Accurate Prediction of the Ammonia Probes of a Variable Proton-to-Electron Mass Ratio
Alec Owens, Sergei N. Yurchenko, Walter Thiel, Vladimir Špirko
AAccurate Prediction of the Ammonia Probes of a VariableProton-to-Electron Mass Ratio
A. Owens , S. N. Yurchenko , W. Thiel and V. ˇSpirko ∗ Max-Planck-Institut f¨ur Kohlenforschung, Kaiser-Wilhelm-Platz 1, 45470 M¨ulheim an der Ruhr,Germany Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London,United Kingdom Academy of Sciences of the Czech Republic, Institute of Organic Chemistry and Biochemistry,Flemingovo n´am. 2, 166 10 Prague 6, Czech Republic Department of Chemical Physics and Optics, Faculty of Mathematics and Physics, Charles Universityin Prague, Ke Karlovu 3, CZ-12116 Prague 2, Czech Republic
Abstract
A comprehensive study of the mass sensitivity of the vibration-rotation-inversiontransitions of NH , NH , ND , and ND is carried out variationally usingthe TROVE approach. Variational calculations are robust and accurate, offeringa new way to compute sensitivity coefficients. Particular attention is paid to the∆ k = ± ν = 0 + , − , + and 1 − states, and the inversion transitions in the ν = 1 state affected by the “giant” l -type doubling effect. These transitions exhibithighly anomalous sensitivities, thus appearing as promising probes of a possible cos-mological variation of the proton-to-electron mass ratio µ . Moreover, a simultaneouscomparison of the calculated sensitivities reveals a sizeable isotopic dependence whichcould aid an exclusive ammonia detection. ∗ The corresponding author: [email protected] a r X i v : . [ a s t r o - ph . C O ] M a y Introduction
Molecular spectroscopy is a well established discipline and the increasing precision of mea-surements has provided the capacity to test fundamental physics. Recently, a set of several“forbidden” ∆ k = ± NH in the ν vibrational state were proposed as a promising tool to probe a possible space-time variation ofthe proton-to-electron mass ratio µ = m p /m e [1, 2]. The anomalous mass dependency of thesetransitions arises from accidental near-degeneracies of the involved energy levels. The sensitivitycoefficient T u,l , defined as T u,l = µE u − E l (cid:16) d E u d µ − d E l d µ (cid:17) , (1)where E u and E l refer to the energy of the upper and lower state respectively, allows one toquantify the effect that a possible variation of µ would have for a given transition. The largerthe magnitude of T u,l , the more favourable a transition is to test for a drifting µ .The so-called ammonia method [3], which was adapted from van Veldhoven et al. [4], relieson the inversion transitions in the vibrational ground state of NH . Constraints on a tem-poral variation of µ have been determined using this method from measurements of the objectB0218+357 at redshift z ∼ .
685 [3, 5, 6], and of the system PKS1830 −
211 at z ∼ .
886 [7]. Amajor source of systematic error when using the ammonia method is the comparison with rota-tional lines from other molecular species, particularly molecules that are non-nitrogen-bearing(see Murphy et al. [5], Henkel et al. [7], and Kanekar [6] for a more complete discussion). Themost stringent limit using ammonia [6] has since been improved upon with methanol absorptionspectra observed in the lensing galaxy PKS1830 −
211 [8]. Three different radio telescopes wereused to measure ten absorption lines with sensitivity coefficients ranging from T = − . − . NH , NH , ND , and ND . A joint comparison of all relevantisotopic transitions could open the door to an all-ammonia detection, and potentially eliminatecertain systematic errors that arise from using alternative reference molecules. We also note thatthe transitions of the N isotopes are optically thin and free of the nuclear quadrupole structures,providing a simpler radiative and line-shape analysis. A rigorous evaluation of the sensitivitycoefficients will hopefully offer new scope for the ammonia method, and guide future measure-ments that could be carried out for example at the Atacama Large Millimeter/submillimeter2rray (ALMA).
