Experimental rovibrational constants and equilibrium structure of phosphorus trifluoride

Journal of Molecular Spectroscopy 305 (2014) 17–21

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Experimental rovibrational constants and equilibrium structure of phosphorus trifluoride Hamid Najib ⇑ Laboratoire de Physique de la Matière Condensée, Équipe de Spectrométrie Physique, Département de Physique, Université Ibn Tofaïl, Faculté des Sciences, B.P. 133, Kénitra 14000, Morocco

a r t i c l e

i n f o

Article history: Received 27 August 2014 In revised form 23 September 2014 Available online 2 October 2014 Keywords: Phosphorus trifluoride Rovibrational constants Harmonic wavenumbers Anharmonic constants Equilibrium structure

a b s t r a c t Thanks to recent high-resolution Fourier transform infrared (FTIR) and pure rotational (RF/CM/MMW) measurements, several experimental values of the rotation–vibration parameters of the oblate molecule PF3 have been extracted, contributing thus to the knowledge of the molecular potential of phosphorus trifluoride. The data used are those of the fundamental, overtone and combination bands studied in the 300–1500 cm1 range. The new values are in good agreement with ones determined at low resolution, but significantly more accurate. The agreement is excellent with the available values determined by ab initio HF-SCF calculations employing the TZP/TZ2P triple-zeta basis. From the recent experimental rovibrational interaction constants aC and aB, new accurate equilibrium rotational constants Ce and Be have been derived for the symmetric top molecule PF3, which were used to derive the equilibrium geometry of this molecule: re(F–P) = 1.560986 (43) Å; he(FPF) = 97.566657 (64)°. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Phosphorus trifluoride PF3 is a heavy four-atom molecule of symmetry C3v point group. It has four normal modes of vibration:  two totally symmetric (A1-type) modes, m1 and m2;  two doubly degenerate (E-type) modes, m3 and m4. Quite recently, we studied by Fourier transform infrared (FTIR) spectroscopy the vibrational excited states v2 = 1 [1] and v4 = 1 [2] at 487 cm1 and 347 cm1 respectively, and v4 = 2 [3] around 693 cm1. Thus, it was possible to determine an accurate rotational ground state (GS) C0 value, which was used to improve the GS structure of the PF3 pyramidal molecule. Some J-dependent molecular parameters of the ground, v2 = 1 and v4 = 1 levels of PF3 have also recently been determined with a good accuracy using millimeter-wave (MMW) spectra [4]. This work is an extension of our high-resolution FTIR investigation on the vibrationally excited states of fluoride molecules, phosphorus trifluoride PF3 [1–3,5–11], nitrogen trifluoride NF3 [12–17], and deuterated species of silyl fluoride SiD3F [18–21]. The objective is to summarize the recent accurate experimental rotation–vibration interaction parameters, to extract harmonic

⇑ Fax: +212 5 37 32 94 33. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jms.2014.09.008 0022-2852/Ó 2014 Elsevier Inc. All rights reserved.

wavenumbers and anharmonic vibrational constants of the potential function of PF3, and to derive an accurate equilibrium geometry of this molecule, from our recent and latest results available in the literature. 2. Spectral region For the PF3 symmetric top gas, we studied several bands in the 300–1500 cm1 range. We used independent and complementary methods, Fourier transform infrared (FTIR); radiofrequency, centimeter-, millimeter- and submillimeter-wave (RF/CM/MMW) spectroscopies. All studied IR spectra were recorded at Wuppertal (Germany) with the Bruker 120 HR interferometer, except for the spectrum of the m2 and m4 bands, between 300 cm1 and 500 cm1, which was recorded at Giessen (Germany). The rotational spectra were performed in Lille (France) with absorption spectrometers and in Kiel (Germany) by means of one- and twodimension (2D) pulse techniques. 3. Theory 3.1. Hamiltonian operator The diagonal and off-diagonal matrix elements of the Hamiltonian operator Hvr that were considered for the all infrared studies reported in this work are given in the Appendix. They

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H. Najib / Journal of Molecular Spectroscopy 305 (2014) 17–21

include contributions of higher order as well as K and J dependences of the interaction parameters.

perpendicular components [3]; and the 2m±2 3 band has not yet been studied at high-resolution.

