The rotational spectrum of ^{15}ND. Isotopicindependent Dunhamtype analysis of the imidogen radical†
Received
16th July 2018
, Accepted 14th September 2018
First published on 17th September 2018
The rotational spectrum of ^{15}ND in its ground electronic X^{3}Σ^{−} state has been observed for the first time. Fortythree hyperfinestructure components belonging to the ground and ν = 1 vibrational states have been recorded with a frequencymodulation millimeter/submillimeterwave spectrometer. These new measurements, together with the ones available for the other isotopologues NH, ND, and ^{15}NH, have been simultaneously analysed using the Dunham model to represent the rovibrational, fine, and hyperfine energy contributions. The leastsquares fit of more than 1500 transitions yielded an extensive set of isotopically independent U_{lm} parameters plus 13 Born–Oppenheimer Breakdown coefficients Δ_{lm}. As an alternative approach, we performed a Dunham analysis in terms of the most abundant isotopologue coefficients Y_{lm} and some isotopically dependent Born–Oppenheimer Breakdown constants δ_{lm} [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 [r^{BO}_{e} = 103.606721(13) pm] and zeropoint 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 firstrow hydrides, is commonly observed in the combustion products of nitrogenbearing 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 widevariety of environments, from the Earths' atmosphere to astronomical objects, such as comets,^{4} many types of stars^{5,6} including the Sun,^{7,8} diffuse clouds,^{9} massive starforming (SF) regions^{10} and, very recently, in prestellar cores.^{11} Also its deuterated counterpart ND has been identified in the ISM, towards the young solarmass protostar IRAS16293^{12} 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 NH_{2}^{+}, intermediates in the synthesis of interstellar ammonia^{13,14} starting with N^{+} or, alternatively, via dissociative recombination of N_{2}H^{+}.^{15} However, the mechanism of NH production in the ISM is still debated,^{10} and grainsurface 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 Nbearing 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, ^{14}N/^{15}N in ammonia (including the ^{15}NH_{2}D species^{21}).
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 ^{15}NH and ND. A detailed description of the spectroscopic studies of imidogen can be found in the latest experimental works on NH,^{22}^{15}NH,^{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 ^{15}ND up to date. In this work, we report the first observation of its pure rotational spectrum in the ground electronic state X^{3}Σ^{−} 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, ^{15}NH and ND, have been analysed in a global multiisotopologue 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 (Δ_{lm}) could be determined from a Dunhamtype analysis. The alternative Dunham approach proposed by Le Roy^{25} has been also employed. In this case, the results are expressed in terms of the parent species coefficients Y_{lm} plus some isotopically dependent BOB constants (δ_{lm}).
Finally, very accurate equilibrium bond distances r_{e} (including the Born–Oppenheimer bond distance r^{BO}_{e}) and zeropoint energies (ZPE) for each isotopologue have been computed from the determined spectroscopic constants.
2 Experiments
The rotational spectrum of ^{15}ND radical in its ground vibronic state X^{3}Σ^{−} has been recorded with a frequencymodulation millimeter/submillimeterwave 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 PhaseLockLoop (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 sinewave modulating at 6 kHz the reference signal of the wideband Gunn synchronizer. The signal was detected by a liquidheliumcooled InSb hot electron bolometer (QMC Instr. Ltd type QFI/2) and then demodulated at 2f by a lockin 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 ^{15}ND radical was formed in a glowdischarge plasma with the same apparatus employed to produce other unstable and rare species (e.g., ND_{2}^{26} and ^{15}N_{2}H^{+}^{27}). The optimum production was attained in a DC discharge of a mixture of ^{15}N_{2} (5–7 mTorr) and D_{2} (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 liquidnitrogen circulation.
3 Analysis
3.1 Effective Hamiltonian
From a spectroscopic point of view, imidogen is a free radical with a X^{3}Σ^{−} 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) schemewhere N represents the pure rotational angular momentum. Each finestructure 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 
F_{1} = J + I_{N}, F = F_{1} + I_{H}.  (2) 
For each isotopologue in a given rovibronic state, the effective Hamiltonian can be written as

