A new analysis of the ν4(E) rovibrational band of the symmetric top molecule PF3

Journal of Molecular Spectroscopy 264 (2010) 37–42

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A new analysis of the m4(E) rovibrational band of the symmetric top molecule PF3 Hicham Msahal, Hamid Najib ⇑, Siham Hmimou ¨l, Faculté des Sciences, B.P. 133, Kénitra 14000, Morocco Equipe de Spectrométrie Physique, Département de Physique, Université Ibn Tofaı

a r t i c l e

i n f o

Article history: Received 12 July 2010 In revised form 16 August 2010 Available online 22 September 2010 Keywords: Phosphorus trifluoride FTIR Fundamental band Reductions Equivalences

a b s t r a c t The high-resolution Fourier transform infrared spectrum of phosphorus trifluoride PF3 have been reinvestigated in the m4 perpendicular band region around 347 cm1. Thanks to recent pure rotational measurements, 595 new infrared transitions of the m4 band have been assigned extending the rotational quantum number values up to Kmax = 66 and Jmax = 67. As a consequence of this extension, a sophisticated model containing a large number of parameters and interaction constants was adopted for the analysis of the IR transitions of the m4 fundamental band of PF3. A merge of the IR transitions and the reported MW/ MM/RF data within the v4 = 1 excited level yielded an accurate rotational ground state C0 value, 0.159970436 (69) cm1, which was used to determine an improved GS structure, r0(P–F) = 1.56324405 (11) Å and ](FPF) = 97.752232 (29)°. All experimental data have been refined applying various reduction forms of the effective rovibrational Hamiltonian developed for an isolated degenerate state of a symmetric top molecule. The v4 = 1 excited state of the PF3 oblate molecule was treated with models taking into account ‘- and k-type intravibrational resonances. Parameters up to sixth order have been accurately determined and the unitary equivalence of the derived parameter sets in different reductions was demonstrated. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction Phosphorus trifluoride PF3 is an oblate symmetric top molecule belonging to C3v point group. It has four vibrational normal modes:  two non-degenerate A1-type modes: m1 and m2;  two degenerate E-type modes: m3 and m4. PF3 was the subject of several studies by rotational and vibrational spectroscopy [1–9]. The more recent ones can be summarised as: Najib [8] extracted several accurate experimental values of the vibrational constants from the recent high-resolution Fourier transform (FTIR) investigations (3  103 cm1) in the spectra of PF3. Taking into account the Fermi resonance established between the fundamental m3 and combination m2 + m4 bands, he corrected the band-centres and the anharmonicity constants of the molecular potential of this molecule. Masoud and Tahere [9] studied the millimetre-wave rotational spectra of the ground, v2 = 1 and v4 = 1 states. They improved the accuracy of some parameters of these levels.

⇑ Corresponding author. Fax: +212 5 37 32 94 33. E-mail addresses: [email protected], [email protected] (H. Najib). 0022-2852/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2010.08.005

The subject of the present study is the m4 fundamental band of PF3 near 347 cm1. The main objective is first to extend the assignment of m4 transitions to high values of the rotational quantum numbers J and K in the our previous FTIR spectrum [1], and second to check the validity of the approach of unitary equivalent sets of parameters in different reductions of the rovibrational Hamiltonian H. Since the sphericity of the oblate molecule PF3 is not too pronounced, c = 2(jC0  B0j)/(C0 + B0) = 0.479, the various reduced forms of H can be applied. It is well known that because of collinearity problems, the molecular parameters of the effective rovibrational Hamiltonian cannot all be determined simultaneously by fitting experimental data. The number of parameters has to be reduced through a unitary transformation called reduction. The reduction method of H was first proposed by Lobodenko et al. [10] and further developed by Watson et al. [11] and Harder [12] for an isolated vt(E) = 1 excited state of C3v molecules. Quite recently, different reductions were proposed by Sarka and Harder [13] for the vt(E) = 2 vibrational state, and extended by Wötzel et al. [14] to the vt(E) = 3 excited state of a symmetric top molecule. More recently, Strˇíteská et al. [15] and Sarka and Strˇíteská [16] suggested reduction schemes for the Hamiltonian with the terms describing the vt(E), vs(A1) Coriolis-interacting states up to the fourth order, and for the Fermi resonance involving degenerate vibrational states of a symmetric top molecule. It was shown that to avoid the ambiguity of H some of the spectroscopic constants have to be constrained to predetermined values, usually to the value zero.