3.2. Relationships linking vibrational constants

5. The rovibrational bands of PF3 observed below 1500 cm1

More recently, we established the following relationships linking harmonic wavenumbers, anharmonicity constants and bandcenters of a symmetric top molecule [5].



m‘i i

0

¼ x0i þ xii þ g ii ‘i

ð1Þ

 0  0  2    ‘ nmi i ¼ n m1 þ n  n xii þ ‘2i  n g ii i

ð2Þ

   0  ‘ 0 ‘ 0 ‘ ‘ nmi i þ mmj j ¼ nmi i þ mmj j þ nmxij þ g ij ‘i ‘j

ð3Þ

5.1. The {m1, m3, m2 + m4} interacting system In the analysis of this system, we used spectra recorded by high-resolution Fourier transform spectrometer [10].  The m1 and m3 fundamental bands are linked by the Coriolis interaction (fy13 fixed at the value: 0.474 cm1, predicted by ab initio calculation [26]).  The m3 fundamental and m2 + m4 combination bands are linked by the Fermi resonance W234 = 2.86 cm1.

i: refers to a A1- or E-type mode of vibration; x0i is the normal wavenumber corresponding to the vibrational quantum number vi;  0 m‘i i is the band-center; ‘i is the vibrational angular momentum:

The experimental band-centers obtained are: (m1)0 = 891.940472 (29) cm1; (m3)0 = 859.219339 (19) cm1 and (m2 + m4)0 = 834.38073 (10) cm1.

|‘i| = vi, vi  2, . . ., 1, or 0 only exists for a doubly degenerate vibration; xij and gij are the anharmonicity constants.

5.2. The m2 fundamental band

4. The ground state constants of PF3 The most accurate experimental ground state (GS) rotational parameters obtained for the symmetric top molecule PF3 are collected in Table 1. They come from different sources:

More recently in 2011, we analyzed the m2 parallel band of PF3 by FTIR and MMW spectroscopies, near 487 cm1 [1]. The v2 = 1 excited state was considered vibrationally isolated and the following band-center was obtained: (m2)0 = 487.715684 (18) cm1. 5.3. The m4 fundamental band

 We recently determined the experimental K-dependent C0 value [2] from effects associated with avoided crossings within the v4 = 1 excited level.  The DK0 quartic centrifugal distortion constant comes from harmonic force field calculations [22].  The HKK0 sextic centrifugal distortion constant remains unknown for PF3.  The J-dependent parameters, namely B0, DJ0, DJK0, HJJ0, HJK0 and HKJ0, and the h3 splitting term were calculated by Cotti et al. [24] using pure rotational spectra. We point out that the DK0 and HKK0 cannot be obtained from normal pure rotational transitions, or from GS combination differences. They are to be determined through perturbation-allowed microwave or infrared transitions. In practice, we used the method called ‘‘loop-method’’ [25]. We successfully applied it to NF3 [15] and SiD3F [20]. We combined the IR data of the mt perpendicular fundamental, the 2mt  mt hot and the 2mt overtone bands. Unfortunately, for PF3, we did not succeed to assign any line of the 2m+2 4 Table 1 Ground state constants (cm1) of PF3. Parameter

Value

Reference

C0 B0 DJ0  107 DJK0  107 DK0  107 HJJ0  1013 HJK0  1012 HKJ0  1012 HKK0  1012 h03

0.159970436 (69) 0.2608469623 (36) 2.619095 (46) 3.92547 (18) 1.73a 5.421 (26) 2.330 (23) 3.102 (46) 0. b 8.333 (24)

[2] [24] [24] [24] [23] [24] [24] [24] – [24]

Errors given in parentheses are 1 SD expressed in units of the last digit quoted. a From harmonic force field calculations, uncertainty not given. b Unknown.