H = H_{rv} + H_{fs} + H_{hfs}  (3) 
where
H_{rv},
H_{fs} and
H_{hfs} are the rovibrational, fine and hyperfinestructure Hamiltonians, respectively:

H_{rv} = G_{v} + B_{v}N^{2} − D_{v}N^{4} + H_{v}N^{6} + L_{v}N^{8} + M_{v}N^{10}  (4) 

 (5) 

 (6) 
Here,
G_{v} is the pure vibrational energy,
B_{v} the rotational constant,
D_{v},
H_{v},
L_{v}, and
M_{v} the centrifugal distortion parameters up to the fifth power in the
N^{2} expansion,
λ_{v} and
λ_{Nv} are the electron spin–spin interaction parameter and its centrifugal distortion coefficient;
γ_{v}, and
γ_{Nv} are the electron spinrotation constant and its centrifugal distortion coefficient, respectively. The constants
b_{F,v} and
c_{v} are the isotropic (Fermi contact interaction) and anisotropic parts of the electron spinnuclear spin coupling, eQq
_{v} represents the electric quadrupole interaction and
C_{I,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
^{15}N and
I = 1 for D and
^{14}N.
3.2 Multiisotopologue Dunham models
In order to treat the data of all the available isotopologues in a global analysis, it is convenient to adopt a Dunhamtype expansion.^{35} The rovibrational energy levels are given by the equation: 
 (7) 
The fine and hyperfinestructure parameters [i.e., λ_{v}, λ_{Nv}, γ_{v}, γ_{Nv}, b_{F,v}, c_{v}, eQq_{v}, and C_{I,v} in eqn (5) and (6)], are given by analogous expansions:

 (8) 
where
y(
v,
N) represents the effective value of the parameter
y in the rovibrational level labeled by (
v,
N), and
_{lm} are the coefficients of its Dunhamtype expansion.
‡
The spectroscopic constants of eqn (4)–(6) can be expressed in terms of the Dunham coefficients Y_{lm} and _{lm}. For example, the constants G_{v}, B_{v}, and D_{v}, of the rovibrational Hamiltonian are given by the following expansions:

 (9a) 

 (9b) 

 (9c) 
Each fine and hyperfinestructure constant is also expressed by suitable expansions. For example, the electron spinrotation constant and its centrifugal dependence in a given vibrational state can be expressed as:

 (10a) 

 (10b) 
where
γ_{l0} and
γ_{l1} are the
_{lm} constants of
eqn (8) relative to the spinrotation interaction. For a given isotopologue
α, a specific set of Dunham constants
Y^{(α)}_{lm} and
^{(α)}_{lm} is defined. Such constants can be described in terms of isotopically invariant parameters using the known reduced mass dependences given by
^{36,37} 
 (11a) 

 (11b) 
where
M^{(α)}_{X} (with X = N, H) are the atomic masses,
μ_{α} is the reduced mass of the
α isotopologue, and
m_{e} is the electron mass.
U_{lm} and
U^{y}_{lm} are isotopically invariant Dunham constants, whereas
Δ^{X}_{lm} and
Δ^{y,X}_{lm} are unitless coefficients which account for the Born–Oppenheimer Breakdown.
^{37,38} In
eqn (11b),
p = 0 for
y =
λ,
b_{F},
c, eQq, while
p = 1 for
y =
γ,
C_{I}. This extra
μ^{−1} factor in the mass scaling is needed to account for the intrinsic
N^{2} dependence of the spinrotation constants.
^{39} Here, the unknowns are the
U_{lm},
U^{y}_{lm} coefficients and the corresponding
Δ^{X}_{lm} and
Δ^{y,X}_{lm} 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 ^{(α)}_{lm} of any other species are obtained by the following mass scaling