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H. Msahal et al. / Journal of Molecular Spectroscopy 264 (2010) 37–42

Many studies were performed [17–21] using this approach. The obtained results demonstrated that the procedure of reduction is an appropriate solution for the indeterminacy problems, and therefore should be used systematically in various fits of experimental spectra. Thus, we successfully applied reduction methods to the v3 = 1 [22], v4 = 2 [23], v1 = v3 = 1, and v3 = 2 [24], and v1 = v4 = 1 [25] excited states of the nitrogen trifluoride 14NF3. In this study, we apply the concept of reduction of the rovibrational Hamiltonian to the v4 = 1 state of the phosphorus trifluoride PF3 molecule. To our knowledge, this approach has not yet been used in the fitting of the infrared experimental data of this molecule. 2. Spectral region

Table 1 Ground state constants (cm1) of PF3. Parameter

3. Theory The diagonal and off-diagonal matrix elements of the Hamiltonian operator that were considered for the ground and v4 = 1 states, which include contributions of higher order as well as K and J dependences of the interaction parameters, are listed in Appendix. According to the theory of equivalent reductions, and in order to prevent correlations in the fitting experimental data, some of the molecular parameters have to be constrained. One of the off-diagonal parameters accounting for D(k  ‘) – 0 interaction can be refined while the others are fixed, usually to zero value. In the case of the isolated v4 = 1 state of PF3, three equivalent D-, Q-, and QD-reduction schemes were proposed [10–12]. – In the D-reduction, the d and e parameters of the (0, 3) k-type interaction are constrained to zero, while the r parameter of the (2, 1) ‘-type resonance is refined. – In the Q-reduction, the r and e parameters are fixed to zero. – In the QD-reduction, the e parameter is refined. Furthermore, the condition rK = 0 and qK = 0, or gJJ = 0, or gJK = 0, or gKK = 0 has to be imposed in all reductions in order to avoid collinearity problems [11,13]. 4. The ground state The most accurate experimental ground state (GS) parameters for the oblate molecule PF3 employed in the present study and reported in Table 1 come from different sources. The J-dependent parameters up to sextic centrifugal distortion constants, and including a very small h3 splitting term, have been determined by Cotti et al. [2] using MW and RF spectra. The experimental C0

Ref. [1]

This study

D0J  107

0.1599635 (11) 0.26084712 (31) 2.616 3 (20)

0.159970436 (69)c

0.2608469623 (36)c 2.619095 (46)c

D0JK  107

3.92547 (18)c

3.916 (5)

5.421 (26)c

4.6 (4)

2.330 (23)c

2.14 (13)

3.102 (46)c

3.31 (20)

C0 B0

D0K  107 H0JJ  1013 H0JK  1012 H0KJ  1012 H0KK  1012 0 h3  1014 a b

In our previous study of the lowest m4 fundamental band of PF3 near 347 cm1 [1], by high-resolution infrared spectroscopy (a resolution of 3  103 cm1; the accuracy of the wavenumber scale ca. 2  104 cm1), it was found that the v4 = 1 state is well isolated, but affected by numerous intravibrational perturbations. Parameters of this level were obtained by least-squares fit, r = 0.30  103 cm1, of 2317 rovibrational lines with Kmax/ Jmax = 55/59. The sign of the strong ‘-doubling constant was determined unambiguously from the intensity simulations. In this study, we were able to expand the IR assignment of m4 lines to high values of the rotational quantum numbers J and K. The predictions were based on our previous results [1] and rotational transitions recently measured in the centimetre-wave (MW), millimeter-wave (MM) and radiofrequency-wave (RF) ranges [7]. About 590 new transitions of the m4 perpendicular band were identified with Kmax/Jmax = 66/67.