The induced frequency shift of a probed transition is given as∆ νν = T u,l ∆ µµ , (2)where ∆ ν = ν obs − ν is the change in the frequency and ∆ µ = µ obs − µ is the change in µ , both with respect to their accepted values ν and µ . Using this relation one can easilyshow that the rotation-inversion transitions associated with the ν vibrational state of ammoniamay exhibit induced frequency shifts more than one order of magnitude larger than the pureinversion transitions in the vibrational ground state, which are currently used in the probingof µ both temporally [3, 5–7, 9] and spatially [10–13]. Various NH ro-inversional transitionshave already been observed extraterrestrially [14–16], whilst others with notable sensitivitiespossess Einstein coefficients comparable to those of the observed transitions. It is legitimatethen to expect their eventual extragalactic detection, and when combined with their enhancedsensitivity, there is scope for a major improvement of the current ammonia analyses.Another promising anomaly exhibited by the spectra of ammonia is caused by the so-called“giant” l -type doubling, which leads to a “reversal” of the inversion doublets in the K = 1levels in the + l component of the ν states of NH and NH . The inversion doublets arereversed because for K = 1, only one of the A or A sublevels is shifted by the Coriolisinteractions, and only the A states have non-zero spin statistical weights (see Fig. 1 and ˇSpirkoet al. [17]). So far these transitions have not been detected extraterrestrially. This is to beexpected since the physical temperatures prevailing in the interstellar medium are too low toprovide significant population of the aforementioned states. However they could be effectivelypopulated by exoergic chemical formation processes, resulting in the detection of highly excitedstates [18, 19]. Interestingly, the ‘highest energy’ ( J, K ) = (18 ,
18) line of NH observedtowards the galactic centre star forming region Sgr B2, corresponds to the state lying 3130 Kabove the ground vibrational state [20].The most common approach to computing sensitivity coefficients for a molecular systemmakes use of an effective Hamiltonian model, and determining how the parameters of this3 ∆ e ∆ o A (4) A (0)A (0) A (4)A (0) A (4)A (4) A (0)odd even J,K=1 +l J + - inversion splitting ∆ e for J even ∆ o for J odd Figure 1: “Reversal” of the inversion doublets in the + l component of the ν level by the “giant” l -type doubling effect. Values in parentheses are the spin statistical weights.model depend on µ [21–27]. For ammonia, the semiclassical Wentzel-Kramers-Brillouin (WKB)approximation has been used to obtain a general relationship to estimate the sensitivity of pureinversion frequencies in the ground vibrational state for NH [3], NH [28], ND [3], and ND [4], whilst rotation-inversion transitions have been considered for the partly deuteratedspecies NH D and ND H by Kozlov, Lapinov, and Levshakov [29].The vibration-rotation-inversion transitions of NH were investigated by ˇSpirko [2], buttheoretical calculations of the sensitivities using perturbation theory may not be entirely robustsince the nominator and denominator in Eq. (1) contain differences of large numbers. We thusfind it worthwhile not only to check the literature data for NH by means of highly accuratevariational calculations, but to extend the treatment to NH , ND and ND , which areequally valid probes of µ . It is also straightforward to incorporate the so far unprobed ν statesinto the present study.The advantage of our variational approach is that along with sensitivity coefficients, reliabletheoretical transition frequencies can be generated if no experimental data is available, and forall selected transitions, Einstein A coefficients can be calculated to guide future observations. The variational nuclear motion program TROVE [30] has provided highly accurate theoreticalfrequency, intensity, and thermodynamic data for both NH and NH [31–36]. We use thepotential energy surface and computational setup as described in Yurchenko et al. [34] and4urchenko [36], which can naturally be extended to treat ND and ND . Here we onlydiscuss the calculation of sensitivity coefficients, for the method used to compute transitionfrequencies and Einstein A coefficients we refer the reader to Yurchenko et al. [31].We rely on the assumption that the baryonic matter may be treated equally [37], i.e. µ isassumed to be proportional to the molecular mass. It is then sufficient to perform a series ofcalculations employing suitably scaled values for the mass of ammonia. We choose the scalingcoefficient f m = { . , . , . , . , . } such that the scaled mass, m (cid:48) NH = f m × m NH . The mass dependency of any energy level can be found by using finite differencesfor (a) the f m = { . , . } , and (b) the f m = { . , . } calculated energies. Both(a) and (b) should yield identical results, with the latter values used to verify the former.Numerical values for the derivatives d E/ d µ are easily determined and then used in Eq. (1),along with accurate experimental values for the transition frequencies, to calculate sensitivitycoefficients. Calculations with f m = 1 . A coefficients.The variational approach is powerful in