The rotational spectrum of 15 ND. Isotopic-independent Dunham-type analysis of the imidogen radical
Mattia Melosso, Luca Bizzocchi, Filippo Tamassia, Claudio Degli Esposti, Elisabetta Canè, Luca Dore
TThe rotational spectrum of ND. Isotopic-independentDunham-type analysis of the imidogen radical † Mattia Melosso a , Luca Bizzocchi b , ∗ , Filippo Tamassia c , Claudio Degli Esposti a , Elisa-betta Canè c , and Luca Dore a , ∗ The rotational spectrum of ND in its ground electronic X Σ − state has been observed for the firsttime. Forty-three hyperfine-structure components belonging to the ground and v = vibrationalstates have been recorded with a frequency-modulation millimeter-/submillimeter-wave spectrom-eter. These new measurements, together with the ones available for the other isotopologues NH,ND, and NH, have been simultaneously analysed using the Dunham model to represent thero-vibrational, fine, and hyperfine energy contributions. The least-squares fit of more than 1500transitions yielded an extensive set of isotopically independent U lm parameters plus 13 Born–Oppenheimer Breakdown coefficients ∆ lm . As an alternative approach, we performed a Dunhamanalysis in terms of the most abundant isotopologue coefficients Y lm and some isotopically de-pendent Born–Oppenheimer Breakdown constants δ lm [R. J. Le Roy, J. Mol. Spectrosc. , 189(1999)]. The two fits provide results of equivalent quality. The Born–Oppenheimer equilibriumbond distance for the imidogen radical has been calculated [ r BO e = . ( ) pm] and zeropoint energies have been derived for all the isotopologues. The imidogen radical has been the subject of many spectro-scopic, computational and astrophysical studies. This diatomicradical belongs to the first-row hydrides, is commonly observed inthe combustion products of nitrogen-bearing compounds , andis also an intermediate in the formation process of ammonia in theinterstellar medium (ISM) . The main isotopologue of imidogen,NH, has been detected in a wide-variety of environments, fromthe Earths’ atmosphere to astronomical objects, such as comets ,many types of stars including the Sun , diffuse clouds , mas-sive star-forming (SF) regions and, very recently, in prestellarcores . Also its deuterated counterpart ND has been identified ∗ Corresponding author. a Dipartimento di Chimica “Giacomo Ciamician”, Università di Bologna, ViaF. Selmi 2, 40126 Bologna (Italy). E-mail: [email protected], [email protected] [email protected] b Center for Astrochemical Studies, Max-Planck-Institut für extraterrestrische Physik,Gießenbachstr. 1, 85748 Garching bei München (Germany) E-mail: [email protected] c Dipartimento di Chimica Industriale “Toso Montanari”, Università di Bologna, Vialedel Risorgimento 4, 40136 Bologna (Italy). E-mail: fi[email protected], [email protected] † Electronic Supplementary Information (ESI) available: The .LIN and .PAR filesfor the SPFIT programm are provided for both the single-species and multi-isotopologues fits. A reformatted list of all the transitions used in the Dunham-typeanalysis, together with their residuals from the final fit, is also included as ESI. SeeDOI: 10.1039/c8cp04498h in the ISM, towards the young solar-mass protostar IRAS16293 and in the prestellar core 16293E .A lot of studies have been devoted to the origin of interstellarimidogen and different formation models have been proposed toexplain its observed abundance in various sources. Two mainformation routes have been devised for the NH radical: fromthe electronic recombination of NH + and NH + , intermediates inthe synthesis of interstellar ammonia starting with N + or,alternatively, via dissociative recombination of N H + . How-ever, the mechanism of NH production in the ISM is still de-bated , and grain-surface processes might also play a significantrole . Imidogen, together with other light hydrides, often ap-pears in the first steps of chemical networks leading to more com-plex N-bearing molecules. Its observation thus provides crucialconstraints for the chemical modeling of astrophysical sources .Also the rare isotopologues of this radical yield important astro-chemical