We recently studied the lowest v4 = 1 excited state of PF3 through FTIR and RF/CM/MMW techniques [2,22]. It was found that this level is vibrationally isolated, but affected by numerous intravibrational perturbations. All experimental data of the m4 band were refined by applying the three equivalent D-, Q- and QD-reduction schemes of the effective rovibrational Hamiltonian, developed for an isolated degenerate level of a C3v symmetric top molecule. The band-center obtained in the D-reduction is: (m4)0 = 347.086183 (11) cm1 5.4. The 2m2 overtone band Because of the low intensity of this band of PF3, we studied the

v2 = 2 excited state under a poor resolution [10] and estimated the band-center at 975.19 cm1 by means of rough contour simulation. 5.5. The 2m4 overtone band More recently in 2013, we investigated the v4 = 2 excited state of the oblate molecule PF3, by means of FTIR and MMW spectroscopies [3]. The v4 = 20 sublevel was directly studied from the GS; whereas, 1 the v4 = 22 sublevel could be reached only through the 2m2 4  m4 hot band. All data of the 2m04 and 2m2 components were refined 4 using the five equivalent D-, Q-, L-, QD- and LD-reductions of the rovibrational Hamiltonian. The fit gave the following experimental band-centers in the D-reduction: (2m04)0 = 692.846957 (37) cm1 0 1 and (2m±2 . 4 ) = 694.695429 (82) cm 5.6. The 3m4 overtone band The v4 = 3 excited state of PF3 was reached only through the 3m4  m4 and 3m4  2m4 hot bands. The fit gave the experimental 0 1 0 band-centers: (3m±1 and (3m±3 4 ) = 1039.07152 (47) cm 4 ) = 1 1042.650 cm (fixed in the fit).

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5.7. The {m1 + m4, m3 + m4} interacting system and m2 + 2m04 parallel component We studied this system by FTIR spectroscopy [8]. It was treated with a model taking into account ‘(2, 2) interaction inside m1 + m4 and between (A1 + A2) and E components of m3 + m4, the ‘-vibrational resonance inside m3 + m4 (A1 + A2), and the Coriolis interaction between m1 + m4 and m3 + m4. For m2 + 2m04, only 120 lines were assigned. Thus, this parallel component was treated alone using an unperturbed model. The band centers determined are: (m3 + m4)0\ = 1205.068123 (23) cm1; (m3 + m4)0// = 1204.789546 (30) cm1; (m1 + m4)0 = 1237.386253 (54) cm1; (m2 + 2m04)0 = 0 1179.06959 (33) cm1 and (m2 + 2m±2 ) = 1180.9186 cm1 4 (constrained in the fit).

 the diagonal Coriolis interaction terms between rotation and vibration of PF3;  the q(2, 2), r(2, 1), t(2, 4) and h3(0, 6) intravibrational interaction terms inside E-type mode of vibration;  the Coriolis between m1 and m3 and Fermi between m3 and m2 + m4 resonance terms. From the following relationships (Eq. (3)):

ðm1 þ m2 Þ0 ¼ ðm1 Þ0 þ ðm2 Þ0 þ x12 ðm1 þ m4 Þ0 ¼ ðm1 Þ0 þ ðm4 Þ0 þ x14 ðm2 þ m3 Þ0 ¼ ðm2 Þ0 þ ðm3 Þ0 þ x23 ðm3 þ m4 Þ0== ¼ ðm3 Þ0 þ ðm4 Þ0 þ x34  g 34

