.In reducing the Hamiltonian, the ∆K=±3 matrix elements are eliminated.So, The reduced Hamiltonian contributes in first order to the A 1 /A 2 splitting of the K = 3 levels: For C 3v symmetric tops the splitting of the K=3 levels is dependent on the magnitude of the sextic constant h 3 , which is usually smaller than H J .Cazzoli 23  This relation can predict known values of h 3 to better than 20%.If the values of rotational constants B and C in ground state are put in Eq 10 then h 3 = 2.836 mHz is obtained for PF 3 , while this value has been determined 2.4982 (73) mHz 7 .Figure 2 shows the variation of k = 3 splitting from J = 26 to J = 82, which increases rapidly as J increases 7 .
it is obvious that submillimeter-wave spectroscopy is one of best method to observe it.

Figure 2.
Variation of K = 3 splitting with J.For the v 4 = 1 state the important parameters in determining form of the spectrum are the l-type doubling constant + t q 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 (3) this results in the positive series being displaced to lower frequency and the negative series to higher frequency.
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.This vibrational state is especially characterized by a very small value of [C-B-Cζ] =26 MHz, which together with a large value of the l-doubling constant q t + =29.5 MHz, leads to strong l-type resonances of states with ∆K=∆1=±2.
A least-squares refinement of the 215 observations was carried out using the programme 12 .The results of fitting are given in Tables 6 and 7, the results are shown in Table 8.The Fortrat-like diagram in this state is shown in Figure 3.
This Table shows that most of the obtained parameters have different values and higher accuracy compare with Ref 10 .

Figure 1 .
Figure 1.Fortrat diagram of PF 3 in ground and v 2 = 1 states.J=20→21.It is generally assumed that rotational Hamiltonian of a symmetric top in its ground vibrational state is diagonal in J and K. But, actually the rotation of the molecule about the b-ax is produces a small distortion moment perpendicular to the symmetry axis and a new term appears in the Hamiltonian:).J (J h )] J (J J )J J ε[(J H and colleagues have found a strong correlation between h 3 and B 4 /(C-B), the result being (in MHz)

Table 1 .
Results of Refinement of Observed Frequencies for PF 3 in ground State.

Table 2 .
Correlation Coefficients for PF 3 in ground state

Table 3 .
Approximate populations of the fundamental vibrational states at 298 K.

Table 4 .
Results of Refinement of Observed Frequencies for PF 3 in v 2 = 1 State

Table 5 .
Results of refinement of observed frequencies for PF 3 in ground and v 2 = 1 state.
* constrained at this value.

Table 6 .
Results of Refinement of Observed Frequencies for PF 3 in v 4 = 1 State Sigma = .144795,215 Transitions in fit, Sigma.w= 1.111612, 202 Degrees of freedom

Table 7 .
Correlation coefficients of PF 3 for the v 4 = 1 state.

Table 8 .
Comparison of rotation-vibration parameters for PF 3 in v 4 = 1 state.
*constrained at this value.