 (12a) 

 (12b) 
Here, ΔM_{X} = M^{(α)}_{X} − M^{(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 δ^{X}_{lm} and δ^{y,X}_{lm} coefficients. These are related to the dimensionless Δ^{X}_{lm} of eqn (11) through the simple relation

 (13) 
Albeit formally equivalent, this latter parametrisation was introduced to overcome a number of deficiencies of the traditional treatment which were pointed out by Watson^{37} 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)}_{10} ≈ ω_{e}, Y^{(1)}_{20} ≈ −ω_{e}x_{e}, Y^{(1)}_{01} ≈ B_{e}, Y^{(1)}_{11} ≈ −α_{e}, etc.). Furthermore, the BOB contributions are accounted for using purely addictive terms thus reducing the correlations among the parameters.
4 Results
4.1
^{15}ND spectrum
For the previously unobserved ^{15}ND species, we have recorded 34 lines for the ground vibrational state and 9 lines for the v = 1 state. They include the complete finestructure of the N = 1 ← 0 transition and the strongest finecomponents 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 leastsquares 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 singlespecies fit, is given in Table 2. In addition, the .LIN and .PAR input files for the SPFIT program are included in the ESI.†

 Fig. 1 Upper panel: Energy levels scheme of ^{15}ND in the ground vibrational state. The hyperfinestructure 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 ^{15}ND 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. Parameter held fixed in the fit. Fit standard deviation. 
B
_{
v
}

MHz 
261083.4809(96) 
253597.797(24) 
D
_{v}

MHz 
14.3906(13) 
14.3906^{a} 
λ
_{v}

MHz 
27544.852(22) 
27544.852^{a} 
γ
_{v}

MHz 
−876.139(15) 
−841.674(46) 
γ
_{
Nv}

MHz 
0.1241(20) 
0.1241^{a} 
b
_{
F,v}(^{15}N) 
MHz 
−26.519(20) 
−25.944(41) 
c
_{v}(^{15}N) 
MHz 
95.154(56) 
94.48(30) 
C
_{
I,v}(^{15}N) 
MHz 
−0.124(14) 
−0.124^{a} 
b
_{
F,v}(D) 
MHz 
−10.062(21) 
−10.524(42) 
c
_{v}(D) 
MHz 
14.236(78) 
13.18(22) 
eQq_{v}(D) 
MHz 
0.271(93) 
0.271^{a} 

σ
_{w}
^{
}


0.84 
rms 
MHz 
0.080 
No. of lines 

34 
9 
Table 2 Observed frequencies and residuals (in MHz) from the singleisotopologue fit of ^{15}ND 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 Multiisotopologue Dunham fit
In this work, we carried out a multiisotopologue Dunham fit of the imidogen radical in its X^{3}Σ^{−} ground electronic state using our newly measured transition frequencies for the doubly substituted ^{15}ND variant plus all the available rotational and rovibrational data for the NH, ^{15}NH 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/σ^{2}. 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 rovibrational transitions which correspond to 1201 distinct frequencies. These data were fitted to the multiisotopologue 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 Dunhamtype parameters (Y_{lm} and _{lm}) of each isotopic species.
Table 3 Summary of the data used for the multiisotopologue fit of imidogen

Pure rotational 
Rovibrational 
No. of lines 
No. of vib states 
Ref. 
No. of lines 
No. of bands 
Ref. 
NH 
96 
2 
FloresMijangos et al.,^{28} Lewen et al.,^{29} TW 
451 
6 
Bernath,^{30} Geller et al.,^{8} Ram and Bernath^{22} 
ND 
144 
6 
Saito and Goto,^{31} Takano et al.,^{32} Dore et al.,^{24} TW 
406 
6 
Ram^{33} 
^{15}NH 
61 
2 
Bailleux et al.,^{34} Bizzocchi et al.^{23} 
— 
— 