Ref. [2]

c

1.73a,c

0b,c 8.333 (24)c

From harmonic force field calculations, uncertainty not given. Fixed to zero because unknown. Values adopted for the upper state least-squares fits.

value has been obtained by Najib et al. [1] from effects associated with avoided crossings within the v4 = 1 state. However, the D0K constant comes from harmonic force field [26], while H0K remains unknown. The major difficulty in the determining of the K-dependent GS parameters for a symmetric top molecule comes from the fact that these constants cannot be obtained from microwave spectra or from ordinary GS combination differences. In practice, two useful methods can be used: – Microwave or infrared ‘‘perturbation-allowed” transitions, but this method is applicable only in favourable cases. – The ‘‘loop-method” [27] combining the IR spectra of the m1 t fun1 damental band, the 2m2 hot band and the 2m2 overtone t  mt t band; unfortunately, the 2m2 and 2m2 perpendicular compo3 4 nents of PF3 have not yet been observed. More recently, Maki et al. [28] proposed a method using several different combinations without the need for localised perturbations. In this study, it was possible to obtain a new experimental value of the GS axial rotational constant C0, with D0K fixed to its harmonic force field value: 1.73  107 cm1, and H0KK fixed to zero. We obtained: C0 = 0.159970436 (69) cm1 by combining the more accurate reported microwave C4 value [7]: 0.159642461 (23) cm1, with aC4 ¼ C 0  C 4 ¼ 3:27945 ð46Þ  104 cm1 determined in the present study by the fit of the merged IR and microwave data (vide infra); the error given in parentheses being the standard deviation of C4 with C0 fixed to the value of Table 1 (column 2). We adopted this new C0 value for the upper state least-squares fits. Our experimental C0 value, which is correlated with D0K , H0KK and C4, is in perfect agreement with the most recent one reported [1] (Table 1, column 2), but more accurate by two orders of magnitude.

5. The m4(E) fundamental band

5.1. Data fits and results The present fit calculations were carried out according to the same reduction schemes of the rovibrational Hamiltonian already performed for the v4 = 1 state of 14NF3 [19] through a vibrationally isolated model. The same FULLSYM program [29] used for this level was adapted for the v4 = 1 state of PF3. The diagonal and off-diagonal matrix elements of the Hamiltonian that were considered for the ground and the excited states are given in the Appendix. The GS parameters were constrained to the values of Table 1.

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H. Msahal et al. / Journal of Molecular Spectroscopy 264 (2010) 37–42

The least-squares fits were started in D-reduction employing only the IR transitions. The body of data constituted of 2912 lines included 595 new transitions identified in this study. A unit weight was ascribed to each pure line, while 0.1 and 0.01 weights were given to blended lines according to their profiles. Moreover, all the transition wavenumbers that differed from the corresponding calculated values by more than 2  103 cm1 were zero weighted. According to the reduction of the rovibrational Hamiltonian, the qK parameter of the ‘(2, 2) interaction was refined. The qJJ and qJK constants of this interaction could not be statistically determined and were fixed to zero. The best results quoted in Table 2 for different reductions were obtained by refining 19 parameters of the m4 band of PF3. In order to obtain an improved set of m4 parameters, we merged our IR data with 534 microwave transitions (Kmax/Jmax = 75/75) obtained with great precision by MW/MM/RF techniques [7,30]. The body of pure rotational data comprised 157 rotational transitions with the selection rule DJ = 1; 363 direct ‘-type resonance transitions with the selection rules DJ = 0, Dk = D‘ = 2; 4D(k  ‘) = 3 ‘‘perturbation-allowed” transitions, and 10 A1/A2 splittings with G = jk  ‘j = 3. Weights of 4000 and 3  107 were assigned to large majority of these transitions according to the ratio of precision between MW/MM/RF and IR measurements. The fitted parameters are listed in Table 3. All the parameters not appearing in Tables 2 and 3, in particular the small octic centrifugal distortion constants (in 1018/ 1017 cm1), are not relevant for this study and therefore were omitted. It should be noted that the DK and HKK parameters for different reductions were refined relative to the D0K fixed to its harmonic force field (Table 2, column 2) and H0KK fixed to zero.