5.8. The {m1 + m2, m2 + m3} interacting system The m1 + m2 band was treated using a model including the Coriolis interaction with the upper levels of the unobserved m2 + m3 perpendicular band of PF3 [7]. We obtained the following results: (m1 + m2)0 = 1377.754023 (71) cm1 and (m2 + m3)0 = 1345.44 cm1. The (m2 + m3)0 band-center value was calculated taking into account the Fermi resonance between m2 + m3 and the unobserved 2m2 + m4 bands, and was constrained in the fit to a value given a small standard deviation. 6. Results and discussion 6.1. Rovibrational constants In Table 2, we summarize the most accurate experimental values, obtained by high-resolution FTIR spectroscopy, of the all bands studied between 300 cm1 and 1500 cm1. We also give the corresponding experimental rotation–vibration constants: aC = C0  C0 and aB = B0  B0 , C0 and B0 being the rotational constants of the upper level. It is worth noting the fair agreement between our values and those available in the literature [28,29], experimentally determined at low resolution. In Table 3, we gathered:

ðm3 þ m4 Þ0? ¼ ðm3 Þ0 þ ðm4 Þ0 þ x34 þ g 34 we determine the x12, x14, x23, x34 and g34 anharmonic constant values (Table 4). For the 2m4 harmonic band, we obtain the system of equations (Eq. (2)): 0

ð2m04 Þ ¼ 2ðm4 Þ0 þ 2x44  2g 44 

2m2 4

0

¼ 2ðm4 Þ0 þ 2x44 þ 2g 44

We deduce: x44 = 0.049135 (43) cm1 and g44 = 0.462118 (30) cm1. For the band-center of 2m2, we derive an estimated value of the x22 constant: x22 = 0.11 cm1. All the harmonic constant values deduced in this work are given in Table 4. Accurate values of x11, x13, x33 and g33 remain unknown. It is unfortunate that no high-resolution study of the {2m1, 2m3, m1 + m3} interacting system of PF3 has ever been published around 1750 cm1. Using Eq. (1), we can also extract values of the harmonic wavenumbers:

x02 ¼ 487:82 cm1 and x04 ¼ 347:499166ð84Þ cm1

Table 2 Band-centers and rovibrational interaction constants (cm1) of PF3. Band or component

Experimental band-center, this work

Exp. band-center [28]

Experimental aC  104, this work

Experimental aB  104, this work

Exp. aB  104 [28,29]

m1 m2 m3 m4 m2 + m4  m4 m2 + m4 2m04 2m±2 4 2m2 3m±1 4 3m±3 4 m2 + 2m04 m2 + 2m4±2 (m3 + m4)// (m3 + m4)\ 2m2 + m4 m2 + m3 m1 + m2

891.940472 (29) 487.715684 (18) 859.219339 (19) 347.086183 (11) – 834.38073 (10) 692.846957 (37) 694.695429 (82) 975.19a 1039.07152 (47) 1042.650b 1179.06959 (33) 1180.9186b 1204.789546 (30) 1205.068123 (23) 1320.7880b 1345.44c 1377.754023 (71)

891.912 (2) 487.718 (1) – – 489.960 (1) – – – – – – – – – – – – –

4.7671 (12) 0.676756 (68) 3.1609 (7) 3.28028 (115)

3.8332 (6) 2.94647 (10) 6.9502 (4) 1.034424 (72)

4.250 (5) 6.6103 (16) 6.522 (24) – 9.8816 (66) – 7.51 (20) – 6.011 (86) 6.093 (81) – 3.242 (11) 5.4183 (47)

2.179 (4) 2.0559 (28) 2.0301 (52) – 3.0665 (32) 2.923 (39) 7.123 (29 – 8.703 (14) 8.726 (15) – 8.7060 (42) 6.8681 (60)

2.120 2.61 7.696 1.031 2.76 – – – – – – – – – – – – –

Errors given in parentheses are 1 SD expressed in units of the last digit quoted. a Value estimated from a contour of 2m2. b Value constrained in the fit. c Fixed to a value given a small SD.

