The rotational spectrum of 15ND. Isotopic-independent Dunham-type analysis of the imidogen radical

Mattia Melosso a, Luca Bizzocchi *b, Filippo Tamassia c, Claudio Degli Esposti a, Elisabetta Canè c and Luca Dore *a
aDipartimento di Chimica “Giacomo Ciamician”, Università di Bologna, Via F. Selmi 2, 40126 Bologna, Italy. E-mail: mattia.melosso2@unibo.it; claudio.degliesposti@unibo.it; luca.dore@unibo.it
bCenter for Astrochemical Studies, Max-Planck-Institut für extraterrestrische Physik, Gießenbachstr. 1, 85748 Garching bei München, Germany. E-mail: bizzocchi@mpe.mpg.de
cDipartimento di Chimica Industriale “Toso Montanari”, Università di Bologna, Viale del Risorgimento 4, 40136 Bologna, Italy. E-mail: filippo.tamassia@unibo.it; elisabetta.cane@unibo.it

Received 16th July 2018 , Accepted 14th September 2018

First published on 17th September 2018


The rotational spectrum of 15ND in its ground electronic X3Σ state has been observed for the first time. Forty-three hyperfine-structure components belonging to the ground and ν = 1 vibrational states have been recorded with a frequency-modulation millimeter-/submillimeter-wave spectrometer. These new measurements, together with the ones available for the other isotopologues NH, ND, and 15NH, have been simultaneously analysed using the Dunham model to represent the ro-vibrational, fine, and hyperfine energy contributions. The least-squares fit of more than 1500 transitions yielded an extensive set of isotopically independent Ulm parameters plus 13 Born–Oppenheimer Breakdown coefficients Δlm. As an alternative approach, we performed a Dunham analysis in terms of the most abundant isotopologue coefficients Ylm and some isotopically dependent Born–Oppenheimer Breakdown constants δlm [R. J. Le Roy, J. Mol. Spectrosc., 1999, 194, 189]. The two fits provide results of equivalent quality. The Born–Oppenheimer equilibrium bond distance for the imidogen radical has been calculated [rBOe = 103.606721(13) pm] and zero-point energies have been derived for all the isotopologues.


1 Introduction

The imidogen radical has been the subject of many spectroscopic, computational and astrophysical studies. This diatomic radical belongs to the first-row hydrides, is commonly observed in the combustion products of nitrogen-bearing compounds,1,2 and is also an intermediate in the formation process of ammonia in the interstellar medium (ISM).3 The main isotopologue of imidogen, NH, has been detected in a wide-variety of environments, from the Earths' atmosphere to astronomical objects, such as comets,4 many types of stars5,6 including the Sun,7,8 diffuse clouds,9 massive star-forming (SF) regions10 and, very recently, in prestellar cores.11 Also its deuterated counterpart ND has been identified in the ISM, towards the young solar-mass protostar IRAS1629312 and in the prestellar core 16293E.11

A lot of studies have been devoted to the origin of interstellar imidogen and different formation models have been proposed to explain its observed abundance in various sources. Two main formation routes have been devised for the NH radical: from the electronic recombination of NH+ and NH2+, intermediates in the synthesis of interstellar ammonia13,14 starting with N+ or, alternatively, via dissociative recombination of N2H+.15 However, the mechanism of NH production in the ISM is still debated,10 and grain-surface processes might also play a significant role.16 Imidogen, together with other light hydrides, often appears in the first steps of chemical networks leading to more complex N-bearing molecules. Its observation thus provides crucial constraints for the chemical modeling of astrophysical sources.17 Also, the rare isotopologues of this radical yield important astrochemical insights. Being proxies for N and D isotopic fractionation processes, they may help to trace the evolution of gas and dust during the star formation, thus shedding light on the link between Solar System materials and the parent ISM.18 This is particularly relevant for nitrogen, whose molecular isotopic compositions exhibits large and still unexplained variations.19,20 Measuring the isotopic ratios in imidogen provides useful complementary information on the already measured H/D, 14N/15N in ammonia (including the 15NH2D species21).

As far as the laboratory work is concerned, there is a substantial amount of spectroscopic data for the most abundant species and less extensive measurements for 15NH and ND. A detailed description of the spectroscopic studies of imidogen can be found in the latest experimental works on NH,2215NH,23 and ND.24 It has to be noticed that no experimental data or theoretical computations were available in literature for the doubly substituted species 15ND up to date. In this work, we report the first observation of its pure rotational spectrum in the ground electronic state X3Σ recorded up to 1.068 THz. A limited number of new transition frequencies for the isotopologues NH and ND in the v = 1 excited state have also been measured in the course of the present investigation. This new set of data, together with the literature data for NH, 15NH and ND, have been analysed in a global multi-isotopologue fit to give a comprehensive set of isotopically independent spectroscopic parameters. Thanks to the high precision of the measurements, several Born–Oppenheimer Breakdown (BOB) constants (Δlm) could be determined from a Dunham-type analysis. The alternative Dunham approach proposed by Le Roy25 has been also employed. In this case, the results are expressed in terms of the parent species coefficients Ylm plus some isotopically dependent BOB constants (δlm).

Finally, very accurate equilibrium bond distances re (including the Born–Oppenheimer bond distance rBOe) and zero-point energies (ZPE) for each isotopologue have been computed from the determined spectroscopic constants.