Table 2 Parameters (cm1) of the

m0 B C DJ  107 DJK  107 DK  107 HJJ  1013 HJK  1012 HKJ  1012 HKK  1012 Cf gJ  106 gK  106 gJJ  1010 gJK  1010 gKK  1010 q  104 qJ  109 qK  109 qJJ  1013 qJK  1012 r  105 t  109 h3  1014 d  107 e  107 No. of IR data Kmax/Jmax r  103

5.2. Discussion The conditions suggested by the theoreticians for the reductions of the Hamiltonian H relative to the vt = 1 isolated degenerate state of a symmetric top molecule were imposed to avoid collinearity problems. As mentioned in Section 2, we were able to assign more than 500 new lines of m4 with Kmax/Jmax = 66/67 thanks to MW/MM/RF accurate information [7]. As a consequence of this extension, it was necessary to adopt for the analysis of the IR transitions a more sophisticated model, containing a number of parameters and interaction constants larger than before. Inspection of Table 2 shows that the parameters are in good agreement with our previous investigation [1], but significantly more accurate. Comparison between the different reductions can be done by checking: – The standard deviations: Tables 2 and 3 show that the refined parameters are covered by quasi-equivalent standard deviations, the common values r = 0.2  103 cm1 for the IR data and r = 1.5 kHz for the rotational data are of the same order of magnitude as the estimated precision of the experimental measurements. – The rotational C, B constants, the Cf term of the z-Coriolis interaction, the quartic centrifugal distortion constants, and the q ‘doubling terms should not be affected by the unitary transformation. Tables 2 and 3 show that these parameters have values very close to each other in the three reductions. – Some relations demonstrating the unitary equivalence of the three reductions: the relations between parameters of the D- and Q-reductions were given by Lobodenko et al. [10], and

v4 = 1 state of PF3, IR data only. Previous study [1] IR only

D-reduction IR only

Q-reduction IR only

QD-reduction IR only

347.085735 (19) 0.26094803 (64) 0.15915173 (57) 2.6759 (25) 4.0793 (69) 1.8294 (56) 4.6a 2.14a 3.31a — 0.1009279 (12) 1.0865 (83) 0.9306 (75) — — — 9.88591 (75) — — — — — — — — — 2317 55/59 0.304

347.086101 (18) 0.260950452 (89) 0.159642243 (75) 2.65187 (67) 4.0128 (13) 1.77805 (96) 6.08 (14) 2.570 (38) 3.232 (31) — 0.1004263 (12) 1.360 (15) 1.241 (15) 0.102 (37) 0.206 (39) 0b 9.83974 (84) 5.442 (42) 4.873 (36) 0c 0c (±)9.98 (21) 4.390 (95) 8.333d 0b 0b 2912 66/67 0.231

347.086097 (19) 0.260950609 (91) 0.159641995 (73) 2.65209 (73) 4.0125 (14) 1.77764 (95) 6.12 (16) 2.559 (40) 3.215 (31) — 0.1004264 (11) 1.0980 (96) 0.979 (10) 0.107 (36) 0.210 (39) 0b 9.83966 (83) 5.877 (34) 5.088 (33) 0c 0c 0b 4.07 (10) 8.333d ()3.050 (82) 0b 2912 66/67 0.225

347.086083 (18) 0.260950480 (83) 0.159642157 (62) 2.65260 (66) 4.0164 (14) 1.78205 (97) 1.71 (26) 1.53 (20) 3.62 (34) 3.18 (16) 0.1004273 (11) 1.0432 (58) 0.9161 (58) 0.0765 (57) 0c 0b 9.84449 (71) 6.096 (32) 5.093 (31) 0c 0c 0b 2.18 (17) 8.333d 0b ()1.586 (36) 2912 66/67 0.230

Errors given in parentheses are 1 SD expressed in units of the last digit quoted. a Fixed to our ground state values [1]. b Imposed by the reduction. c Fixed to zero because not significant. d Fixed to the ground state value (Table 1 of this study).