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H. Najib / Journal of Molecular Spectroscopy 305 (2014) 17–21

Table 3 Coriolis, Fermi, ‘(2, 2), ‘(2, 1), ‘(2, 4) and k(0, 6) interaction terms (cm1) of PF3. Interaction term Cf

z

gJ  106 gK  106 q  104 qJ  109 qK  109 r  105 rJ  109 rK  109 t  109 h3  109

m3 band

m4 band

0.0692558 (12) 2.949 (21)

0.100433831 (48) 1.33689 (18)

2.630 (21) 0.1590 (18) 12.63 (10) – 14.998 (27) 7.03 (14) 0.38 (28) 6.66 (14) –

1.20404 (23) 9.83796522 (37) 5.38429 (14) 4.87363 (94) 9.85057 (18) 0.4656 (24 – 4.346695 (61) 8.1446 (54)

Coriolis between m1 and m3 pffiffiffi W C ¼ 2B1y13 X13 ¼ 0:155 cm1 Fermi between m3 and m2 + m4 W 234 ¼ 2:86 cm1

Table 4 Anharmonicity constants (cm1) of PF3. Anharmonicity constant

Experimental value this Work

Experimental Value [28–30]

Ab initio 6-31G** [27]

Ab initio TZP/TZ2P [27]

x11 x12 x13 x14 x22 x23 x24 x33 x34 x44 g33 g34 g44

– 1.902217 (107) – 1.640402 (40) 0.11 1.48 0.42113 (12) – 1.376687 (56) 0.049135 (43) – 0.139288 (109) 0.462118 (30)

1.42 2.18

1.32 1.45 4.38 1.44 0.15 1.98 0.39 2.43 1.42 0.03 1.39 0.12 0.36

1.51 1.76 5.33 1.64 0.02 1.74 0.90 2.75 1.39 0.25 1.59 0.11 0.46

3.37

0.45

Numbers in parentheses are one standard deviation in units of the last digit quoted.

where IC,B are moments of inertia. We find:

The reference [29] reported: 0 1

1

0 2

x ¼ 897:08 cm ; x ¼ 489:83  x23 cm

1

r0 ðF  PÞ ¼ 1:56324405 ð11Þ Å

x03 ¼ 867:81 cm1 ; x04 ¼ 348:49  x34 cm1

h0 ðFPFÞ ¼ 97:752232 ð29Þ

One can notice that our values of the anharmonic constants of the symmetric top molecule PF3 are generally in perfect agreement with the previous ones determined at low resolution, but significantly more accurate. We can also compare our constants to the ab initio calculations using the Hartree–Fock SCF method and employing two different basis sets: the usual 6-31G** basis and a more advanced TZP/ TZ2P triple-zeta basis [27]. The agreement is excellent with those determined in the second basis. The large difference concerns mainly the x22 anharmonic constant. We think that our experimental value is more credible, which was given through an observed contour. Furthermore, it is important to note the sign positive of the theoretical value in the classical basis. It is well known in molecular spectroscopy that for a symmetric top molecule, the anharmonic constant xij is negative, as a rule.

r0(F–P) is the GS bond length; h0(FPF) is the GS angle. The agreement is excellent with our previous values [11]:

6.2. Experimental equilibrium structure of PF3

6.2.1. Ground state structure From the values of the rotational constants C0 and B0 given in Table 1, we can determine an improved ground state structure of the symmetric top PF3. We employ the formulae quoted in Ref. [31] (Table 3.1), and the conversion factor: 5

C; BðMHzÞ ¼ 5:05376  10 =IC;B

r0 ðF  PÞ ¼ 1:563230 ð10Þ and h0 ðFPFÞ ¼ 97:759 ð2Þ : From the relations : C 0 ¼ C e 

X

X

i

i

aCi g i =2 and B0 ¼ Be 

aBi g i =2

where gi = 1 or 2 for A1- or E-type mode of vibration, which give the rotational constants in the GS in terms of the equilibrium constants Ce and Be and the corresponding rovibrational interaction constants aC and aB, the higher terms than aC and aB, i.e., the cubic and higher order terms, being neglected; we can deduce:

C e ¼ 0:16088676 ð69Þ cm1 and Be ¼ 0:26177751 ð7Þ cm1 : Note that the electronic corrections have also not been considered in this work. Their effects are negligible on the structure of PF3. We can also derive the equilibrium geometry of the PF3 pyramidal molecule:

re ðF  PÞ ¼ 1:560986 ð43Þ Å he ðFPFÞ ¼ 97:566657 ð64Þ One can notice that the new values are significantly more accurate than our previous values calculated in Ref. [10]:

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hv ; ‘  1; J; kjHvr =hcjv ; ‘  1; J; k  6i

r e ðF  PÞ ¼ 1:56099 ð14Þ Å

¼ h3 F  ðJ; kÞF  ðJ; k  1ÞF  ðJ; k  2ÞF  ðJ; k  3ÞF  ðJ; k

he ðFPFÞ ¼ 97:57 ð4Þ

 4ÞF  ðJ; k  5Þ – The k (0, 3) interaction (‘-dependent d resonance):

7. Conclusion In the present work, several accurate experimental values of the rovibrational interaction and the harmonic and anharmonic vibrational constants have been extracted from our most recent FTIR and RF/CM/MMW investigations in the spectra of the symmetric top molecule PF3. We also determined an improvement equilibrium structure of this molecule. Our results contribute incontestably to the experimental knowledge of the function potential of phosphorus trifluoride, which helps to improve theoretical models. Appendix

hv ; ‘  1; J; kjHvr =hcjv ; ‘  1; J; k  3i ¼ ‘½d þ dJJ ðJ þ 1ÞF  ðJ; kÞF  ðJ; k  1ÞF  ðJ; k  2Þ – The k (0, 3) interaction (‘-independent e resonance):

hv ; ‘  1; J; kjHvr =hcjv ; ‘  1; J; k  3i ¼ ð2k  3Þ½e þ eJJ ðJ þ 1ÞF  ðJ; kÞF  ðJ; k  1ÞF  ðJ; k  2Þ – The Coriolis resonance, with phase conventions according to Ref. [34]:

hv 1 ; ‘; J; kjHvr =hcjv 3 ; ‘  1; J; k  1i ¼

The following expressions are the diagonal and off-diagonal matrix elements of the Hamiltonian operator that are used in the all IR studies reported in this work:

¼ WC – The Fermi resonance:

hv 3 ; ‘; J; kjHvr =hcjv 2 þ v 4 ; ‘; J; k  1i ¼ W 234

– 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 = |k| and d ¼ h3 JðJ þ 1Þ½JðJ þ 1Þ  2½JðJ þ 1Þ  6 (see Ref. [32]). – 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‘ – The ‘ (2, 2) interaction (q resonance), employing the phase convention of Cartwright and Mills [33]:

hv ; ‘  1; J; kjHvr =hcjv ; ‘  1; J; k  2i 2

2

¼ 1=4fq þ qJJ ðJ þ 1Þ þ qK ½k þ ðk  2Þ  þ qJJ J 2 ðJ þ 1Þ2 2

pffiffiffi y 2B113 X13 F  ðJ; kÞ

2

þ qJK JðJ þ 1Þ½k þ ðk  2Þ g  ½ðv  ‘ þ 2Þðv  ‘Þ1=2 F  ðJ; kÞF  ðJ; k  1Þ with F±(J, k) = [J(J + 1)  k(k ± 1)]1/2. – The ‘ (2, 1) interaction (r resonance):

hv ; ‘  1; J; kjHvr =hcjv ; ‘  1; J; k  1i 3

3

¼ fð2k  1Þ½r þ r JJ ðJ þ 1Þ þ r K ½k þ ðk  1Þ g  ½ðv  ‘ þ 2Þðv  ‘Þ1=2 F  ðJ; kÞ – The ‘ (2, 4) interaction (t resonance):

hv ; ‘  2; J; kjHvr =hcjv ; ‘; J; k  4i ¼ 1=2½t þ t JJ ðJ þ 1ÞF  ðJ; kÞF  ðJ; k  1ÞF  ðJ; k  2ÞF  ðJ; k  3Þ – The k (0, 6) interaction (h3 resonance):

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