insights. Being proxies for N and D isotopic fractiona-tion processes, they may help to trace the evolution of gas anddust during the star formation, thus shedding light on the link be-tween Solar System materials and the parent ISM . This is par-ticularly relevant for nitrogen, whose molecular isotopic composi-tions exhibits large and still unexplained variations . Measur-ing the isotopic ratios in imidogen provides useful complementaryinformation on the already measured H/D, N/ N in ammonia(including the NH D species ).As far as the laboratory work is concerned, there is a substan- a r X i v : . [ a s t r o - ph . GA ] N ov ial amount of spectroscopic data for the most abundant speciesand less extensive measurements for NH and ND. A detailed de-scription of the spectroscopic studies of imidogen can be found inthe latest experimental works on NH , NH , and ND . Ithas to be noticed that no experimental data or theoretical com-putations were available in literature for the doubly substitutedspecies ND up to date. In this work, we report the first ob-servation of its pure rotational spectrum in the ground electronicstate X Σ − recorded up to 1.068 THz. A limited number of newtransition frequencies for the isotopologues NH and ND in the v = excited state have also been measured in the course of thepresent investigation. This new set of data, together with theliterature data for NH, NH and ND, have been analysed in aglobal multi-isotopologue fit to give a comprehensive set of iso-topically independent spectroscopic parameters. Thanks to thehigh precision of the measurements, several Born–OppenheimerBreakdown (BOB) constants ( ∆ lm ) could be determined from aDunham-type analysis. The alternative Dunham approach pro-posed by Le Roy has been also employed. In this case, theresults 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 (includingthe Born–Oppenheimer bond distance r BO e ) and Zero-Point Ener-gies (ZPE) for each isotopologue have been computed from thedetermined spectroscopic constants. The rotational spectrum of ND radical in its ground vibronicstate X Σ − has been recorded with a frequency-modulationmillimeter-/submillimeter-wave spectrometer. The primarysource of radiation was constituted by a series of Gunn diodes(Radiometer Physics GmbH, J. E. Carlstrom Co.) emitting in therange 80–134 GHz, whose frequency is stabilized by a Phase-Lock-Loop (PLL) system. The PLL allowed the stabilizationof the Gunn oscillator with respect to a frequency synthesizer(Schomandl ND 1000), which was driven by a 5 MHz rubidiumfrequency standard. Higher frequencies were obtained by usingpassive multipliers (RPG, × and × ). The frequency modulationof the output radiation was realized by sine-wave modulating at6 kHz the reference signal of the wide-band Gunn synchronizer.The signal was detected by a liquid-helium-cooled InSb hot elec-tron bolometer (QMC Instr. Ltd. type QFI/2) and then demod-ulated at 2 f by a lock-in amplifier. The experimental uncertain-ties of present measurements are between 40 and 80 kHz in mostcases, up to 500 kHz for a few disturbed lines.The ND radical was formed in a glow-discharge plasma withthe same apparatus employed to produce other unstable and rarespecies (e.g., ND and N H + ). The optimum productionwas attained in a DC discharge of a mixture of N (5–7 mTorr)and D (1–2 mTorr) in Ar as buffer gas (15 mTorr). Typically, avoltage of 1 kV and a current of 60 mA were employed. The ab-sorption cell was cooled down at ca. − ◦ C by liquid-nitrogencirculation.
Table 1
Spectroscopic constants determined for ND in the ground and v = vibrational states. Constant Unit v = v = B v / MHz . ( ) . ( ) D v / MHz . ( ) . a λ v / MHz . ( ) . a γ v / MHz − . ( ) − . ( ) γ Nv / MHz . ( ) . a b F , v ( N ) / MHz − . ( ) − . ( ) c v ( N ) / MHz . ( ) . ( ) C I , v ( N ) / MHz − . ( ) − . a b F , v ( D ) / MHz − . ( ) − . ( ) c v ( D ) / MHz . ( ) . ( ) eQq v ( D ) / MHz . ( ) . a σ b w rms / MHz 0.080no. of lines 34 9 Notes.
Number in parentheses are the σ statistical errors in unit of thelast quoted digit. ( a ) Parameter held fixed in the fit. ( b ) Fit standarddeviation.