2 Experiments

The rotational spectrum of 15ND radical in its ground vibronic state X3Σ has been recorded with a frequency-modulation millimeter-/submillimeter-wave spectrometer. The primary source of radiation was constituted by a series of Gunn diodes (Radiometer Physics GmbH, J. E. Carlstrom Co.) emitting in the range 80–134 GHz, whose frequency is stabilized by a Phase-Lock-Loop (PLL) system. The PLL allowed the stabilization of the Gunn oscillator with respect to a frequency synthesizer (Schomandl ND 1000), which was driven by a 5 MHz rubidium frequency standard. Higher frequencies were obtained by using passive multipliers (RPG, ×6 and ×9). The frequency modulation of the output radiation was realized by sine-wave modulating at 6 kHz the reference signal of the wide-band Gunn synchronizer. The signal was detected by a liquid-helium-cooled InSb hot electron bolometer (QMC Instr. Ltd type QFI/2) and then demodulated at 2f by a lock-in amplifier. The experimental uncertainties of present measurements are between 40 and 80 kHz in most cases, up to 500 kHz for a few disturbed lines.

The 15ND radical was formed in a glow-discharge plasma with the same apparatus employed to produce other unstable and rare species (e.g., ND226 and 15N2H+[thin space (1/6-em)]27). The optimum production was attained in a DC discharge of a mixture of 15N2 (5–7 mTorr) and D2 (1–2 mTorr) in Ar as buffer gas (15 mTorr). Typically, a voltage of 1 kV and a current of 60 mA were employed. The absorption cell was cooled down at ca. −190 °C by liquid-nitrogen circulation.

3 Analysis

3.1 Effective Hamiltonian

From a spectroscopic point of view, imidogen is a free radical with a X3Σ ground electronic state and exhibits a fine structure due to the dipole–dipole interaction of the two unpaired electron spins and to the magnetic coupling of the molecular rotation with the total electron spin. The couplings of the various angular momenta in NH are described more appropriately by Hund's case (b) scheme
 
J = N + S,(1)
where N represents the pure rotational angular momentum. Each fine-structure level is thus labeled by J, N quantum numbers, where J = N + 1, N, N − 1. For N = 0, only one component (J = 1) exists. Inclusion of the nitrogen and hydrogen hyperfine interactions leads to the couplings
 
F1 = J + IN, F = F1 + IH.(2)

For each isotopologue in a given ro-vibronic state, the effective Hamiltonian can be written as

 
H = Hrv + Hfs + Hhfs(3)
where Hrv, Hfs and Hhfs are the ro-vibrational, fine- and hyperfine-structure Hamiltonians, respectively:
 
Hrv = Gv + BvN2DvN4 + HvN6 + LvN8 + MvN10(4)
 
image file: c8cp04498h-t1.tif(5)
 
image file: c8cp04498h-t2.tif(6)
Here, Gv is the pure vibrational energy, Bv the rotational constant, Dv, Hv, Lv, and Mv the centrifugal distortion parameters up to the fifth power in the N2 expansion, λv and λNv are the electron spin–spin interaction parameter and its centrifugal distortion coefficient; γv, and γNv are the electron spin-rotation constant and its centrifugal distortion coefficient, respectively. The constants bF,v and cv are the isotropic (Fermi contact interaction) and anisotropic parts of the electron spin-nuclear spin coupling, eQqv represents the electric quadrupole interaction and CI,v is the nuclear spin–rotation parameter. In eqn (6) the index i runs over the different nuclei present in a given isotopologue. The four nuclear spins are: I = 1/2 for H and 15N and I = 1 for D and 14N.

3.2 Multi-isotopologue Dunham models

In order to treat the data of all the available isotopologues in a global analysis, it is convenient to adopt a Dunham-type expansion.35 The ro-vibrational energy levels are given by the equation:
 
image file: c8cp04498h-t3.tif(7)

The fine- and hyperfine-structure parameters [i.e., λv, λNv, γv, γNv, bF,v, cv, eQqv, and CI,v in eqn (5) and (6)], are given by analogous expansions:

 
image file: c8cp04498h-t4.tif(8)
where y(v, N) represents the effective value of the parameter y in the ro-vibrational level labeled by (v, N), and [scr Y, script letter Y]lm are the coefficients of its Dunham-type expansion.

The spectroscopic constants of eqn (4)–(6) can be expressed in terms of the Dunham coefficients Ylm and [scr Y, script letter Y]lm. For example, the constants Gv, Bv, and Dv, of the ro-vibrational Hamiltonian are given by the following expansions:

 
image file: c8cp04498h-t5.tif(9a)
 
image file: c8cp04498h-t6.tif(9b)
 
image file: c8cp04498h-t7.tif(9c)

Each fine- and hyperfine-structure constant is also expressed by suitable expansions. For example, the electron spin-rotation constant and its centrifugal dependence in a given vibrational state can be expressed as:

 
image file: c8cp04498h-t8.tif(10a)
 
image file: c8cp04498h-t9.tif(10b)
where γl0 and γl1 are the [scr Y, script letter Y]lm constants of eqn (8) relative to the spin-rotation interaction. For a given isotopologue α, a specific set of Dunham constants Y(α)lm and [scr Y, script letter Y](α)lm is defined. Such constants can be described in terms of isotopically invariant parameters using the known reduced mass dependences given by36,37
 
image file: c8cp04498h-t10.tif(11a)
 
image file: c8cp04498h-t11.tif(11b)
where M(α)X (with X = N, H) are the atomic masses, μα is the reduced mass of the α isotopologue, and me is the electron mass. Ulm and Uylm are isotopically invariant Dunham constants, whereas ΔXlm and Δy,Xlm are unitless coefficients which account for the Born–Oppenheimer Breakdown.37,38 In eqn (11b), p = 0 for y = λ, bF, c, eQq, while p = 1 for y = γ, CI. This extra μ−1 factor in the mass scaling is needed to account for the intrinsic N2 dependence of the spin-rotation constants.39 Here, the unknowns are the Ulm, Uylm coefficients and the corresponding ΔXlm and Δy,Xlm BOB corrections.