that it allows a comprehensive treatment of amolecule to be undertaken. All possible transitions and their mass dependence can be cal-culated. This permits a simple exploration of the sensitivities for any molecule, provided thenecessary steps have been taken to perform accurate variational calculations in the first place.As a cross-check, we also employ the nonrigid inverter theory [17, 38] to compute sensitivitycoefficients as was done by ˇSpirko [2]. In the following we evaluate both approaches. Note thatthe standard Herzberg convention [39] is used to label the vibrational states of ammonia withthe normal mode quantum numbers v , v , v , v , l and l . The ν state corresponds to thesingly excited inversion mode v = 1, whilst ν is the singly excited asymmetric bending mode v = | l | = 1 (see Down et al. [40] for more details). The variationally calculated sensitivities for NH and NH are listed in Tables 1 to 5. Theresults are consistent with previous perturbative values [2] obtained using the nonrigid invertertheory approach [17, 38], and ‘Born-Oppenheimer’ estimates from Jansen et al. [1] (subsequentlyreferred to as JBU). For transitions involving the ν vibrational states shown in Tables 1, 2 and 3,the agreement is near quantitative with the exception of the “forbidden” combination difference | a, J =3 , K =3 , v =1 (cid:105) - | s, J =3 , K =0 , v =1 (cid:105) . The profoundly different sensitivities for these5able 1: The rotation-inversion frequencies ( ν ), Einstein coefficients ( A ), and sensitivities ( T )of NH and their NH counterparts in the ν vibrational state. Γ (cid:48) p (cid:48) J (cid:48) K (cid:48) v (cid:48) Γ (cid:48)(cid:48) p (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) v (cid:48)(cid:48) ν /MHz A /s − T NH E (cid:48)(cid:48) s 2 1 1 E (cid:48) a 1 1 1 140142 a × − b ) A (cid:48)(cid:48) a 0 0 1 A (cid:48) s 1 0 1 466244 c × − -6.587(-6.409) NH E (cid:48)(cid:48) s 2 1 1 E (cid:48) a 1 1 1 175053 0.2939 × − A (cid:48)(cid:48) a 0 0 1 A (cid:48) s 1 0 1 430038 0.1425 × − -6.894(-6.908) NH : Einstein coefficients from Yurchenko et al. [34]; a Astronomical observation from Mauersberger, Henkel,and Wilson [14] and Schilke et al. [15]; b JBU sensitivity coefficient reaches a value of 18.8 (see Jansen et al.[1]); c Astronomical observation from Schilke et al. [16]; values in parentheses from ˇSpirko [2], obtained using thenonrigid inverter theory. NH : Frequencies and Einstein coefficients from Urban et al. [41] and Yurchenko [36], respectively; values inparentheses obtained using the nonrigid inverter theory with the frequencies from Urban et al. [41]. transitions when going from NH to NH is of particular interest. A possible variation of µ requires the measurement of at least two transitions with differing sensitivities. In the caseof | a, J =3 , K =3 , v =1 (cid:105) - | s, J =3 , K =0 , v =1 (cid:105) , both isotopologues possess a large value of T .Importantly though they are of opposite sign, thus enabling a conclusive detection with regardto these particular transitions. An all-ammonia observation of a drifting µ would circumventsome of the intrinsic difficulties that appear when using other reference molecules [5–7], whichmay not necessarily reside at the same location in space.The inversion frequencies in the ground vibrational state, Table 4, have comparable sensitiv-ities for both NH and NH , and this also holds true for the ro-inversional transitions shownin Table 5, demonstrating the validity of NH as a probe of µ . The sensitivity coefficients ofthe ν transitions shown in Table 6, although promising, do not acquire the impressive magni-tudes of their ν counterparts. However the appearance of positive and negative values could beof real use in constraining µ .Because of the substantial differences in size of the inversion splittings, the mass sensitivityof the ND and ND transitions exhibit centrifugal distortion and Coriolis interaction de-pendence significantly different from that exhibited by NH and NH (see Tables 7, 8, 9, 10and Fig. 2). The effects of these interactions are non negligible and must be included in anycritical analysis. As only a small fraction of the total presence of ammonia in the interstellarmedium is heavy ammonia, a detection of ‘higher energy’ transitions is rather improbable. Allthe ammonia isotopomers appear as suitable targets for terrestrial studies however, such as those6able 2: The wavenumbers ( ν ), wavelengths ( λ ), Einstein coefficients ( A ), and sensitivities( T ) for transitions between the ground and ν vibrational state of NH and their NH counterparts. Γ (cid:48) p (cid:48) J (cid:48) K (cid:48) v (cid:48) Γ (cid:48)(cid:48) p (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) v (cid:48)(cid:48) ν /cm − λ / µ m A /s − T NH A (cid:48) s 6 6 1 A (cid:48)(cid:48) a 6 6 0 927.3230 10.7837 0.1316 × +2 -0.367(-0.356) E (cid:48) s 2 2 1 E (cid:48)(cid:48) a 2 2 0 931.3333 10.7373 0.1030 × +2 -0.371(-0.366) E (cid:48)(cid:48) s 2 1 1 E (cid:48) a 1 1 0 971.8821 10.2893 0.5238 × +1 -0.399(-0.394) E (cid:48)(cid:48) s 1 1 1 E (cid:48) a 2 1 0 891.8820 11.2122 0.6795 × +1 -0.344(-0.339) A (cid:48) s 1 0 1 A (cid:48)(cid:48) a 2 0 0 892.1567 11.2088 0.9054 × +1 -0.344(-0.339) A (cid:48)(cid:48) s 3 3 1 A (cid:48) a 3 3 0 930.7571 10.7439 0.1158 × +2 -0.370(-0.366) NH A (cid:48) s 6 6 1 A (cid:48)(cid:48) a 6 6 0 923.4541 10.8289 0.1290 × +2 -0.365(-0.365) E (cid:48) s 2 2 1 E (cid:48)(cid:48) a 2 2 0 927.4034 10.7828 0.1010 × +2 -0.373(-0.373) E (cid:48)(cid:48) s 2 1 1 E (cid:48) a 1 1 0 967.8597 10.3321 0.5133 × +1 -0.400(-0.400) E (cid:48)(cid:48) s 1 1 1 E (cid:48) a 2 1 0 888.0413 11.2607 0.6664 × +1 -0.345(-0.345) A (cid:48) s 1 0 1 A (cid:48)(cid:48) a 2 0 0 888.3174 11.2572 0.8878 × +1 -0.346(-0.346) A (cid:48)(cid:48) s 3 3 1 A (cid:48) a 3 3 0 926.8378 10.7894 0.1135 × +2 -0.372(-0.372) NH : Wavenumbers and Einstein coefficients from Urban et al. [42] and Yurchenko et al. [34], respectively;Astronomical observations reported in Betz et al. [43] and Evans et al. [44]; values in parentheses from ˇSpirko[2], obtained using the nonrigid inverter theory. NH : Wavenumbers provided by Fusina, Di Lonardo & Predoi-Cross (in preparation), Einstein coefficientsfrom Yurchenko [36]; values in parentheses obtained using the nonrigid inverter theory with the frequencies fromFusina, Di Lonardo & Predoi-Cross (in preparation). reported by van Veldhoven et al. [4], Bethlem et al. [28], and Quintero-Perez et al. [51].It is expected that any variation in the fundamental constants will be confirmed, or refuted,over a series of independent measurements on a variety of molecular absorbers. As a relevantastrophysical molecule, and with certain inversion transitions already detected extraterrestri-ally [57–59], NH has potential to aid this search along with the already established probes of NH . For the deuterated species ND and ND , it is perhaps more likely that their use willbe restricted to precision measurements in the laboratory, despite possessing larger sensitivitycoefficients for the pure inversion frequencies in the ground vibrational state. A comprehensive study of the vibration-rotation-inversion transitions of all stable, symmetrictop isotopomers of ammonia has been performed. The variational method offers a new androbust approach to computing sensitivity coefficients. The calculated mass sensitivities provideperspectives for the further development of the ammonia method, used in the probing of thecosmological variability of the proton-to-electron mass ratio. Most notably the reliance onother reference molecular species, which is the main source of systematic error, can be avoided.7able 3: The vibration-rotation-inversion transitions associated with the | a, J, K = 3 , v = 1 (cid:105) - | s, J, K =0 , v =1 (cid:105) resonances. Γ (cid:48) p (cid:48) J (cid:48) K (cid:48) v (cid:48) Γ (cid:48)(cid:48) p (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) v (cid:48)(cid:48) ν /MHz A /s − T Obs. Ref. NH A (cid:48) a 3 3 1 A (cid:48)(cid:48) s 3 3 0 29000313.7 0.1176 × +2 -0.484(-0.484) b A (cid:48) s 3 0 1 A (cid:48)(cid:48) s 3 3 0 28997430.0 0.2025 × -0.405(-0.405) b a 3 3 1 s 3 0 1 2883.7 -790.6(-1001 a ) b A (cid:48) a 3 3 1 A (cid:48)(cid:48) a 2 0 1 772594.9 0.6018 × − -0.868(-0.868) c A (cid:48) s 3 0 1 A (cid:48)(cid:48) a 2 0 1 769710.2 0.3471 × − c a 3 3 1 s 3 0 1 2884.7 -790.3(-1001) c A (cid:48) a 3 3 1 A (cid:48)(cid:48) s 3 3 1 1073050.7 0.1634 × − -3.350(-3.353) c A (cid:48) s 3 0 1 A (cid:48)(cid:48) s 3 3 1 1070166.6 0.2765 × − -1.228(-1.229) c a 3 3 1 s 3 0 1 2884.1 -790.5(-1001) c A (cid:48) a 5 3 1 A (cid:48)(cid:48) s 5 3 0 28971340.5 0.4692 × +1 -0.484(-0.484) d A (cid:48) s 5 0 1 A (cid:48)(cid:48) s 5 3 0 29050552.5 0.2147 × − -0.408(-0.408) d a 5 3 1 s 5 0 1 79212.0 27.38(27.35) d A (cid:48) a 5 3 1 A (cid:48)(cid:48) s 5 3 1 979649.1 0.5141 × − -3.425(-3.427) d A (cid:48) s 5 0 1 A (cid:48)(cid:48) s 5 3 1 1058861.1 0.3714 × − -1.120(-1.120) d a 5 3 1 s 5 0 1 79212.0 27.38(27.35) d A (cid:48) a 5 3 1 A (cid:48)(cid:48) a 4 0 1 1956241.1 0.4129 × − -0.988(-0.988) d A (cid:48) s 5 0 1 A (cid:48)(cid:48) a 4 0 1 2035453.1 0.7023 × − d a 5 3 1 s 5 0 1 79212.0 27.38(29.35) d A (cid:48) a 7 3 1 A (cid:48)(cid:48) s 7 3 0 28934099.5 0.2399 × +1 -0.480(-0.480) d A (cid:48) s 7 0 1 