From a spectroscopic point of view, imidogen is a free radicalwith a X Σ − ground electronic state and exhibits a fine structuredue to the dipole-dipole interaction of the two unpaired electronspins and to the magnetic coupling of the molecular rotation withthe total electron spin. The couplings of the various angular mo-menta in NH are described more appropriately by Hund’s case ( b )scheme J = N + S , (1)where N represents the pure rotational angular momentum. Eachfine-structure level is thus labeled by J , N quantum numbers,where J = N + , N , N − . For N = , only one component ( J = )exists. Inclusion of the nitrogen and hydrogen hyperfine interac-tions leads to the couplings F = J + I N , F = F + I H . (2)For each isotopologue in a given ro-vibronic state, the effectiveHamiltonian can be written as H = H rv + H fs + H hfs (3)where H rv , H fs and H hfs are the ro-vibrational, fine- andhyperfine-structure Hamiltonians, respectively: H rv = G v + B v N − D v N + H v N + L v N + M v N (4) H fs = (cid:16) λ v + λ Nv N (cid:17)(cid:16) S z − S (cid:17) + (cid:16) γ v + γ Nv N (cid:17) N · S (5) hfs = ∑ i b F , v ( i ) I i · S + ∑ i c v ( i ) (cid:18) I iz S z − I i · S (cid:19) + ∑ i eQq v ( i ) ( I iz − I i ) I i ( I i − ) + ∑ i C I , v ( i ) I i · N (6)Here, G v is the pure vibrational energy, B v the rotational con-stant, D v , H v , L v , and M v the centrifugal distortion parametersup to the fifth power in the ˆ N expansion, λ v and λ Nv are theelectron spin–spin interaction parameter and its centrifugal dis-tortion coefficient; γ v , and γ Nv are the electron spin–rotation con-stant and its centrifugal distortion coefficient, respectively. Theconstants b F , v and c v are the isotropic (Fermi contact interaction)and anisotropic parts of the electron spin–nuclear spin coupling, eQq v represents the electric quadrupole interaction and C I , v is thenuclear spin–rotation parameter. In Eq. (6) the index i runs overthe different nuclei present in a given isotopologue. The four nu-clear spins are: I = 1/2 for H and N and I = 1 for D and N. In order to treat the data of all the available isotopologues in aglobal analysis, it is convenient to adopt a Dunham-type expan-sion . The ro-vibrational energy levels are given by the equa-tion: E rv ( v , N ) = ∑ l , m Y lm (cid:0) v + (cid:1) l [ N ( N + )] m . (7)The fine- and hyperfine-structure parameters [i.e., λ v , λ Nv , γ v , γ Nv , b F , v , c v , eQq v , and C I , v in Eqs. (5)–(6)], are given by analogousexpansions: y ( v , N ) = ∑ l , m Y lm (cid:0) v + (cid:1) l [ N ( N + )] m , (8)where y ( v , N ) represents the effective value of the parameter y inthe ro-vibrational level labeled by ( v , N ) , and Y lm are the coeffi-cients of its Dunham-type expansion ∗ .The spectroscopic constants of Eqs. (4)–(6) can be expressed interms of the Dunham coefficients Y lm and Y lm . For example, theconstants G v , B v , and D v , of the ro-vibrational Hamiltonian aregiven by the following expansions: G v = ∑ l = Y l (cid:0) v + (cid:1) l , (9a) B v = ∑ l = Y l (cid:0) v + (cid:1) l , (9b) D v = ∑ l = Y l (cid:0) v + (cid:1) l . (9c)Each fine- and hyperfine-structure constant is also expressed bysuitable expansions. For example, the electron spin-rotation con-stant and its centrifugal dependence in a given vibrational state v ∗ This ( v , N )-factorisation is possible because all the angular momentum operatorsmultiplying the coefficients of Eqs. (5) and (6) commute with purely vibrationaloperators and with N . can be expressed as: γ v = ∑ l = γ l (cid:0) v + (cid:1) l , (10a) γ Nv = ∑ l = γ l (cid:0) v + (cid:1) l , (10b)where γ l and γ l are the Y lm constants of Eq. 8 relative to thespin-rotation interaction. For a given isotopologue α , a specificset of Dunham constants Y ( α ) lm and Y ( α ) lm is defined. Such constantscan be described in terms of isotopically invariant parameters us-ing the known reduced mass dependences given by Y ( α ) lm = U lm µ − ( l / + m ) α (cid:34) + m e (cid:32) ∆ N lm M ( α ) N + ∆ H lm M ( α ) H (cid:33)(cid:35) , (11a) Y ( α ) lm = U ylm µ − ( l / + m + p ) α (cid:34) + m e (cid:32) ∆ y , N lm M ( α ) N + ∆ y , H lm M ( α ) H (cid:33)(cid:35) , (11b)where M ( α ) X (with X = N , H ) are the atomic masses, µ α is