An alternative approach has been proposed by Le Roy,25 where one isotopologue (usually the most abundant one) is chosen as reference species (α = 1), and the Dunham parameters Y(α)lm and [scr Y, script letter Y](α)lm of any other species are obtained by the following mass scaling

 
image file: c8cp04498h-t12.tif(12a)
 
image file: c8cp04498h-t13.tif(12b)

Here, ΔMX = M(α)XM(1)X (with X = N, H) are the mass differences produced by the isotopic substitution, with respect to the reference species, and the BOB corrections are described by the new δXlm and δy,Xlm coefficients. These are related to the dimensionless ΔXlm of eqn (11) through the simple relation

 
image file: c8cp04498h-t14.tif(13)

Albeit formally equivalent, this latter parametrisation was introduced to overcome a number of deficiencies of the traditional treatment which were pointed out by Watson37 and Tiemann,40 and its features are discussed in great detail in the original paper.25 An obvious advantage of the alternative mass scaling of eqn (12) is that the fitted coefficients are all expressed in frequency units and are directly linked to the familiar spectroscopic parameters of the reference isotopologue (e.g., Y(1)10ωe, Y(1)20 ≈ −ωexe, Y(1)01Be, Y(1)11 ≈ −αe, etc.). Furthermore, the BOB contributions are accounted for using purely addictive terms thus reducing the correlations among the parameters.

4 Results

4.1 15ND spectrum

For the previously unobserved 15ND species, we have recorded 34 lines for the ground vibrational state and 9 lines for the v = 1 state. They include the complete fine-structure of the N = 1 ← 0 transition and the strongest fine-components of the N = 2 ← 1 transition for the ground state (see Fig. 1), and the ΔJ = 0, +1 components of the N = 1 ← 0 transition for the v = 1 state. The corresponding transition frequencies were fitted to the Hamiltonian of eqn (4)–(6) using the SPFIT analysis program.41 Because of the small number of transitions detected for the v = 1 state, some of the spectroscopic parameters for this state could not be directly determined in the least-squares fit and were constrained to the corresponding ground state values. The two sets of constants for v = 0 and v = 1 states are reported in Table 1. The list of observed frequencies, along with the residuals from the single-species fit, is given in Table 2. In addition, the .LIN and .PAR input files for the SPFIT program are included in the ESI.
image file: c8cp04498h-f1.tif
Fig. 1 Upper panel: Energy levels scheme of 15ND in the ground vibrational state. The hyperfine-structure is not shown. The arrows mark the transitions observed in this study: ΔJ = +1 (red), ΔJ = 0 (blue), and ΔJ = −1 (green). Lower panel: Spectral recordings for the transitions marked with the labels a, b, c and d showing the corresponding hyperfine structure. The brown sticks represent the positions and the intensities of the hyperfine components computed from the spectroscopic parameters of Table 1.
Table 1 Spectroscopic constants determined for 15ND in the ground and v = 1 vibrational states
Constant Unit v = 0 v = 1
Notes. Number in parentheses are the 1σ statistical errors in unit of the last quoted digit.a Parameter held fixed in the fit.b Fit standard deviation.
B v MHz 261083.4809(96) 253597.797(24)
D v MHz 14.3906(13) 14.3906a
λ v MHz 27544.852(22) 27544.852a
γ v MHz −876.139(15) −841.674(46)
γ Nv MHz 0.1241(20) 0.1241a
b F,v(15N) MHz −26.519(20) −25.944(41)
c v(15N) MHz 95.154(56) 94.48(30)
C I,v(15N) MHz −0.124(14) −0.124a
b F,v(D) MHz −10.062(21) −10.524(42)
c v(D) MHz 14.236(78) 13.18(22)
eQqv(D) MHz 0.271(93) 0.271a
σ w 0.84
rms MHz 0.080
No. of lines 34 9


Table 2 Observed frequencies and residuals (in MHz) from the single-isotopologue fit of 15ND in the ground and first vibrational excited states
State N J F 1 F N′′ J′′ F 1′′ F′′ Obs. freq. Obs.-calc. Rel. weight
Notes. Number in parentheses are the experimental uncertainties in units of the last quoted digit. The relative weight is given only for blended transitions.
v = 0 1 0 0.5 1.5 0 1 0.5 0.5 487528.290(80) 0.119 0.95
1 0 0.5 0.5 0 1 0.5 0.5 487528.290(80) 0.119 0.05
1 0 0.5 1.5 0 1 0.5 1.5 487547.166(80) 0.012 0.79
1 0 0.5 0.5 0 1 0.5 1.5 487547.166(80) 0.012 0.21
1 0 0.5 1.5 0 1 1.5 1.5 487578.185(500) −0.229 0.21
1 0 0.5 0.5 0 1 1.5 1.5 487578.185(500) −0.229 0.79
1 0 0.5 1.5 0 1 1.5 2.5 487593.045(500) −0.332
1 2 2.5 3.5 0 1 1.5 2.5 517707.856(50) 0.002
1 2 2.5 2.5 0 1 1.5 1.5 517709.082(50) −0.060
1 2 2.5 1.5 0 1 1.5 0.5 517709.837(50) 0.024
1 2 1.5 2.5 0 1 0.5 1.5 517712.966(50) −0.008
1 2 2.5 1.5 0 1 1.5 1.5 517721.385(50) 0.046
1 2 2.5 2.5 0 1 1.5 2.5 517724.182(50) 0.052
1 2 1.5 1.5 0 1 0.5 1.5 517730.725(50) −0.032
1 2 1.5 2.5 0 1 1.5 2.5 517759.241(50) 0.038
1 1 0.5 1.5 0 1 0.5 0.5 541723.559(80) −0.009 0.85
1 1 0.5 0.5 0 1 0.5 0.5 541723.559(80) −0.009 0.15
1 1 0.5 1.5 0 1 0.5 1.5 541742.758(80) −0.015 0.43
1 1 0.5 0.5 0 1 0.5 1.5 541742.758(80) −0.015 0.57
1 1 1.5 1.5 0 1 0.5 0.5 541751.550(80) 0.064 0.74
1 1 1.5 0.5 0 1 0.5 0.5 541751.550(80) 0.064 0.26
1 1 0.5 1.5 0 1 1.5 0.5 541762.512(80) −0.042 0.31
1 1 0.5 0.5 0 1 1.5 0.5 541762.512(80) −0.042 0.69
1 1 1.5 2.5 0 1 0.5 1.5 541769.817(80) −0.122 0.98
1 1 1.5 1.5 0 1 0.5 1.5 541769.817(80) −0.122 0.02
1 1 0.5 1.5 0 1 1.5 1.5 541773.823(80) 0.053 0.88
1 1 0.5 0.5 0 1 1.5 1.5 541773.823(80) 0.053 0.12
1 1 0.5 1.5 0 1 1.5 2.5 541788.538(80) −0.154
1 1 1.5 1.5 0 1 1.5 0.5 541790.239(80) −0.069 0.37
1 1 1.5 0.5 0 1 1.5 0.5 541790.239(80) −0.069 0.63
1 1 1.5 2.5 0 1 1.5 1.5 541801.642(80) −0.006 0.12
1 1 1.5 1.5 0 1 1.5 1.5 541801.642(80) −0.006 0.64
1 1 1.5 0.5 0 1 1.5 1.5 541801.642(80) −0.006 0.24
1 1 1.5 2.5 0 1 1.5 2.5 541816.257(80) 0.027 0.84
1 1 1.5 1.5 0 1 1.5 2.5 541816.257(80) 0.027 0.16
2 1 1.5 0.5 1 1 0.5 0.5 1009356.833(40) −0.043 0.78
2 1 1.5 0.5 1 1 0.5 1.5 1009356.833(40) −0.043 0.22
2 3 3.5 3.5 1 2 2.5 2.5 1041497.348(40) 0.028 0.33
2 3 3.5 4.5 1 2 2.5 3.5 1041497.348(40) 0.028 0.44
2 3 3.5 2.5 1 2 2.5 1.5 1041497.348(40) 0.028 0.24
2 3 2.5 2.5 1 2 1.5 1.5 1041499.205(40) −0.006 0.31
2 3 2.5 3.5 1 2 1.5 2.5 1041499.205(40) −0.006 0.50
2 3 2.5 1.5 1 2 1.5 0.5 1041499.205(40) −0.006 0.19
2 3 3.5 2.5 1 2 2.5 2.5 1041510.251(40) −0.026 0.38
2 3 2.5 1.5 1 2 1.5 1.5 1041510.251(40) −0.026 0.62
2 3 2.5 2.5 1 2 1.5 2.5 1041516.279(40) −0.058
2 3 2.5 3.5 1 2 2.5 3.5 1041550.924(40) 0.018 0.60
2 3 2.5 1.5 1 2 2.5 1.5 1041550.924(40) 0.018 0.16
2 3 2.5 2.5 1 2 2.5 2.5 1041550.924(40) 0.018 0.24
2 2 1.5 1.5 1 1 1.5 0.5 1043854.642(40) −0.025 0.07
2 2 1.5 2.5 1 1 1.5 1.5 1043854.642(40) −0.025 0.08
2 2 1.5 0.5 1 1 1.5 0.5 1043854.642(40) −0.025 0.10
2 2 1.5 1.5 1 1 1.5 1.5 1043854.642(40) −0.025 0.18
2 2 1.5 0.5 1 1 1.5 1.5 1043854.642(40) −0.025 0.08
2 2 1.5 2.5 1 1 1.5 2.5 1043854.642(40) −0.025 0.41
2 2 1.5 1.5 1 1 1.5 2.5 1043854.642(40) −0.025 0.08
2 2 2.5 1.5 1 1 1.5 0.5 1043869.811(40) 0.031 0.17
2 2 2.5 2.5 1 1 1.5 1.5 1043869.811(40) 0.031 0.28
2 2 2.5 1.5 1 1 1.5 1.5 1043869.811(40) 0.031 0.05
2 2 2.5 3.5 1 1 1.5 2.5 1043869.811(40) 0.031 0.44
2 2 2.5 2.5 1 1 1.5 2.5 1043869.811(40) 0.031 0.05
2 2 2.5 1.5 1 1 1.5 2.5 1043869.811(40) 0.031 0.01
2 2 1.5 1.5 1 1 0.5 0.5 1043882.279(40) 0.047 0.19
2 2 1.5 0.5 1 1 0.5 0.5 1043882.279(40) 0.047 0.16
2 2 1.5 2.5 1 1 0.5 1.5 1043882.279(40) 0.047 0.50
2 2 1.5 1.5 1 1 0.5 1.5 1043882.279(40) 0.047 0.15
2 2 1.5 0.5 1 1 0.5 1.5 1043882.279(40) 0.047 0.02
2 1 0.5 1.5 1 0 0.5 0.5 1063506.127(40) 0.013 0.32
2 1 0.5 1.5 1 0 0.5 1.5 1063506.127(40) 0.013 0.68
2 1 1.5 2.5 1 0 0.5 1.5 1063578.610(40) −0.036
2 2 2.5 3.5 1 2 2.5 3.5 1067978.220(40) 0.058
v = 1 1 2 2.5 3.5 0 1 1.5 2.5 502775.779(60) −0.063
1 2 2.5 2.5 0 1 1.5 1.5 502777.216(60) 0.054 0.63
1 2 2.5 1.5 0 1 1.5 0.5 502777.216(60) 0.054 0.37
1 2 1.5 2.5 0 1 0.5 1.5 502780.683(60) 0.052
1 2 2.5 1.5 0 1 1.5 1.5 502789.681(60) −0.062 0.35
1 2 1.5 0.5 0 1 0.5 0.5 502789.681(60) −0.062 0.65
1 2 2.5 2.5 0 1 1.5 2.5 502792.545(60) 0.031
1 2 1.5 1.5 0 1 0.5 1.5 502798.877(60) 0.009
1 2 1.5 2.5 0 1 1.5 2.5 502826.422(60) −0.021
1 1 1.5 2.5 0 1 0.5 1.5 526777.091(60) −0.019
1 1 1.5 2.5 0 1 1.5 2.5 526822.941(60) 0.019