40

H. Msahal et al. / Journal of Molecular Spectroscopy 264 (2010) 37–42 Table 3 Parameters (cm1) of the

m0 B C DJ  107 DJK  107 DK  107 HJJ  1013 HJK  1012 HKJ  1012 HKK  1012 Cf gJ  106 gK  106 gJJ  1012 gJK  1012 gKK  1012 sJJJ  1014 sJJK  1014 sJKK  1014 sKKK  1014 q  104 qJ  109 qK  109 qJJ  1014 qJK  1014 r  105 rJ  1010 h3  1014 t  109 tJ  1014 d  107 dJ  1012 e  107 eJ  1012 No. of data Kmax/Jmax r(IR)  103 r(MW/MM/RF) (kHz)

v4 = 1 state of PF3, IR + pure rotational data. Previous study [1]

D-reduction

Q-reduction

QD-reduction

347.086156 (16) 0.26095049 (7) 0.1596355 (1) 2.650 (3) 3.944 (5) 1.666 (6) 4.6a 2.14a 3.31a — 0.1004411 (6) 1.359 (8) 1.201 (8) — — — — — — — 9.83665 (7) 5.334 (7) 9.77c — — (±)9.805 (4) — — 4.313 (4) — — — — — 2317 IR + 140 MW 55/59 0.307 —

347.086183 (11) 0.2609504047 (36) 0.159642408 (46) 2.651439 (98) 4.00796 (43) 1.77464 (45) 6.004 (35) 2.436 (15) 3.106 (16) — 0.100433831 (48) 1.33689 (18) 1.20404 (23) 0.929 (32) 0b 6.526 (42) 0b 1.323 (11) 2.669 (22) 1.346 (11) 9.83796522 (37) 5.38429 (14) 4.87363 (94) 2.205 (13) 6.607 (54) (±)9.85057 (18) ()4.656 (24) 8.1446 (54) 4.346695 (61) 2.545 (12) 0b 0b 0b 0b 2912 IR + 534 MW/MM/RF 75/75 0.268 1.56

347.086183 (11) 0.2609505331 (36) 0.159642152 (45) 2.651448 (97) 4.00798 (43) 1.77463 (45) 6.006 (35) 2.437 (15) 3.107 (16) — 0.100433769 (48) 1.08004 (18) 0.94718 (23) 2.051 (32) 0b 5.408 (41) 0b 1.305 (11) 2.636 (22) 1.330 (11) 9.83796944 (37) 5.80249 (15) 5.08229 (92) 2.219 (13) 6.534 (54) 0b 0b 8.1501 (63) 3.928505 (62) 2.529 (13) ()3.207058 (62) (±)0.2892 (89) 0b 0b 2912 IR + 534 MW/MM/RF 75/75 0.267 1.56

347.086178 (11) 0.2609505330 (37) 0.159642174 (47) 2.650726 (101) 4.00355 (45) 1.77090 (47) 0.219 (37) 2.732 (19) 5.459 (34) 3.977 (21) 0.100433669 (49) 1.07821 (19) 0.94496 (24) 1.854 (50) 0b 5.541 (82) 0b 1.605 (11) 3.221 (22) 1.615 (11) 9.83796941 (38) 5.80242 (15) 5.08324 (99) 2.048 (13) 7.984 (56) 0b 0b 34.3914 (65) 3.928464 (64) 2.684 (13) 0b 0b ()1.617373 (32) (±)1.4013 (47) 2912 IR + 534 MW/MM/RF 75/75 0.275 1.59

Errors given in parentheses are one standard deviation expressed in units of the last digit quoted. a Fixed to our ground state values [1]. b Imposed by the reduction. c Fixed to the value quoted in Ref. [30].