thereduced mass of the α isotopologue, and m e is the electronmass. U lm and U ylm are isotopically invariant Dunham constants,whereas ∆ X lm and ∆ y , X lm are unitless coefficients which account forthe Born–Oppenheimer Breakdown . In Eq. (11b), p = for y = λ , b F , c , eQq , while p = for y = γ , C I . This extra µ − factor inthe mass scaling is needed to account for the intrinsic N depen-dence of the spin–rotation constants . Here, the unknowns arethe U lm , U ylm coefficients and the corresponding ∆ X lm and ∆ y , X lm BOBcorrections.An alternative approach has been proposed by LeRoy , whereone isotopologue (usually the most abundant one) is chosen asreference species ( α = ), and the Dunham parameters Y ( α ) lm and Y ( α ) lm of any other species are obtained by the following mass scal-ing Y ( α ) lm = (cid:34) Y ( ) lm + ∆ M N M ( α ) N δ N lm + ∆ M H M ( α ) H δ H lm (cid:35)(cid:18) µ µ α (cid:19) ( l / + m ) , (12a) Y ( α ) lm = (cid:34) Y ( ) lm + ∆ M N M ( α ) N δ y , N lm + ∆ M H M ( α ) H δ y , H lm (cid:35)(cid:18) µ µ α (cid:19) ( l / + m + p ) . (12b)Here, ∆ M X = M ( α ) X − M ( ) X (with X = N , H ) are the mass differencesproduced by the isotopic substitution, with respect to the refer-ence 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 ∆ X lm of Eqs. 11 through the simple relation ∆ X lm = δ X lm M ( ) X m e (cid:16) Y ( ) lm + δ N lm + δ H lm (cid:17) − . (13)Albeit formally equivalent, this latter parametrisation was intro-duced to overcome a number of deficiencies of the traditionaltreatment which were pointed out by Watson and Tiemann ,and its features are discussed in great detail in the original pa-per . An obvious advantage of the alternative mass scalingof Eqs. (12) is that the fitted coefficients are all expressed in requency units and are directly linked to the familiar spectro-scopic parameters of the reference isotopologue (e.g., Y ( ) ≈ ω e , Y ( ) ≈ − ω e x e , Y ( ) ≈ B e , Y ( ) ≈ − α e , etc.). Furthermore, the BOBcontributions are accounted for using purely addictive terms thusreducing the correlations among the parameters. ND spectrum
For the previously unobserved ND species, we have recorded34 lines for the ground vibrational state and 9 lines for the v = state. They include the complete fine-structure of the N = ← transition and the strongest fine-components of the N = ← transition for the ground state (see Figure 1), and the ∆ J = , + components of the N = ← transition for the v = state. Thecorresponding transition frequencies were fitted to the Hamilto-nian of Eqs. (4)–(6) using the SPFIT analysis program . Be-cause of the small number of transitions detected for the v = state, some of the spectroscopic parameters for this state couldnot be directly determined in the least-squares fit and were con-strained to the corresponding ground state values. The two setsof constants for v = and v = states are reported in Table 1.The list of observed frequencies, along with the residuals fromthe single-species fit, is given in Table 2. In addition, the .LINand .PAR input files for the SPFIT programm are included in thesupplementary material † . In this work, we carried out a multi-isotopologue Dunham fit ofthe imidogen radical in its X Σ − ground electronic state using ournewly measured transition frequencies for the doubly substituted ND variant plus all the available rotational and ro-vibrationaldata for the NH, NH and ND species. To take into account thedifferent experimental precision, each datum was given a weightinversely proportional to the square of its estimated measurementerror, w = / σ . The σ values adopted for the present measure-ments have been discussed in § 2 , while for literature data, weretained the values provided in each original work.The content of the data set and the relevant bibliographic ref-erences are summarised in Table 3. In total, the data set contains1563 ro-vibrational transitions which correspond to 1201 distinctfrequencies. These data were fitted to the multi-isotopologuemodel described in §§ 3.1–3.2, using both traditional [Eqs. (11)]and LeRoy [Eqs. (12)] mass scaling schemes to describe theDunham-type parameters ( Y lm and Y lm ) of each isotopic species.The analysis was performed using a custom P YTHON codewhich