4.2 Multi-isotopologue Dunham fit

In this work, we carried out a multi-isotopologue Dunham fit of the imidogen radical in its X3Σ ground electronic state using our newly measured transition frequencies for the doubly substituted 15ND variant plus all the available rotational and ro-vibrational data for the NH, 15NH and ND species. To take into account the different experimental precision, each datum was given a weight inversely proportional to the square of its estimated measurement error, w = 1/σ2. The σ values adopted for the present measurements have been discussed in Section 2, while for literature data, we retained the values provided in each original work.

The content of the data set and the relevant bibliographic references are summarised in Table 3. In total, the data set contains 1563 ro-vibrational transitions which correspond to 1201 distinct frequencies. These data were fitted to the multi-isotopologue model described in Sections 3.1 and 3.2, using both traditional [eqn (11)] and Le Roy [eqn (12)] mass scaling schemes to describe the Dunham-type parameters (Ylm and [scr Y, script letter Y]lm) of each isotopic species.

Table 3 Summary of the data used for the multi-isotopologue fit of imidogen
Pure rotational Ro-vibrational
No. of lines No. of vib states Ref. No. of lines No. of bands Ref.
NH 96 2 Flores-Mijangos et al.,28 Lewen et al.,29 TW 451 6 Bernath,30 Geller et al.,8 Ram and Bernath22
ND 144 6 Saito and Goto,31 Takano et al.,32 Dore et al.,24 TW 406 6 Ram33
15NH 61 2 Bailleux et al.,34 Bizzocchi et al.23
15ND 43 2 This work (TW)


The analysis was performed using the SPFIT program:41 the complex parameter file (.PAR) and the line file (.LIN), as well as the output file (.FIT), were processed with a custom Python code, which accounts for the mass constraints of the Dunham constants and tracks the isotopologue-dependent experimental and derived quantities. The atomic masses used were taken from the Wang et al.42 compilation. The optimised parameters are reported in Tables 4 and 5, while the complete list of all the fitted data, together with the residuals from the multi-isotopologue analysis, is provided as ESI (the .LIN and .PAR files are also provided).

Table 4 Ro-vibrational Dunham Ylm constants and isotopically invariant Ulm parameters determined in the multi-isotopologue fit for imidogen radical
l m Y lm U lm
Units Value Units Value
Notes. The Dunham constants Ylm are referred to the most abundant NH isotopologue. The BOB coefficients ΔXlm are dimensionless. Number in parentheses are the 1σ statistical errors in units of the last quoted digit.
1 0 cm−1 3282.3629(39) cm−1 u1/2 3184.2027(35)
2 0 cm−1 −78.6810(46) cm−1 u −73.9831(43)
3 0 cm−1 0.2223(25) cm−1 u3/2 0.2027(23)
4 0 cm−1 −0.02953(68) cm−1 u2 −0.02610(60)
5 0 cm−1 −0.000263(88) cm−1 u5/2 −0.000225(75)
6 0 cm−1 −0.0001393(45) cm−1 u5/2 −0.0001158(37)
0 1 MHz 499690.529(84) cm−1 u 15.7043731(39)
1 1 MHz −19494.41(34) cm−1 u3/2 −0.593919(10)
2 1 MHz 67.19(48) cm−1 u2 0.001981(14)
3 1 MHz −7.69(31) cm−1 u5/2 −2.201(87) × 10−4
4 1 MHz −1.579(94) cm−1 u3 −4.37(26) × 10−5
5 1 MHz 0.130(14) cm−1 u7/2 3.48(37) × 10−6
6 1 MHz −0.01456(76) cm−1 u7/2 −3.79(20) × 10−7
0 2 MHz −51.44722(91) cm−1 u2 −0.00152786(15)
1 2 MHz 0.8253(24) cm−1 u5/2 2.3594(68) × 10−5
2 2 MHz −0.0642(15) cm−1 u3 −1.779(42) × 10−6
3 2 MHz 0.00269(37) cm−1 u7/2 7.24(99) × 10−8
4 2 MHz −0.001460(28) cm−1 u4 −3.806(72) × 10−8
0 3 MHz 0.0037950(79) cm−1 u3 1.0931(36) × 10−7
1 3 MHz −1.339(45) × 10−4 cm−1 u7/2 −3.60(12) × 10−9
2 3 MHz −1.25(18) × 10−5 cm−1 u4 −3.26(47) × 10−10
3 3 MHz −4.26(31) × 10−6 cm−1 u9/2 −1.075(79) × 10−10
0 4 MHz −4.47(16) × 10−7 cm−1 u4 −1.166(42) × 10−11
1 4 MHz −3.49(29) × 10−8 cm−1 u9/2 −8.80(74) × 10−13
0 5 MHz 3.72(88) × 10−11 cm−1 u9/2 9.1(22) × 10−16