Watson et al. [11]. Those between the Q- and QD-reductions were proposed by Harder [12]. As can be seen in Table 4, the relations referring to Eqs. (1)–(5) are nicely fulfilled, with some reservations concerning the two last terms of Eq. (5). This agreement confirms the unitary equivalence of reductions D, Q and QD for fitting the m4 band of PF3.

mulae quoted in Ref. [32] and the conversion factor B, C (MHz) = 5.05376  105/IB,C, where IB,C are moments of inertia. We obtain:

The positive sign of q was obtained from our previous intensity calculations on the m4 fundamental band [1], employing the phase convention of Cartwright and Mills [31]. The signs of r, d and e are arbitrary and cannot be determined from the experimental data. The relative signs of these parameters are however well determined. They should be compatible with the reduction theory: according to Eq. (1) of Table 4, since (C  B + 2Cf)/q is negative, r and d must have opposite signs. Thus r has been arbitrarily chosen positive, and with this assumption d is negative. A similar discussion can be made for d and e whose signs depend on the values of (C  B) and Cf (see Eq. (4), Table 4). Because these latter and d are negative, e has to be negative, too, when r is taken to be positive. The sign of the h3 excited coefficient, which cannot be determined, is assumed to be positive as for the ground state [2]. An improved ground state structure of the phosphorus trifluoride PF3 may be derived from the recent B0 value given in Table 1 and the C0 constant determined in this study, employing the for-

r0(P–F) is the GS bond length and ](FPF) is the GS angle. These new values are in good agreement with the most recent ones reported [1]:

2

2

I0B ¼ 64:626126633 ð91Þ uaÅ ;

I0C ¼ 105:379027 ð46Þ uaÅ

r 0 ðP—FÞ ¼ 1:56324405 ð11Þ Å;

]ðFPFÞ ¼ 97:752232 ð29Þ

r0 ðP—FÞ ¼ 1:563230 ð10Þ Å; ]ðFPFÞ ¼ 97:759 ð2Þ but significantly more accurate. 6. Conclusion In the present study, we reanalysed our high-resolution FTIR spectrum of the oblate symmetric top molecule PF3 in the m4 fundamental band near 347 cm1. Helped by the MW/MM/RF measurements, the extension of IR assignments in the v4 = 1 state yielded improved values of its parameters. The experimental data of the m4 perpendicular band were fitted using different reduction forms of the rovibrational Hamiltonian. The standard deviations of the fits are comparable and the

41

H. Msahal et al. / Journal of Molecular Spectroscopy 264 (2010) 37–42 Table 4 Verification of unitary equivalence for PF3 in the Equation

Value (cm1)

Quantity

rD

(1)

Q

d F q

(2)

gDJ  gQJ Q ðgD K  gK Þ Q 2

8F ðdq2Þ

1/4(tD  tQ) 1=4ðqD J



qQJ Þ

9.98 (21)  105 9.34 (25)  105

9.85057 (18)  105 9.85056 (19)  105

2

¼ 1=4fq þ qJ JðJ þ 1Þ þ qK ½k þ ðk  2Þ2 2

þ qJJ J 2 ðJ þ 1Þ2 þ qJK JðJ þ 1Þ½k þ ðk  2Þ2 g

0.262 (24)  106

0.25676 (36)  106

6

6

0.25686 (46)  10

0.263 (11)  106

0.256893 (95)  106

0.232 (12)  106

0.256893 (10)  106

0.08 (4)  109 0.108 (19)  109

0.104546 (31)  109 0.104550 (73)  109

0.107 (4)  10

0.104546 (38)  10

0.095 (50)  109

0.104546 (40)  109

eQD

1.586 (36)  107 1.538 (41)  107

1.617373 (32)  107 1.617486 (31)  107

0.441 (42)  1012 0.45 (2)  1012

0.5847 (72)  1012

dQ ðCBÞ 2Cf Q HQD JJ  H JJ Q 1=9ðHQD JK  HJK Þ

0.45 (2)  1012

0.5711 (33)  1012

QD 2ðh3  ðeQD Þ2 2 CB

—

0.524 82 (25)  1012

0.496 (45)  1012

0.51642 (20)  1012

0.3021614 (25)

0.30217565 (14)

F = C  B + 2Cf

with F±(J, k) = [J(J+1)  k(k ± 1)]1/2. – The ‘(2, 1) interaction (r resonance):

hv;‘  1; J; k j H=hc j v;‘  1; J; k  1i 3

¼ fð2k  1Þ½r þ rJ JðJ þ 1Þ þ r K ½k þ ðk  1Þ3 g  ½ðv  ‘ þ 2Þðv  ‘Þ 1=2 F  ðJ; kÞ – The ‘(2, 4) interaction (t resonance):

hv;‘  2; J; k j H=hc j v;‘; J; k  4i ¼ 1=2½t þ t J JðJ þ 1Þ F  ðJ; kÞF  ðJ; k  1ÞF  ðJ; k  2ÞF  ðJ; k  3Þ