uses the SPFIT program as computational core. Briefly,the scripting routine reads the atomic masses, the spin multiplic-ities, and the Y lm constants for the reference species. Then, theSPFIT parameter file (.PAR) is set up by defining several setsof spectroscopic constants (one for each isotopologue/vibrationalstate), taking into account the mass scaling factors. The SPFITlines file (.LIN) is created by colletting the experimental data. Inthis process, half integer quantum numbers are rounded up anda "quantum number state" is assigned to each isotopologue in agiven vibrational state, in conformity with the SPFIT format. At N = 0 N = 1 N = 2 J = 1 J = 0 J = 2 J = 1 J = 1 J = 3 J = 2 »» E n e r g y / c m - a) b) c) d) a) b) c) d) Fig. 1
Upper panel : energy levels scheme of ND in the ground vi-brational state. The hyperfine-structure is not shown. The arrows markthe transitions observed in this study: ∆ J = + (red), ∆ J = (blue), and ∆ J = − (green). Lower panel : spectral recordings for the transitionsmarked with the labels a , b , c and d showing the corresponding hyperfinestructure. The brown sticks represent the positions and the intensities ofthe hyperfine components computed from the spectroscopic parametersof Table 1. the end of the least-squares optimisation, the SPFIT output ispost-processed, and the final parameters list is reformatted in thesame fashion of the initial input data set. The atomic masses usedwere taken from the Wang et al. compilation. The optimisedparameters are reported in Tables 4 and 5, while the complete listof all the fitted data, together with the residuals from the multi-isotopologue analysis, is provided as supplementary material (the.LIN and .PAR files are also provided) † . From the multi-isotopologue analysis we obtained a highlysatisfactory fit. Its quality can be evaluated in several ways.First of all, we were able to reproduce the input data withintheir estimated uncertainties: the overall standard deviation ofthe weighted fit is σ = . , and the root-mean-square devia-tions of the residuals computed separately for the rotational andro-vibrational data are of the same order of magnitude of thecorresponding measurements error, RMS ROT = 0.107 MHz andRMS
VIBROT = 3.4 × − cm − , respectively. Then, the varioussets of Y lm for a given m constitute a series whose coefficientsdecrease in magnitude for increasing values of the index l , asexpected for a rapidly converging Dunham-type expansion. In eneral, most of the determined coefficients have a relative errorlower than %. Higher errors are observed only for those con-stants with high l -index and this is due to the smaller numberof transitions available for highly vibrationally excited states. Fi-nally, the Kratzer and Pekeris relation can also be used as ayardstick to asses the correct treatment of the Born–OppenheimerBreakdown effects. Using the formula Y (cid:39) Y Y , (14)we obtained for Y a value of 51.54051 MHz which compareswell with the fitted one of 51.44722(91) MHz. The precision yielded by the high-resolution spectroscopic tech-nique led to a very accurate determination of the equilibriumbond length r e for the imidogen radical. The rotational mea-surements of a diatomic molecule in its ground vibrational state( v = ) allow the determination of precise value of r , which in-cludes the zero point vibrational contributions and differs from r e . This latter is determinable from the rotational spectrum in atleast one vibrationally excited state. In the present analysis, dataof four isotopic species in several vibrational excited states havebeen combined, allowing for a very precise determination of r e for each isotopologue α . The equilibrium bond distance is givenby: r ( α ) e = (cid:115) N a h π B ( α ) e µ a . (15)where N a h is the molar Planck constant. Actually, the values of B ( α ) e differ from those of Y ( α ) obtained from the Dunham-typeanalysis. This discrepancy should be ascribed to a small contribu-tion, expressed by : Y = B e + ∆ Y ( Dunh ) = B e + β (cid:18) B e ω e (cid:19) , (16)with β = Y (cid:32) Y Y (cid:33) + a (cid:18) Y Y (cid:19) − a − a + a , (17)and a = Y √− Y Y − . (18)From Eq. (15), it is evident that the bond length r e assumes differ-ent values for each isotopologue. On