X l m Units δ X lm Δ X lm
N 0 1 MHz 75.71(11) −3.8592(58)
N 1 1 MHz −3.45(13) −4.50(17)
H 1 0 cm−1 1.6117(14) −0.90162(78)
H 2 0 cm−1 −0.01098(24) −0.2565(55)
H 0 1 MHz 1005.124(35) −3.68744(13)
H 1 1 MHz −0.3733(47) −3.2030(29)
H 0 2 MHz −34.054(31) −13.23(17)
H 0 3 MHz 1.48(11) × 10−4 −69.1(51)


Table 5 Fine and hyperfine Dunham [scr Y, script letter Y]lm constants and isotopically invariant Uylm parameters determined in the multi-isotopologue fit for NH
Dunham type Isotopically invariant
Notes. The Dunham constants [scr Y, script letter Y]lm are referred to the most abundant NH isotopologue. The BOB coefficients ΔXlm are dimensionless. Number in parentheses are the 1σ statistical errors in units of the last quoted digit.
Fine structure parameters
λ 00 MHz 27573.424(23) U λ 00 MHz u 0.9174536(52)
λ 10 MHz 16.200(46) U λ 10 MHz u1/2 5.864(20) × 10−4
λ 20 MHz −14.645(22) U λ 20 MHz u −4.5922(68) × 10−4
λ 01 MHz 0.0109(38) U λ 01 MHz u3/2 3.4(12) × 10−7
γ 00 MHz −1688.280(31) U γ 00 MHz u −0.0528438(13)
γ 10 MHz 88.172(93) U γ 10 MHz u3/2 0.0026750(27)
γ 20 MHz −1.387(79) U γ 20 MHz u3/2 −4.06(23) × 10−5
γ 30 MHz 0.370(21) U γ 30 MHz u3/2 1.054(61) × 10−5
γ 01 MHz 0.4631(31) U γ 01 MHz u2 1.3656(91) × 10−5
γ 11 MHz −0.0291(81) U γ 11 MHz u3/2 −8.4(23) × 10−7
γ 21 MHz 0.0152(45) U γ 21 MHz u2 4.2(13) × 10−7
γ 31 MHz −0.00333(72) U γ 31 MHz u2 −9.0(19) × 10−8
δ λ,N00 MHz −3.59(16) Δ λ,N00 0.837(37)
δ λ,H00 MHz −65.250(40) Δ λ,H00 1.08957(67)
δ λ,N10 MHz 1.934(40) Δ λ,H10 −49.0(10)
δ γ,H00 MHz 1.712(18) Δ γ,H00 3.515(38)
δ γ,H10 MHz −0.091(17) Δ γ,H10 3.83(66)
Hyperfine structure parameters
b F,00(H) MHz −64.194(13) image file: c8cp04498h-t26.tif cm−1 −0.00214129(45)
b F,10(H) MHz −3.785(15) image file: c8cp04498h-t27.tif cm−1 −1.2243(48) × 10−4
c 00(H) MHz 92.216(61) U c 00(H) cm−1 0.0030760(20)
c 10(H) MHz −3.391(58) U c 10(H) cm−1 −1.097(19) × 10−4
C 00(H) MHz −0.068(11) U C 00(H) cm−1 −2.14(33) × 10−6
b F,00(D) MHz −9.836(12) image file: c8cp04498h-t28.tif cm−1 −3.2810(41) × 10−4
b F,10(D) MHz −0.632(10) image file: c8cp04498h-t29.tif cm−1 u1/2 −2.044(34) × 10−5
c 00(D) MHz 14.233(74) U c 00(D) cm−1 4.746(25) × 10−4
c 10(D) MHz −0.372(99) U c 10(D) cm−1 u1/2 −1.19(32) × 10−5
c 20(D) MHz −0.117(32) U c 10(D) cm−1 u1/2 −3.66(100) × 10−6
eQq00(D) MHz 0.080(32) U eQq00(D) cm−1 2.7(11) × 10−6
b F,00(14N) MHz 19.084(18) image file: c8cp04498h-t30.tif cm−1 6.3660(59) × 10−4
b F,10(14N) MHz −0.421(31) image file: c8cp04498h-t31.tif cm−1 u1/2 −1.365(99) × 10−5
b F,20(14N) MHz −0.088(13) image file: c8cp04498h-t32.tif cm−1 u1/2 −2.76(41) × 10−6
c 00(14N) MHz −68.135(31) U c 00(14N) cm−1 u1/2 −0.0022727(10)
c 10(14N) MHz 0.467(26) U c 10(14N) cm−1 u3/2 1.509(83) × 10−5
eQq00(14N) MHz −3.367(54) U eQq00(14N) cm−1 −1.123(18) × 10−4
eQq10(14N) MHz 0.395(38) U eQq10(14N) cm−1 u1/2 1.28(12) × 10−5
C 00(14N) MHz 0.172(11) U C 00(14N) cm−1 u1/2 5.42(36) × 10−6
C 10(14N) kHz −0.0293(85) U C 10(14N) cm−1 u3/2 −9.0(26) × 10−7
b F,00(15N) MHz −26.848(16) image file: c8cp04498h-t33.tif cm−1 −8.9556(52) × 10−4
b F,10(15N) MHz 0.860(18) image file: c8cp04498h-t34.tif cm−1 u1/2 2.789(58) × 10−5
c 00(15N) MHz 95.428(49) U c 00(15N) cm−1 0.0031831(16)
c 10(15N) MHz −0.556(54) U c 10(15N) cm−1 −1.80(17) × 10−5
C 00(15N) MHz −0.259(22) U C 00(15N) cm−1 u1/2 −8.18(71) × 10−6
C 10(15N) MHz 0.099(24) U C 00(15N) cm−1 u1/2 3.06(74) × 10−6