A (cid:48)(cid:48) s 7 3 0 29118808.5 0.1095 × − -0.416(-0.416) d a 7 3 1 s 7 0 1 184709.0 9.561(9.582) d A (cid:48) a 9 3 1 A (cid:48)(cid:48) s 9 3 0 28892089.9 0.1444 × +1 -0.475(-0.475) d A (cid:48) s 9 0 1 A (cid:48)(cid:48) s 9 3 0 29194454.6 0.1029 × − -0.425(-0.425) d a 9 3 1 s 9 0 1 302364.7 4.350(4.363) d NH A (cid:48) a 3 3 1 A (cid:48)(cid:48) s 3 3 0 28843885.0 0.1171 × +2 -0.486(-0.486) e A (cid:48) s 3 0 1 A (cid:48)(cid:48) s 3 3 0 28872669.9 0.2187 × − -0.403(-0.403) e a 3 3 1 s 3 0 1 28784.9 82.96(81.69) e A (cid:48) a 3 3 1 A (cid:48)(cid:48) a 2 0 1 774222.8 0.7160 × − -0.999(-0.999 ) f A (cid:48) s 3 0 1 A (cid:48)(cid:48) a 2 0 1 802986.7 0.4035 × − f a 3 3 1 s 3 0 1 28763.9 83.02(81.69) f A (cid:48) a 3 3 1 A (cid:48)(cid:48) s 3 3 1 1035207.4 0.1491 × − -3.473(-3.476) f A (cid:48) s 3 0 1 A (cid:48)(cid:48) s 3 3 1 1063971.3 0.3245 × − -1.228(-1.135) f a 3 3 1 s 3 0 1 28763.9 83.02(81.69) f A (cid:48) a 5 3 1 A (cid:48)(cid:48) s 5 3 0 28817906.5 0.4598 × +1 -0.483(-0.483) e A (cid:48) s 5 0 1 A (cid:48)(cid:48) s 5 3 0 28927141.3 0.7768 × − -0.409(-0.409) e a 5 3 1 s 5 0 1 109234.8 19.02(19.02) e A (cid:48) a 5 3 1 A (cid:48)(cid:48) s 5 3 1 943226.9 0.4588 × − -3.453(-3.455) f A (cid:48) s 5 0 1 A (cid:48)(cid:48) s 5 3 1 1052459.7 0.1548 × − -1.120(-1.121) f a 5 3 1 s 5 0 1 109232.8 19.04(19.02) f A (cid:48) a 5 3 1 A (cid:48)(cid:48) a 4 0 1 1955711.7 0.1882 × − -0.988(-0.988) f A (cid:48) s 5 0 1 A (cid:48)(cid:48) a 4 0 1 2064944.5 0.7369 × − f a 5 3 1 s 5 0 1 109232.8 19.05(19.02) f A (cid:48) a 7 3 1 A (cid:48)(cid:48) s 7 3 0 28784706.6 0.2399 × +1 -0.479(-0.479) e A (cid:48) s 7 0 1 A (cid:48)(cid:48) s 7 3 0 28997286.1 0.1095 × − -0.418(-0.418) e a 7 3 1 s 7 0 1 212579.5 7.898(7.073) e A (cid:48) a 9 3 1 A (cid:48)(cid:48) s 9 3 0 28747714.9 0.1444 × +1 -0.479(-0.475) e A (cid:48) s 9 0 1 A (cid:48)(cid:48) s 9 3 0 29075088.5 0.1029 × − -0.418(-0.427) e a 9 3 1 s 9 0 1 327373.6 3.782(3.782) e NH : Einstein coefficients from Yurchenko et al. [34]; a JBU sensitivity coefficient reaches a value of -938 (seeJansen et al. [1]); values in parentheses obtained using the nonrigid inverter theory with the calculated TROVEfrequencies; b Fichoux et al. [45]; c Belov et al. [46]; d Urban et al. [42]. NH : Einstein coefficients from Yurchenko [36]; values in parentheses obtained using the nonrigidinverter theory with the calculated TROVE frequencies; e Fusina, Di Lonardo & Predoi-Cross (in preparation); f Urban et al. [41]. ν ), Einstein coefficients ( A ), and sensitivities ( T ) of NH andtheir NH counterparts in the ground vibrational state. J K ν /MHz A /s − T J K ν /MHz A /s − T NH × − -4.310(-4.365) 4 3 22688.3 0.1311 × − -4.289(-4.514)2 1 23098.8 0.5123 × − -4.297(-4.413) 4 4 24139.4 0.2797 × − -4.317(-4.471)2 2 23722.5 0.2216 × − -4.311(-4.385) 5 1 19838.3 0.6540 × − -4.220(-4.700)3 2 22834.2 0.9902 × − -4.288(-4.464) 5 2 20371.5 0.2828 × − -4.231(-4.546)3 3 23870.1 0.2538 × − -4.312(-4.419) 5 3 21285.3 0.7239 × − -4.257(-4.634)4 1 21134.3 0.1182 × − -4.249(-4.568) 5 4 22653.0 0.1546 × − -4.282(-4.592)4 2 21703.4 0.5114 × − -4.262(-4.545) 5 5 24533.0 0.3053 × − -4.327(-4.509) NH × − -4.352(-4.333) 4 3 21637.9 0.1149 × − -4.330(-4.309)2 1 22044.2 0.4521 × − -4.341(-4.321) 4 4 23046.0 0.2469 × − -4.360(-4.341)2 2 22649.8 0.1958 × − -4.349(-4.330) 5 1 18871.5 0.5729 × − -4.264(-4.239)3 2 21783.9 0.8730 × − -4.333(-4.312) 5 2 19387.4 0.2480 × − -4.278(-4.254)3 3 22789.4 0.2241 × − -4.356(-4.337) 5 3 20272.1 0.6358 × − -4.299(-4.276)4 1 20131.4 0.1039 × − -4.293(-4.270) 5 4 21597.9 0.1360 × − -4.330(-4.309)4 2 20682.8 0.4498 × − -4.306(-4.284) 5 5 23422.0 0.2695 × − -4.366(-4.347) NH : Frequencies and Einstein coefficients from Lovas et al. [47] and Yurchenko et al. [34], respectively; valuesin parentheses from ˇSpirko [2], obtained using the nonrigid inverter theory. NH : Frequencies and Einstein coefficients from Urban et al. [41] and Yurchenko [36], respectively; values inparentheses obtained using the nonrigid inverter theory with the frequencies from Urban et al. [41]. Table 5: The rotation-inversion frequencies ( ν ), Einstein coefficients ( A ), and sensitivities ( T )of NH and their NH counterparts in the ground vibrational state. Γ (cid:48) p (cid:48) J (cid:48) K (cid:48) v (cid:48) Γ (cid:48)(cid:48) p (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) v (cid:48)(cid:48) ν /MHz A /s − T NH A (cid:48) s 1 0 0 A (cid:48)(cid:48) a 0 0 0 572498 0.1561 × − -0.860(-0.862) A (cid:48)(cid:48) a 2 0 0 A (cid:48) s 1 0 0 1214859 0.1791 × − -1.060(-1.063) E (cid:48) a 2 1 0 E (cid:48)(cid:48) s 1 1 0 1215245 0.1344 × − -1.061(-1.064) NH A (cid:48) s 1 0 0 A (cid:48)(cid:48) a 0 0 0 572112 0.1557 × − -0.865(-0.866) A (cid:48)(cid:48) a 2 0 0 A (cid:48) s 1 0 0 1210889 0.1774 × − -1.058(-1.058) E (cid:48) a 2 1 0 E (cid:48)(cid:48) s 1 1 0 1211277 0.1331 × − -1.059(-1.059) NH : Frequencies and Einstein coefficients from Persson et al. [48] and Yurchenko et al. [34], respectively;values given in parentheses from ˇSpirko [2], obtained using the nonrigid inverter theory. NH : Frequencies and Einstein coefficients from Urban et al. [41] and Yurchenko [36], respectively; values inparentheses obtained using the nonrigid inverter theory with the frequencies from Urban et al. [41]. ν ), Einstein coefficients ( A ), and sensitivities ( T ) of NH and NH in the ν vibrational state. J K l ν /MHz A /s − T J K l ν /MHz A /s − T NH × − -4.268 4 3 1 57132 0.1968 × − × − -2.234 4 2 -1 47526 0.5467 × − -1.5502 2 -1 32111 0.5514 × − -4.250 4 2 1 46515 0.4020 × − -0.2472 2 1 40189 0.1056 × − -2.381 4 1 -1 57681 0.2548 × − -0.2202 1 -1 36797 0.2085 × − -3.133 4 1 1 145888 a × − -0.9622 1 1 20655 0.3743 × − × − -4.2643 3 -1 31893 0.6081 × − -4.259 5 5 1 68699 0.6198 × − × − -0.667 5 4 -1 39071 0.8020 × − -2.8323 2 -1 37500 0.4424 × − -2.961 5 4 1 73534 0.4807 × − × − -1.023 5 3 -1 48346 0.8610 × − -1.5063 1 -1 44755 0.1908 × − -1.687 5 3 1 64906 0.1799 × − a × − -0.482 5 2 -1 58699 0.6967 × − -0.1814 4 -1 31884 0.6482 × − -4.258 5 2 1 44876 0.2025 × − × − a × − × − -2.855 5 1 -1 380542 a × − -0.178 NH × − -4.291 4 3 1 51989 0.1501 × − × − -2.410 4 2 -1 44599 0.4524 × − -1.7652 2 -1 30825 0.4880 × − -4.271 4 2 1 43225 0.3278 × − -0.7282 2 1 37900 0.8883 × − -2.722 4 1 -1 53406 0.2029 × − -0.5582 1 -1 34950 0.1788 × − -3.273 4 1 1 146961 a × − -0.9832 1 1 21904 0.4450 × − × − -4.2803 3 -1 30606 0.5377 × − -4.281 5 5 1 61128 a × − × − -1.358 5 4 -1 37071 0.6856 × − -2.9373 2 -1 35551 0.3772 × − -3.082 5 4 1 65945 a × − × − -1.494 5 3 -1 45373 0.7129 × − -1.6893 1 -1 41947 0.1574 × − -1.941 5 3 1 59236 a × − a × − -0.731 5 2 -1 54322 0.5536 × − -0.4404 4 -1 30591 0.5729 × − -4.281 5 2 1 42037 a × − -0.2424 4 1 50530 0.2502 × − a × − × − -2.978 5 1 1 369287 a × − -0.379 Frequencies from Cohen and Poynter [49] and Cohen [50]; a TROVE calculated value T J
Strongly anomalous values ND : T J,K=314 ND : T J,K=-315 ND : T J,K=315 ND : T J,K=-3 ND : T = 176.2 ND : T = 10.3 -5-4.9-4.8-4.7-4.6-4.5-4.4-4.3 3 5 7 9 11 13 15 17 19 T J ND : T J,K=314 ND : T J,K=-315 ND : T J,K=315 ND : T J,K=-3
Figure 2: The sensitivities, T , of the inversion transitions of the ( J, K = ±
3) rotational statesof ND and ND in the ground (left panel) and ν (right panel) vibrational states.Table 7: Inversion frequencies ( ν ), Einstein coefficients ( A ), and sensitivities ( T ) of ND and ND in the ground vibrational state. J K ν /MHz A /s − T J K ν /MHz A /s − T ND × − -5.541(-5.528) 4 3 1558.600 0.4897 × − -5.533(-5.520)2 1 1568.357 0.1849 × − -5.556(-5.542) 4 -3 1558.178 0.4893 × − -5.534(-5.521)2 2 1591.695 0.7721 × − -5.543(-5.530) 4 4 1612.997 0.9623 × − -5.536(-5.525)3 1 1537.915 0.8725 × − -5.526(-5.511) 5 1 1450.435 a × − -5.511(-5.493)3 2 1560.774 0.3644 × − -5.537(-5.523) 5 2 1471.785 0.1226 × − -5.504(-5.487)3 3 1599.645 0.8810 × − -5.571(-5.559) 5 3 1507.525 0.2960 × − -5.553(-5.537)3 -3 1599.704 0.8811 × − -5.571(-5.559) 5 -3 1509.218 0.2969 × − -5.499(-5.484)4 1 1498.270 0.4848 × − -5.503(-5.487) 5 4 1561.146 0.5827 × − -5.524(-5.511)4 2 1520.537 0.2025 × − -5.493(-5.478) 5 5 1631.784 0.1036 × − -5.561(-5.551) ND × − -5.600(-5.577) 4 3 1401.312 0.3578 × − -5.600(-5.577)2 1 1410.980 0.1354 × − -5.613(-5.589) 4 -3 1400.878 0.3575 × − -5.602(-5.578)2 2 1432.641 0.5661 × − -5.604(-5.581) 4 2 1366.027 0.1476 × − -5.586(-5.561)3 1 1382.510 0.5374 × − -5.585(-5.560) 5 1 1300.841 a × − -5.562(-5.534)3 2 1403.684 0.2665 × − -5.566(-5.542) 5 2 1320.460 a × − -5.575(-5.547)3 3 1439.719 0.5458 × − -5.601(-5.579) 5 3 1353.451 0.2153 × − -5.585(-5.559)3 -3 1439.783 0.6459 × − -5.601(-5.579) 5 -3 1355.161 0.2162 × − -5.551(-5.526)4 1 1345.533 a × − -5.564(-5.538) 5 4 1403.179 0.4254 × − -5.606(-5.583)4 2 1366.027 0.1476 × − -5.586(-5.561) 5 5 1468.666 0.7595 × − -5.639(-5.619) Unless stated otherwise, ND and ND frequencies from Fusina and Murzin [52] and Fusina et al. [53],respectively; values in parentheses obtained using the nonrigid inverter theory with the calculated TROVEfrequencies; the K = − A (cid:48) , A (cid:48)(cid:48) species), the K = 3 values refer to transitions between levels with spin statistical weight = 1 ( A (cid:48) , A (cid:48)(cid:48) ) species); a Urban et al. [54]. ν ), Einstein coefficients ( A ), and sensitivities ( T )of ND and ND in the ground vibrational state.