the contrary, by substitutingthe product B ( α ) e µ a with U , one obtains an isotopically indepen-dent equilibrium bond length r BO e . In the present case, r BO e takesthe value of 103.606721(13) pm. In Table 6, this result is com-pared with the equilibrium bond distances calculated from the B e of each isotopologue NH, NH, ND, and ND. In this case, B e was obtained by correcting the corresponding Y constant ac-cording to Eqs.. (16)–(18). It should be noticed that the valuesdiffer at sub-picometre level but these differences, even if small,are detectable thanks to the high-precision of rotational measure-ments. The experimental value derived for r BO e has been comparedwith a theoretically best estimate obtained following the prescrip-tions of Refs. 46,47. A composite calculation have been carriedout considering basis-set extrapolation, core-correlation effects,and inclusion of higher-order corrections due to the use of thefull coupled-cluster singles and doubles, augmented by a pertur-bative treatment of triple excitation [CCSD(T)] modelfc-CCSD(T)/cc-pV ∞ Z + ∆ core/cc-pCV5Z + ∆ T/cc-pVTZ . The computation have been performed using CFOUR . Fromthis theoretical procedure we obtained r theor e = . pm (seealso Table 6), which is in very good agreement with the experi-mentally derived value, the discrepancy being 15 fm. The BOB coefficients ∆ Xlm determined in the present analysisaccount for the small inaccuracies of the Born–Oppenheimer ap-proximation in describing the ro-vibrational energies of the imi-dogen radical. For the rotational constant ( ≈ Y ), it is possibleto identify three different contributions to the corresponding BOBparameter ∆ X = (cid:16) ∆ X (cid:17) ad + (cid:16) ∆ X (cid:17) nad + (cid:16) ∆ X (cid:17) Dunh = (cid:16) ∆ X (cid:17) ad + ( µ g J ) X (cid:48) m p + ∆ Y ( Dunh ) µ m e B e , (19)namely an adiabatic contribution, a non-adiabatic term, and aDunham correction, respectively. The last two terms on the rightside of Eq. (19) can be computed from purely experimental quan-tities: (cid:0) ∆ X (cid:1) Dunh arises from the use of a Dunham expansion andcontains the term ∆ Y ( Dunh ) of Eq. (16), whereas (cid:0) ∆ X (cid:1) nad dependson the mixing of the electronic ground state with nearby elec-tronic excited states, and can be estimated from the molecularelectric dipole moment µ and the rotational g J factors . Theadiabatic term can be simply computed as the difference betweenthe experimental ∆ X and the terms (cid:0) ∆ X (cid:1) Dunh and (cid:0) ∆ X (cid:1) nad .Tiemann et al. found that the adiabatic term (cid:0) ∆ X (cid:1) ad basi-cally depends on the corresponding X atom rather than on theparticular molecular species. Hence, it is interesting to derive thiscontribution in order to compare the results obtained for differ-ent molecules and to verify the reliability of the empirical fittingprocedure.All the contributions of Eq. (19) are collected in Table 7. Thenon-adiabatic contribution has been computed using the litera-ture value of the dipole moment µ = . D and the groundstate g J value estimated from a laser magnetic resonance study , g J = . .From the adiabatic contribution to the Born–OppenheimerBreakdown coefficients for the rotational constants, (cid:0) ∆ X (cid:1) ad , onemay derive the corresponding correction to the equilibrium bonddistance, a quantity which can also be accessed by ab initio com-putations. From our Eq. (11a) and Eq. (6) of Ref. 52, the follow- ng equality is obtained ∆ R ad = − r e (cid:20) m e M N (cid:16) ∆ N01 (cid:17) ad + m e M H (cid:16) ∆ H01 (cid:17) ad (cid:21) . (20)The adiabatic correction to the equilibrium bond distance, ∆ R ad ,can be theoretically estimated through the computation of theadiabatic bond distance, i.e., the minimum of the potential givenby the sum of the Born–Oppenheimer potential augmented by thediagonal Born–Oppenheimer corrections (DBOC) . The differ-ence between the equilibrium bond distances calculated with andwithout DBOC, with tight convergence limits, performed at theCCSD/cc-pCV n Z level ( n = , , ), yielded ∆ R ad = . pm. Thisvalue is in very good agreement with the purely experimentalone obtained by Eq. (20) which results . pm, thus providinga confirmation for the validity of our data