^{15}ND 
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 isotopologuedependent 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 multiisotopologue analysis, is provided as ESI† (the .LIN and .PAR files are also provided).
Table 4 Rovibrational Dunham Y_{lm} constants and isotopically invariant U_{lm} parameters determined in the multiisotopologue fit for imidogen radical
l

m

Y
_{
lm
}

U
_{
lm
}

Units 
Value 
Units 
Value 
Notes. The Dunham constants Y_{lm} are referred to the most abundant NH isotopologue. The BOB coefficients Δ^{X}_{lm} 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} u^{1/2} 
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} u^{3/2} 
0.2027(23) 
4 
0 
cm^{−1} 
−0.02953(68) 
cm^{−1} u^{2} 
−0.02610(60) 
5 
0 
cm^{−1} 
−0.000263(88) 
cm^{−1} u^{5/2} 
−0.000225(75) 
6 
0 
cm^{−1} 
−0.0001393(45) 
cm^{−1} u^{5/2} 
−0.0001158(37) 
0 
1 
MHz 
499690.529(84) 
cm^{−1} u 
15.7043731(39) 
1 
1 
MHz 
−19494.41(34) 
cm^{−1} u^{3/2} 
−0.593919(10) 
2 
1 
MHz 
67.19(48) 
cm^{−1} u^{2} 
0.001981(14) 
3 
1 
MHz 
−7.69(31) 
cm^{−1} u^{5/2} 
−2.201(87) × 10^{−4} 
4 
1 
MHz 
−1.579(94) 
cm^{−1} u^{3} 
−4.37(26) × 10^{−5} 
5 
1 
MHz 
0.130(14) 
cm^{−1} u^{7/2} 
3.48(37) × 10^{−6} 
6 
1 
MHz 
−0.01456(76) 
cm^{−1} u^{7/2} 
−3.79(20) × 10^{−7} 
0 
2 
MHz 
−51.44722(91) 
cm^{−1} u^{2} 
−0.00152786(15) 
1 
2 
MHz 
0.8253(24) 
cm^{−1} u^{5/2} 
2.3594(68) × 10^{−5} 
2 
2 
MHz 
−0.0642(15) 
cm^{−1} u^{3} 
−1.779(42) × 10^{−6} 
3 
2 
MHz 
0.00269(37) 
cm^{−1} u^{7/2} 
7.24(99) × 10^{−8} 
4 
2 
MHz 
−0.001460(28) 
cm^{−1} u^{4} 
−3.806(72) × 10^{−8} 
0 
3 
MHz 
0.0037950(79) 
cm^{−1} u^{3} 
1.0931(36) × 10^{−7} 
1 
3 
MHz 
−1.339(45) × 10^{−4} 
cm^{−1} u^{7/2} 
−3.60(12) × 10^{−9} 
2 
3 
MHz 
−1.25(18) × 10^{−5} 
cm^{−1} u^{4} 
−3.26(47) × 10^{−10} 
3 
3 
MHz 
−4.26(31) × 10^{−6} 
cm^{−1} u^{9/2} 
−1.075(79) × 10^{−10} 
0 
4 
MHz 
−4.47(16) × 10^{−7} 
cm^{−1} u^{4} 
−1.166(42) × 10^{−11} 
1 
4 
MHz 
−3.49(29) × 10^{−8} 
cm^{−1} u^{9/2} 
−8.80(74) × 10^{−13} 
0 
5 
MHz 
3.72(88) × 10^{−11} 
cm^{−1} u^{9/2} 
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 _{lm} constants and isotopically invariant U^{y}_{lm} parameters determined in the multiisotopologue fit for NH
Dunham type 
Isotopically invariant 
Notes. The Dunham constants _{lm} are referred to the most abundant NH isotopologue. The BOB coefficients Δ^{X}_{lm} 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 u^{1/2} 
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 u^{3/2} 
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 u^{3/2} 
0.0026750(27) 
γ
_{20}