5 Discussion

5.1 Spectroscopic parameters

From the multi-isotopologue analysis we obtained a highly satisfactory fit. Its quality can be evaluated in several ways. First of all, we were able to reproduce the input data within their estimated uncertainties: the overall standard deviation of the weighted fit is σ = 0.89, and the root-mean-square deviations of the residuals computed separately for the rotational and ro-vibrational data are of the same order of magnitude of the corresponding measurements error, RMSROT = 0.107 MHz and RMSVIBROT = 3.4 × 10−3 cm−1, respectively. Then, the various sets of Ylm for a given m constitute a series whose coefficients decrease in magnitude for increasing values of the index l, as expected for a rapidly converging Dunham-type expansion. In general, most of the determined coefficients have a relative error lower than 5%. Higher errors are observed only for those constants with high l-index and this is due to the smaller number of transitions available for highly vibrationally excited states. Finally, the Kratzer43 and Pekeris44 relation can also be used as a yardstick to asses the correct treatment of the Born–Oppenheimer Breakdown effects. Using the formula45
 
image file: c8cp04498h-t15.tif(14)
we obtained for Y02 a value of 51.54051 MHz which compares well with the fitted one of 51.44722(91) MHz.

5.2 Equilibrium bond distance

The precision yielded by the high-resolution spectroscopic technique led to a very accurate determination of the equilibrium bond length re for the imidogen radical. The rotational measurements of a diatomic molecule in its ground vibrational state (v = 0) allow the determination of precise value of r0, which includes the zero-point vibrational contributions and differs from re. This latter is determinable from the rotational spectrum in at least one vibrationally excited state. In the present analysis, data of four isotopic species in several vibrational excited states have been combined, allowing for a very precise determination of re for each isotopologue α. The equilibrium bond distance is given by:
 
image file: c8cp04498h-t16.tif(15)
where Nah is the molar Planck constant. Actually, the values of B(α)e differ from those of Y(α)01 obtained from the Dunham-type analysis. This discrepancy should be ascribed to a small contribution, expressed by:45
 
image file: c8cp04498h-t17.tif(16)
with
 
image file: c8cp04498h-t18.tif(17)
and
 
image file: c8cp04498h-t19.tif(18)

From eqn (15), it is evident that the bond length re assumes different values for each isotopologue. On the contrary, by substituting the product B(α)eμa with U01, one obtains an isotopically independent equilibrium bond length rBOe. In the present case, rBOe takes the value of 103.606721(13) pm. In Table 6, this result is compared with the equilibrium bond distances calculated from the Be of each isotopologue NH, 15NH, ND, and 15ND. In this case, Be was obtained by correcting the corresponding Y01 constant according to eqn (16)–(18). It should be noticed that the values differ at sub-picometre level but these differences, even if small, are detectable thanks to the high-precision of rotational measurements.

Table 6 Born–Oppenheimer and equilibrium bond distances (in pm) from the individual isotopologues (see text)
Species r e r e rBOe
NH 103.716377(16) 0.109656
15NH 103.715864(16) 0.109143
ND 103.665420(10) 0.058699
15ND 103.664908(10) 0.058187
r BOe = 103.606721(13)
r theore = 103.5915


The experimental value derived for rBOe has been compared with a theoretically best estimate obtained following the prescriptions of ref. 46 and 47. A composite calculation have been carried out considering basis-set extrapolation, core-correlation effects, and inclusion of higher-order corrections due to the use of the full coupled-cluster singles and doubles, augmented by a perturbative treatment of triple excitation [CCSD(T)] model

fc-CCSD(T)/cc-pV∞Z + Δcore/cc-pCV5Z + ΔT/cc-pVTZ.

The computation have been performed using CFOUR.48 From this theoretical procedure we obtained rtheore = 103.5915 pm (see also Table 6), which is in very good agreement with the experimentally derived value, the discrepancy being 15 fm.

5.3 Born–Oppenheimer breakdown

The BOB coefficients ΔXlm determined in the present analysis account for the small inaccuracies of the Born–Oppenheimer approximation in describing the ro-vibrational energies of the imidogen radical. For the rotational constant (≈Y01), it is possible to identify three different contributions to the corresponding BOB parameter49
 
image file: c8cp04498h-t20.tif(19)
namely an adiabatic contribution, a non-adiabatic term, and a Dunham correction, respectively. The last two terms on the right side of eqn (19) can be computed from purely experimental quantities: (ΔX01)Dunh arises from the use of a Dunham expansion and contains the term ΔY(Dunh)01 of eqn (16), whereas (ΔX01)nad depends on the mixing of the electronic ground state with nearby electronic excited states, and can be estimated from the molecular electric dipole moment μ and the rotational gJ factors.38 The adiabatic term can be simply computed as the difference between the experimental ΔX01 and the terms (ΔX01)Dunh and (ΔX01)nad.