0.5743 (37)  1012

Q 1=15ðHQD KJ  HKJ Þ Q h3 Þ

 ½ðv  ‘ þ 2Þðv  ‘Þ 1=2 F  ðJ; kÞF  ðJ; k  1Þ

9

2  ðd q Þ

Q

(5)

IR + MW/MM/RF

9

D 2 q ðrF 2Þ

(4)

hv;‘  1; J; k j H=hc j v;‘  1; J; k  2i

IR only

0.262 (25)  10

D 2

8 ðr F Þ

(3)

– The ‘(2, 2) interaction (q resonance) within the v4 = 1±1 state, employing the phase convention of Cartwright and Mills [31]:

v4 = 1 excited state.

– The k (0, 6) interaction (h3 resonance):

hv;‘  1; J; k j H=hc j v;‘  1; J; k  6i ¼ h3 F  ðJ; kÞF  ðJ; k  1ÞF  ðJ; k  2ÞF  ðJ; k  3ÞF  ðJ; k  4ÞF  ðJ; k  5Þ – The k (0, 3) interaction (‘-dependent d resonance):

theoretical relations between parameters of different reductions are quite fulfilled. A merge of the IR transitions and the recent v4 = 1 MW/MM/RF data yielded an accurate rotational ground state C0 value and an improved GS structure.

hv;‘  1; J; k j H=hc j v ; ‘  1; J; k  3i ¼ ‘½d þ dJ JðJ þ 1Þ F  ðJ; kÞF  ðJ; k  1ÞF  ðJ; k  2Þ – The k (0, 3) interaction (‘-independent e resonance):

hv;‘  1; J; k j H=hc j v;‘  1; J; k  3i ¼ ð2k  3Þ½e þ eJ JðJ þ 1Þ F  ðJ; kÞF  ðJ; k

Acknowledgment

 1ÞF  ðJ; k  2Þ:

Prof. H. Najib thanks Prof. L. Halonen, Helsinki (Finland), for the use of his FULLSYM program. References

Appendix The following expressions are the diagonal and off-diagonal matrix elements of the Hamiltonian operator that are used [19]: – The ground state energy:

E0 ðJ; KÞ ¼ ðC 0  B0 ÞK 2 þ B0 JðJ þ 1Þ  D0J J 2 ðJ þ 1Þ2  D0JK JðJ þ 1ÞK 2  D0K K 4 þ H0JJ J 3 ðJ þ 1Þ3 þ H0JK J 2 ðJ þ 1Þ2 K 2 þ H0KJ JðJ þ 1ÞK 4 þ H0KK K 6 þ d 0

where K = jkj and d ¼ h3 JðJ þ 1Þ½JðJ þ 1Þ  2 ½JðJ þ 1Þ  6 (see Ref. [33]). – The excited state energy:

Eðv ; ‘; J; KÞ ¼ m0 þ ðC  BÞK 2 þ BJðJ þ 1Þ  DJ J 2 ðJ þ 1Þ2  DJK JðJ þ 1ÞK 2  DK K 4 þ HJJ J 3 ðJ þ 1Þ3 þ HJK J 2 ðJ þ 1Þ2 K 2 þ HKJ JðJ þ 1ÞK 4 þ HKK K 6 h þ 2Cfz þ gJ JðJ þ 1Þ þ gK K 2 þ gJJ J 2 ðJ þ 1Þ2 þ gJK JðJ þ 1ÞK 2 þ gKK K 4 þ sJJJ J 3 ðJ þ 1Þ3

i þ sJJK J 2 ðJ þ 1Þ2 K 2 þ sJKK JðJ þ 1ÞK 4 þ sKKK K 6 k‘ ‘ = ± 1 for the

v4 = 1±1 state.

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