MHz 
−1.387(79) 
U
^{
γ
}_{20}

MHz u^{3/2} 
−4.06(23) × 10^{−5} 
γ
_{30}

MHz 
0.370(21) 
U
^{
γ
}_{30}

MHz u^{3/2} 
1.054(61) × 10^{−5} 
γ
_{01}

MHz 
0.4631(31) 
U
^{
γ
}_{01}

MHz u^{2} 
1.3656(91) × 10^{−5} 
γ
_{11}

MHz 
−0.0291(81) 
U
^{
γ
}_{11}

MHz u^{3/2} 
−8.4(23) × 10^{−7} 
γ
_{21}

MHz 
0.0152(45) 
U
^{
γ
}_{21}

MHz u^{2} 
4.2(13) × 10^{−7} 
γ
_{31}

MHz 
−0.00333(72) 
U
^{
γ
}_{31}

MHz u^{2} 
−9.0(19) × 10^{−8} 
δ
^{
λ,N}_{00}

MHz 
−3.59(16) 
Δ
^{
λ,N}_{00}


0.837(37) 
δ
^{
λ,H}_{00}

MHz 
−65.250(40) 
Δ
^{
λ,H}_{00}


1.08957(67) 
δ
^{
λ,N}_{10}

MHz 
1.934(40) 
Δ
^{
λ,H}_{10}


−49.0(10) 
δ
^{
γ,H}_{00}

MHz 
1.712(18) 
Δ
^{
γ,H}_{00}


3.515(38) 
δ
^{
γ,H}_{10}

MHz 
−0.091(17) 
Δ
^{
γ,H}_{10}


3.83(66) 

Hyperfine structure parameters 
b
_{
F,00}(H) 
MHz 
−64.194(13) 

cm^{−1} 
−0.00214129(45) 
b
_{
F,10}(H) 
MHz 
−3.785(15) 

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) 

cm^{−1} 
−3.2810(41) × 10^{−4} 
b
_{
F,10}(D) 
MHz 
−0.632(10) 

cm^{−1} u^{1/2} 
−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} u^{1/2} 
−1.19(32) × 10^{−5} 
c
_{20}(D) 
MHz 
−0.117(32) 
U
^{
c
}_{10}(D) 
cm^{−1} u^{1/2} 
−3.66(100) × 10^{−6} 
eQq_{00}(D) 
MHz 
0.080(32) 
U
^{eQq}_{00}(D) 
cm^{−1} 
2.7(11) × 10^{−6} 
b
_{
F,00}(^{14}N) 
MHz 
19.084(18) 

cm^{−1} 
6.3660(59) × 10^{−4} 
b
_{
F,10}(^{14}N) 
MHz 
−0.421(31) 

cm^{−1} u^{1/2} 
−1.365(99) × 10^{−5} 
b
_{
F,20}(^{14}N) 
MHz 
−0.088(13) 

cm^{−1} u^{1/2} 
−2.76(41) × 10^{−6} 
c
_{00}(^{14}N) 
MHz 
−68.135(31) 
U
^{
c
}_{00}(^{14}N) 
cm^{−1} u^{1/2} 
−0.0022727(10) 
c
_{10}(^{14}N) 
MHz 
0.467(26) 
U
^{
c
}_{10}(^{14}N) 
cm^{−1} u^{3/2} 
1.509(83) × 10^{−5} 
eQq_{00}(^{14}N) 
MHz 
−3.367(54) 
U
^{eQq}_{00}(^{14}N) 
cm^{−1} 
−1.123(18) × 10^{−4} 
eQq_{10}(^{14}N) 
MHz 
0.395(38) 
U
^{eQq}_{10}(^{14}N) 
cm^{−1} u^{1/2} 
1.28(12) × 10^{−5} 
C
_{00}(^{14}N) 
MHz 
0.172(11) 
U
^{
C
}_{00}(^{14}N) 
cm^{−1} u^{1/2} 
5.42(36) × 10^{−6} 
C
_{10}(^{14}N) 
kHz 
−0.0293(85) 
U
^{
C
}_{10}(^{14}N) 
cm^{−1} u^{3/2} 
−9.0(26) × 10^{−7} 
b
_{
F,00}(^{15}N) 
MHz 
−26.848(16) 