Tiemann et al.40 found that the adiabatic term (ΔX01)ad basically depends on the corresponding X atom rather than on the particular molecular species. Hence, it is interesting to derive this contribution in order to compare the results obtained for different molecules and to verify the reliability of the empirical fitting procedure.

All the contributions of eqn (19) are collected in Table 7. The non-adiabatic contribution has been computed using the literature value of the dipole moment50μ = 1.389 D and the ground state gJ value estimated from a laser magnetic resonance study,51gJ = 0.001524.

Table 7 Contributions of the Born–Oppenheimer Breakdown coefficients to the U01 constant
Atom Δ 01 (exp) Adiabatic Non-adiabatic Dunham
N −3.8592 −0.6515 −3.1326 −0.0751
H −3.6874 −1.0379 −2.5744 −0.0751


From the adiabatic contribution to the Born–Oppenheimer Breakdown coefficients for the rotational constants, (ΔX01)ad, one may derive the corresponding correction to the equilibrium bond distance, a quantity which can also be accessed by ab initio computations. From our eqn (11a) and eqn (6) of ref. 52, the following equality is obtained

 
image file: c8cp04498h-t21.tif(20)

The adiabatic correction to the equilibrium bond distance, ΔRad, can be theoretically estimated through the computation of the adiabatic bond distance, i.e., the minimum of the potential given by the sum of the Born–Oppenheimer potential augmented by the diagonal Born–Oppenheimer corrections (DBOC).52 The difference between the equilibrium bond distances calculated with and without DBOC, with tight convergence limits, performed at the CCSD/cc-pCVnZ level (n = 3, 4, 5), yielded ΔRad = 0.026 pm. This value is in very good agreement with the purely experimental one obtained by eqn (20) which results 0.020 pm, thus providing a confirmation for the validity of our data treatment.

5.4 Zero-point energy

The results of our analysis make possible to estimate the zero-point energy (ZPE) for each isotopologue from the Dunham's constants Ylm with m = 0, namely:
 
image file: c8cp04498h-t22.tif(21)

As we determined anharmonicity constants up to the sixth order, the ZPE is derived with a negligible truncation bias53 from the expression:

 
image file: c8cp04498h-t23.tif(22)

The Y00 constant present in the Dunham-type expansions is not experimentally accessible. Its value can be estimated, to a good approximation, through53

 
image file: c8cp04498h-t24.tif(23)

The value for the main isotopologue NH is 1.9987(12) cm−1.

The values obtained for the ZPE of the four isotopologues are collected in Table 8. For comparison, the values of literature are also reported. Our results agree well with those reported in the literature,53 but our precision is more than one order of magnitude higher. The errors on our ZPE values are ca. 1 × 10−3 cm−1 and were calculated taking into account the error propagation

 
σf2 = gTVg(24)
where σf2 is the variance in the function f (i.e., eqn (22) in the present case) of the set of parameters Yl0, whose variance–covariance matrix is V, with the ith element in the vector g being image file: c8cp04498h-t25.tif.

Table 8 Zero-point energies (in cm−1) of imidogen isotopologues
Species This work Ref. 53 Ref. 54
Notes. Number in parentheses are the 1σ statistical errors in unit of the last quoted digit.a From ref. 22.b Computed.c From ref. 24.
NH 1623.5359(17) 1623.6(6) 1621.5a
15NH 1619.9485(17) 1617.9b
ND 1190.0859(11) 1190.13(5) 1189.5c
15ND 1185.1413(11) 1183.6b


Discrepancies of ∼2 cm−1 are observed by comparing our data with those reported in ref. 54 because their definition of the ZPE does not include the term Y00, which is non-negligible for light molecules.53 These newly determined values should be used in the calculation of the exoergicity values ΔE of chemical reactions relevant in fractionation processes.

6 Conclusions

In this work the pure rotational spectrum of 15ND in its ground electronic X3Σ state has been recorded for the first time using a frequency-modulation submillimeter-wave spectrometer. A global fit, including all previously reported rotational and ro-vibrational data for the other isotopologues of the imidogen radical, has been performed and yielded a comprehensive set of Dunham coefficients. Moreover, the Born–Oppenheimer Breakdown constants have been determined for 13 parameters and also the adiabatic contribution of the terms ΔN01 and ΔH01 were evaluated and compared to theoretical estimates. The present analysis enables to predict rotational and ro-vibrational spectra of any isotopic variant of NH at a high level of accuracy and to assist further astronomical searches of imidogen. From our results, very accurate values of the equilibrium bond distances re and the vibrational zero-point energies for the different isotopologues have been derived.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by Italian MIUR (2015F59J3R) (PRIN 2015 “STARS in the CAOS”) and by the University of Bologna (RFO funds). Open Access funding provided by the Max Planck Society.

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Footnotes

Electronic supplementary information (ESI) available: The .LIN and .PAR files for the SPFIT program are provided for both the single-species and multi-isotopologue fits. A reformatted list of all the transitions used in the Dunham-type analysis, together with their residuals from the final fit. See DOI: 10.1039/c8cp04498h
This (v, N)-factorisation is possible because all the angular momentum operators multiplying the coefficients of eqn (5) and (6) commute with purely vibrational operators and with N2.

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