Γ (cid:48) p (cid:48) J (cid:48) K (cid:48) v (cid:48) Γ (cid:48)(cid:48) p (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) v (cid:48)(cid:48) ν /MHz A /s − T ND A (cid:48)(cid:48) a 1 0 0 A (cid:48) s 0 0 0 309909 a × − -1.022 A (cid:48)(cid:48) a 2 0 0 A (cid:48) s 1 0 0 618075 a × − -1.009 E (cid:48) a 2 1 0 E (cid:48)(cid:48) s 1 1 0 618124 a × − -1.009 A (cid:48) s 1 0 0 A (cid:48)(cid:48) a 0 0 0 306737 a × − -0.973 A (cid:48) s 2 0 0 A (cid:48)(cid:48) a 1 0 0 614933 a × − -0.985 E (cid:48)(cid:48) s 2 1 0 E (cid:48) a 1 1 0 614968 a × − -0.985 A (cid:48)(cid:48) a 3 0 0 A (cid:48) s 2 0 0 925947 0.8681 × − -1.005 E (cid:48) a 3 1 0 E (cid:48)(cid:48) s 2 1 0 926018 0.7717 × − -1.005 E (cid:48)(cid:48) a 3 2 0 E (cid:48) s 2 2 0 926228 0.4824 × − -1.005 A (cid:48) s 3 0 0 A (cid:48)(cid:48) a 2 0 0 922857 0.8591 × − -0.989 E (cid:48)(cid:48) s 3 1 0 E (cid:48) a 2 1 0 922911 0.7637 × − -0.989 E (cid:48) s 3 2 0 E (cid:48)(cid:48) a 2 2 0 923076 0.4773 × − -0.999 ND A (cid:48)(cid:48) a 1 0 0 A (cid:48) s 0 0 0 308606 a × − -1.020 A (cid:48)(cid:48) a 2 0 0 A (cid:48) s 1 0 0 615628 a × − -1.008 E (cid:48) a 2 1 0 E (cid:48)(cid:48) s 1 1 0 615677 a × − -1.009 A (cid:48) s 1 0 0 A (cid:48)(cid:48) a 0 0 0 305750 a × − -0.975 A (cid:48) s 2 0 0 A (cid:48)(cid:48) a 1 0 0 612801 a × − -0.987 E (cid:48)(cid:48) s 2 1 0 E (cid:48) a 1 1 0 612836 a × − -0.987 A (cid:48)(cid:48) a 3 0 0 A (cid:48) s 2 0 0 922356 0.8582 × − -1.004 E (cid:48) a 3 1 0 E (cid:48)(cid:48) s 2 1 0 922426 0.7628 × − -1.004 E (cid:48)(cid:48) a 3 2 0 E (cid:48) s 2 2 0 922636 0.4768 × − -1.004 A (cid:48) s 3 0 0 A (cid:48)(cid:48) a 2 0 0 919577 0.8501 × − -0.990 E (cid:48)(cid:48) s 3 1 0 E (cid:48) a 2 1 0 919632 0.7556 × − -0.990 E (cid:48) s 3 2 0 E (cid:48)(cid:48) a 2 2 0 919800 0.4723 × − -0.990 Unless stated otherwise, frequencies from [54]; a Helminger and Gordy [55] and Helminger et al. [56]. ν ), Einstein coefficients ( A ), and sensitivities ( T )of ND and ND in the ν vibrational state.Γ (cid:48) p (cid:48) J (cid:48) K (cid:48) v (cid:48) Γ (cid:48)(cid:48) p (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) v (cid:48)(cid:48) ν /MHz A /s − T ND A (cid:48)(cid:48) a 1 0 1 A (cid:48) s 0 0 1 412847 0.4983 × − -2.030 A (cid:48)(cid:48) a 2 0 1 A (cid:48) s 1 0 1 718585 0.3131 × − -1.585 E (cid:48) a 2 1 1 E (cid:48)(cid:48) s 1 1 1 719092 0.2352 × − -1.588 A (cid:48) s 1 0 1 A (cid:48)(cid:48) a 0 0 1 200763 0.5423 × − A (cid:48) s 2 0 1 A (cid:48)(cid:48) a 1 0 1 508364 0.1082 × − -0.170 E (cid:48)(cid:48) s 2 1 1 E (cid:48) a 1 1 1 507940 0.8088 × − -0.166 A (cid:48)(cid:48) a 3 0 1 A (cid:48) s 2 0 1 1023449 0.9673 × − -1.404 E (cid:48) a 3 1 1 E (cid:48)(cid:48) s 2 1 1 1023971 0.8608 × − -1.405 E (cid:48)(cid:48) a 3 2 1 E (cid:48) s 2 2 1 1025546 0.5399 × − -1.411 A (cid:48) s 3 0 1 A (cid:48)(cid:48) a 2 0 1 816294 0.4830 × − -0.491 E (cid:48)(cid:48) s 3 1 1 E (cid:48) a 2 1 1 815898 0.4286 × − -0.488 E (cid:48) s 3 2 1 E (cid:48)(cid:48) a 2 2 1 814696 0.2663 × − -0.480 A (cid:48)(cid:48) a 4 0 1 A (cid:48) s 3 0 1 1327334 0.2188 × − -1.304 E (cid:48) a 4 1 1 E (cid:48)(cid:48) s 3 1 1 1327865 0.2053 × − -1.305 E (cid:48)(cid:48) a 4 2 1 E (cid:48) s 3 2 1 1329473 0.1647 × − -1.309 A (cid:48) a 4 3 1 A (cid:48)(cid:48) s 3 3 1 1332194 0.9646 × − -1.317 A (cid:48) a 4 -3 1 A (cid:48)(cid:48) s 3 -3 1 1332194 0.9646 × − -1.317 A (cid:48) s 4 0 1 A (cid:48)(cid:48) a 3 0 1 1124392 0.1315 × − -0.637 E (cid:48)(cid:48) s 4 1 1 E (cid:48) a 3 1 1 1124025 0.1231 × − -0.636 E (cid:48) s 4 2 1 E (cid:48)(cid:48) a 3 2 1 1122914 0.9805 × − -0.630 A (cid:48)(cid:48) s 4 3 1 A (cid:48) a 3 3 1 1121023 0.5679 × − -0.621 A (cid:48)(cid:48) s 4 -3 1 A (cid:48) a 3 -3 1 1121023 0.5679 × − -0.621 A (cid:48)(cid:48) a 5 0 1 A (cid:48) s 4 0 1 1630141 0.4149 × − -1.239 E (cid:48) a 5 1 1 E (cid:48)(cid:48) s 4 1 1 1630681 0.3986 × − -1.240 E (cid:48)(cid:48) a 5 2 1 E (cid:48) s 4 2 1 1632314 0.3494 × − -1.243 A (cid:48) a 5 3 1 A (cid:48)(cid:48) s 4 3 1 1635074 0.2671 × − -1.249 A (cid:48) a 5 -3 1 A (cid:48)(cid:48) s 4 -3 1 1635075 0.2671 × − -1.249 E (cid:48)(cid:48) a 5 4 1 E (cid:48) s 4 4 1 1639027 0.1509 × − -1.258 A (cid:48) s 5 0 1 A (cid:48)(cid:48) a 4 0 1 1432485 0.2790 × − -0.722 E (cid:48)(cid:48) s 5 1 1 E (cid:48) a 4 1 1 1432151 0.2676 × − -0.721 E (cid:48) s 5 2 1 E (cid:48)(cid:48) a 4 2 1 1431137 0.2333 × − -0.717 A (cid:48)(cid:48) s 5 3 1 A (cid:48) a 4 3 1 1429410 0.1768 × − -0.710 A (cid:48)(cid:48) s 5 -3 1 A (cid:48) a 4 -3 1 1429409 0.1768 × − -0.710 E (cid:48) s 5 4 1 E (cid:48)(cid:48) a 4 4 1 1426908 0.9864 × − -0.700 ND A (cid:48)(cid:48) a 1 0 1 A (cid:48) s 0 0 1 402779 0.4636 × − -1.979 A (cid:48)(cid:48) a 2 0 1 A (cid:48) s 1 0 1 707552 0.2995 × − -1.551 E (cid:48) a 2 1 1 E (cid:48)(cid:48) s 1 1 1 708033 0.2250 × − -1.554 A (cid:48) s 1 0 1 A (cid:48)(cid:48) a 0 0 1 208813 0.6139 × − A (cid:48) s 2 0 1 A (cid:48)(cid:48) a 1 0 1 515358 0.1131 × − -0.241 E (cid:48)(cid:48) s 2 1 1 E (cid:48) a 1 1 1 514961 0.8458 × − -0.237 Frequencies from Urban et al. [54]. ν ), wavelengths ( λ ), Einstein coefficients ( A ), and sensitivities ( T )for transitions between the ground and ν vibrational state of ND and ND . Γ (cid:48) p (cid:48) J (cid:48) K (cid:48) v (cid:48) Γ (cid:48)(cid:48) p (cid:48)(cid:48) J (cid:48)(cid:48) K (cid:48)(cid:48) v (cid:48)(cid:48) ν /cm − λ / µ m A /s − T ND A (cid:48)(cid:48) a 1 0 1 A (cid:48) s 0 0 0 759.3704 13.1688 0.1955 × +1 -0.475 A (cid:48)(cid:48) a 2 0 1 A (cid:48) s 1 0 0 769.5283 12.9950 0.2444 × +1 -0.482 E (cid:48) a 2 1 1 E (cid:48)(cid:48) s 1 1 0 769.5306 12.9949 0.1834 × +1 -0.482 A (cid:48) s 1 0 1 A (cid:48)(cid:48) a 0 0 0 755.7906 13.2312 0.1948 × +1 -0.454 A (cid:48) s 2 0 1 A (cid:48)(cid:48) a 1 0 0 765.9901 13.0550 0.2434 × +1 -0.461 E (cid:48)(cid:48) s 2 1 1 E (cid:48) a 1 1 0 765.9767 13.0552 0.1827 × +1 -0.461 E (cid:48) a 1 1 1 E (cid:48)(cid:48) s 1 1 0 749.0866 13.3496 0.2810 × +1 -0.468 E (cid:48) a 2 1 1 E (cid:48)(cid:48) s 2 1 0 748.9645 13.3518 0.9344 × -0.468 E (cid:48)(cid:48) a 2 2 1 E (cid:48) s 2 2 0 748.9671 13.3517 0.3744 × +1 -0.468 E (cid:48)(cid:48) s 1 1 1 E (cid:48) a 1 1 0 745.4912 13.4140 0.2798 × +1 -0.446 E (cid:48)(cid:48) s 2 1 1 E (cid:48) a 2 1 0 745.4112 13.4154 0.9305 × -0.446 E (cid:48) s 2 2 1 E (cid:48)(cid:48) a 2 2 0 745.3664 13.4162 0.3729 × +1 -0.446 A (cid:48)(cid:48) a 0 0 1 A (cid:48) s 1 0 0 738.8622 13.5343 0.5381 × +1 -0.461 A (cid:48)(cid:48) a 1 0 1 A (cid:48) s 2 0 0 728.5209 13.7264 0.3427 × +1 -0.453 E (cid:48) a 1 1 1 E (cid:48)(cid:48) s 2 1 0 728.5205 13.7264 0.2572 × +1 -0.453 A (cid:48) s 0 0 1 A (cid:48)(cid:48) a 1 0 0 735.2618 13.6006 0.5358 × +1 -0.439 A (cid:48) s 1 0 1 A (cid:48)(cid:48) a 2 0 0 724.9421 13.7942 0.3412 × +1 -0.431 E (cid:48)(cid:48) s 1 1 1 E (cid:48) a 2 1 0 724.9258 13.7945 0.2560 × +1 -0.431 ND A (cid:48)(cid:48) a 1 0 1 A (cid:48) s 0 0 0 752.9702 13.2807 0.1888 × +1 -0.475 A (cid:48)(cid:48) a 2 0 1 A (cid:48) s 1 0 0 763.1000 13.1044 0.2359 × +1 -0.482 E (cid:48) a 2 1 1 E (cid:48)(cid:48) s 1 1 0 763.0998 13.1044 0.1770 × +1 -0.482 A (cid:48) s 1 0 1 A (cid:48)(cid:48) a 0 0 0 749.6973 13.3387 0.1881 × +1 -0.455 A (cid:48) s 2 0 1 A (cid:48)(cid:48) a 1 0 0 759.8667 13.1602 0.2351 × +1 -0.463 E (cid:48)(cid:48) s 2 1 1 E (cid:48) a 1 1 0 759.8517 13.1605 0.1764 × +1 -0.462 E (cid:48) a 1 1 1 E (cid:48)(cid:48) s 1 1 0 742.7222 13.4640 0.2713 × +1 -0.468 E (cid:48) a 2 1 1 E (cid:48)(cid:48) s 2 1 0 742.6101 13.4660 0.9023 × -0.468 E (cid:48)(cid:48) a 2 2 1 E (cid:48) s 2 2 0 742.6053 13.4661 0.3616 × +1 -0.468 E (cid:48)(cid:48) s 1 1 1 E (cid:48) a 1 1 0 739.4346 13.5238 0.2702 × +1 -0.448 E (cid:48)(cid:48) s 2 1 1 E (cid:48) a 2 1 0 739.3626 13.5252 0.8988 × -0.448 E (cid:48) s 2 2 1 E (cid:48)(cid:48) a 2 2 0 739.3131 13.5261 0.3602 × +1 -0.447 A (cid:48)(cid:48) a 0 0 1 A (cid:48) s 1 0 0 732.5333 13.6513 0.5197 × +1 -0.461 A (cid:48)(cid:48) a 1 0 1 A (cid:48) s 2 0 0 722.2354 13.8459 0.3311 × +1 -0.452 E (cid:48) a 1 1 1 E (cid:48)(cid:48) s 2 1 0 722.2324 13.8460 0.2484 × +1 -0.453 A (cid:48) s 0 0 1 A (cid:48)(cid:48) a 1 0 0 729.2409 13.7129 0.5176 × +1 -0.440 A (cid:48) s 1 0 1 A (cid:48)(cid:48) a 2 0 0 718.9634 13.9089 0.3296 × +1 -0.432 E (cid:48)(cid:48) s 1 1 1 E (cid:48) a 2 1 0 718.9456 13.9093 0.2474 × +1 -0.432 Wavenumbers from Urban et al. [54]. +2 , D +2 and He +2 for instance [60], it can be detected in a wide variety of regions [61], andat redshifts which dramatically enhance spectral shifts (see Riechers et al. [62] and also Eq. (1)of ˇSpirko [2]). The accuracy of the predicted sensitivities seems to fulfil the requirements neededfor a reliable analysis of spectral data obtained at ‘rotational’ resolution. To go beyond thislimit, one should account for the hyperfine interactions and this requires a correct description ofthe ‘hyperfine’ effects, which in turn should respect both the centrifugal distortion and Coriolisinteraction [63]. A study along these lines is in progress in our laboratory. We also note thatthe ammonia rovibrational dynamics show the same characteristics as those of other invertingmolecules, notably the hydronium cation (see Kozlov et al. [23]), thus calling for a rigorousinvestigation into such systems. Acknowledgements
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