cm^{−1} 
−8.9556(52) × 10^{−4} 
b
_{
F,10}(^{15}N) 
MHz 
0.860(18) 

cm^{−1} u^{1/2} 
2.789(58) × 10^{−5} 
c
_{00}(^{15}N) 
MHz 
95.428(49) 
U
^{
c
}_{00}(^{15}N) 
cm^{−1} 
0.0031831(16) 
c
_{10}(^{15}N) 
MHz 
−0.556(54) 
U
^{
c
}_{10}(^{15}N) 
cm^{−1} 
−1.80(17) × 10^{−5} 
C
_{00}(^{15}N) 
MHz 
−0.259(22) 
U
^{
C
}_{00}(^{15}N) 
cm^{−1} u^{1/2} 
−8.18(71) × 10^{−6} 
C
_{10}(^{15}N) 
MHz 
0.099(24) 
U
^{
C
}_{00}(^{15}N) 
cm^{−1} u^{1/2} 
3.06(74) × 10^{−6} 
5 Discussion
5.1 Spectroscopic parameters
From the multiisotopologue 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 rootmeansquare deviations of the residuals computed separately for the rotational and rovibrational data are of the same order of magnitude of the corresponding measurements error, RMS_{ROT} = 0.107 MHz and RMS_{VIBROT} = 3.4 × 10^{−3} cm^{−1}, respectively. Then, the various sets of Y_{lm} 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 Dunhamtype 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 lindex and this is due to the smaller number of transitions available for highly vibrationally excited states. Finally, the Kratzer^{43} and Pekeris^{44} relation can also be used as a yardstick to asses the correct treatment of the Born–Oppenheimer Breakdown effects. Using the formula^{45} 
 (14) 
we obtained for Y_{02} 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 highresolution spectroscopic technique led to a very accurate determination of the equilibrium bond length r_{e} 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 r_{0}, which includes the zeropoint vibrational contributions and differs from r_{e}. 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 r_{e} for each isotopologue α. The equilibrium bond distance is given by: 
 (15) 
where N_{a}h is the molar Planck constant. Actually, the values of B^{(α)}_{e} differ from those of Y^{(α)}_{01} obtained from the Dunhamtype analysis. This discrepancy should be ascribed to a small contribution, expressed by:^{45} 
 (16) 
with 
 (17) 
and 
 (18) 
From eqn (15), it is evident that the bond length r_{e} assumes different values for each isotopologue. On the contrary, by substituting the product B^{(α)}_{e}μ_{a} with U_{01}, one obtains an isotopically independent equilibrium bond length r^{BO}_{e}. In the present case, r^{BO}_{e} takes the value of 103.606721(13) pm. In Table 6, this result is compared with the equilibrium bond distances calculated from the B_{e} of each isotopologue NH, ^{15}NH, ND, and ^{15}ND. In this case, B_{e} was obtained by correcting the corresponding Y_{01} constant according to eqn (16)–(18). It should be noticed that the values differ at subpicometre level but these differences, even if small, are detectable thanks to the highprecision of rotational measurements.
Table 6 Born–Oppenheimer and equilibrium bond distances (in pm) from the individual isotopologues (see text)
Species 
r
_{e}

r
_{
e
} − r^{BO}_{e} 
NH 
103.716377(16) 
0.109656 
^{15}NH 
103.715864(16) 
0.109143 
ND 
103.665420(10) 
0.058699 
^{15}ND 
103.664908(10) 
0.058187 

r
^{BO}_{e} = 103.606721(13) 
r
^{theor}_{e} = 103.5915 
The experimental value derived for r^{BO}_{e} 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 basisset extrapolation, corecorrelation effects, and inclusion of higherorder corrections due to the use of the full coupledcluster singles and doubles, augmented by a perturbative treatment of triple excitation [CCSD(T)] model
fcCCSD(T)/ccpV∞Z + Δcore/ccpCV5Z + ΔT/ccpVTZ. 
The computation have been performed using CFOUR.^{48} From this theoretical procedure we obtained r^{theor}_{e} = 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 Δ^{X}_{lm} determined in the present analysis account for the small inaccuracies of the Born–Oppenheimer approximation in describing the rovibrational energies of the imidogen radical. For the rotational constant (≈Y_{01}), it is possible to identify three different contributions to the corresponding BOB parameter^{49} 
 (19) 
namely an adiabatic contribution, a nonadiabatic term, and a Dunham correction, respectively. The last two terms on the right side of eqn (19) can be computed from purely experimental quantities: (Δ^{X}_{01})^{Dunh} arises from the use of a Dunham expansion and contains the term ΔY^{(Dunh)}_{01} of eqn (16), whereas (Δ^{X}_{01})^{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 g_{J} factors.^{38} The adiabatic term can be simply computed as the difference between the experimental Δ^{X}_{01} and the terms (Δ^{X}_{01})^{Dunh} and (Δ^{X}_{01})^{nad}.
Tiemann et al.^{40} found that the adiabatic term (Δ^{X}_{01})^{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 nonadiabatic contribution has been computed using the literature value of the dipole moment^{50}μ = 1.389 D and the ground state g_{J} value estimated from a laser magnetic resonance study,^{51}g_{J} = 0.001524.
Table 7 Contributions of the Born–Oppenheimer Breakdown coefficients to the U_{01} constant
Atom 
Δ
_{01} (exp) 
Adiabatic 
Nonadiabatic 
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, (Δ^{X}_{01})^{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

 (20) 
The adiabatic correction to the equilibrium bond distance, ΔR_{ad}, 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/ccpCVnZ level (n = 3, 4, 5), yielded ΔR_{ad} = 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 Zeropoint energy
The results of our analysis make possible to estimate the zeropoint energy (ZPE) for each isotopologue from the Dunham's constants Y_{lm} with m = 0, namely: 
 (21) 
As we determined anharmonicity constants up to the sixth order, the ZPE is derived with a negligible truncation bias^{53} from the expression:

 (22) 
The Y_{00} constant present in the Dunhamtype expansions is not experimentally accessible. Its value can be estimated, to a good approximation, through^{53}

 (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
where
σ_{f}^{2} is the variance in the function
f (
i.e.,
eqn (22) in the present case) of the set of parameters
Y_{l0}, whose variance–covariance matrix is
V, with the
ith element in the vector
g being
.
Table 8 Zeropoint 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. From ref. 22. Computed. From ref. 24. 
NH 
1623.5359(17) 
1623.6(6) 
1621.5^{a} 
^{15}NH 
1619.9485(17) 

1617.9^{b} 
ND 
1190.0859(11) 
1190.13(5) 
1189.5^{c} 
^{15}ND 
1185.1413(11) 

1183.6^{b} 
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 Y_{00}, which is nonnegligible 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 ^{15}ND in its ground electronic X^{3}Σ^{−} state has been recorded for the first time using a frequencymodulation submillimeterwave spectrometer. A global fit, including all previously reported rotational and rovibrational 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 Δ^{N}_{01} and Δ^{H}_{01} were evaluated and compared to theoretical estimates. The present analysis enables to predict rotational and rovibrational 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 r_{e} and the vibrational zeropoint 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 singlespecies and multiisotopologue fits. A reformatted list of all the transitions used in the Dunhamtype 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 N^{2}. 

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