treatment. The results of our analysis make possible to estimate the zero-point energy (ZPE) for each isotopologue from the Dunham’s con-stants Y lm with m = , namely: G ( ) = ∑ l = Y l (cid:18) (cid:19) l . (21)As we determined anharmonicity constants up to the sixth order,the ZPE is derived with a negligible truncation bias from theexpression: G ( ) = Y + Y − Y + Y + Y + Y + Y . (22)The Y constant present in the Dunham-type expansions is notexperimentally accessible. Its value can be estimated, to a goodapproximation, through Y (cid:39) B e + α e ω e B e + α e ω e B e − ω e x e . (23)The value for the main isotopologue NH is 1.9987(12) cm − .The values obtained for the ZPE of the four isotopologues are col-lected in Table 8. For comparison, the values of literature are alsoreported. Our results agree well with those reported in the liter-ature , but our precision is more than one order of magnitudehigher. The errors on our ZPE values are ca. × − cm − andwere calculated taking into account the error propagation σ f = g T V g (24)where σ f is the variance in the function f (i.e., Eq. (22) inthe present case) of the set of parameters Y l , whose variance-covariance matrix is V , with the i th element in the vector g being ∂ f ∂ Y i .Discrepancies of ∼ cm − are observed by comparing our datawith those reported in Ref. 54 because their definition of the ZPEdoes not include the term Y , which is non-negligible for lightmolecules . These newly determined values should be used inthe calculation of the exoergicity values ∆ E of chemical reactionsrelevant in fractionation processes. In this work the pure rotational spectrum of ND in its groundelectronic X Σ − state has been recorded for the first time using afrequency-modulation submillimeter-wave spectrometer. A globalfit, including all previously reported rotational and ro-vibrationaldata for the other isotopologues of the imidogen radical, has beenperformed and yielded a comprehensive set of Dunham coeffi-cients. Moreover, the Born–Oppenheimer Breakdown constantshave been determined for 13 parameters and also the adiabaticcontribution of the terms ∆ N and ∆ H were evaluated and com-pared to theoretical estimates. The present analysis enables topredict rotational and ro-vibrational spectra of any isotopic vari-ant of NH at a high level of accuracy and to assist further astro-nomical searches of imidogen. From our results, very accuratevalues of the equilibrium bond distances r e and the vibrationalZero-Point Energies for the different isotopologues have been de-rived. Acknowledgements
This work was supported by Italian MIUR (PRIN 2015 "STARS inthe CAOS") and by the University of Bologna (RFO funds).
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Observed frequencies and residuals (in MHz) from the single-isotopologue fit of ND in the ground and first vibrational excited states.
State N (cid:48) J (cid:48) F (cid:48) F (cid:48) N ” J ” F ” F ” Obs. Freq. Obs.- Calc. Rel. weight v = . ( ) .
119 0 . . ( ) .
119 0 . . ( ) .
012 0 . . ( ) .
012 0 . . ( ) − .
229 0 . . ( ) − .
229 0 . . ( ) − . . ( ) . . ( ) − . . ( ) . . ( ) − . . ( ) . . ( ) . . ( ) − . . ( ) . . ( ) − .
009 0 . . ( ) − .
009 0 . . ( ) − .
015 0 . . ( ) − .
015 0 . . ( ) .
064 0 . . ( ) .
064 0 . . ( ) − .
042 0 . . ( ) − .
042 0 . . ( ) − .
122 0 . . ( ) − .
122 0 . . ( ) .
053 0 . . ( ) .
053 0 . . ( ) − . . ( ) − .
069 0 . . ( ) − .
069 0 . . ( ) − .
006 0 . . ( ) − .
006 0 . . ( ) − .
006 0 . . ( ) .
027 0 . . ( ) .
027 0 . . ( ) − .
043 0 . . ( ) − .
043 0 . . ( ) .
028 0 . . ( ) .
028 0 . . ( ) .
028 0 . . ( ) − .
006 0 . . ( ) − .
006 0 . . ( ) − .
006 0 . . ( ) − .
026 0 . . ( ) − .
026 0 . . ( ) − . . ( ) .
018 0 . . ( ) .
018 0 . . ( ) .
018 0 . . ( ) − .
025 0 . . ( ) − .
025 0 . . ( ) − .
025 0 . . ( ) − .
025 0 . . ( ) − .
025 0 . . ( ) − .
025 0 . . ( ) − .
025 0 . . ( ) .
031 0 . . ( ) .
031 0 . . ( ) .
031 0 . . ( ) .
031 0 . . ( ) .
031 0 . . ( ) .
031 0 . . ( ) .
047 0 . . ( ) .
047 0 . . ( ) .
047 0 . . ( ) .
047 0 . . ( ) .
047 0 . . ( ) .
013 0 . . ( ) .
013 0 . . ( ) − . . ( ) . v = . ( ) − . . ( ) .
054 0 . . ( ) .
054 0 . . ( ) . . ( ) − .
062 0 . . ( ) − .
062 0 . . ( ) . . ( ) . . ( ) − . . ( ) − . . ( ) . Notes.
Number in parentheses are the experimental uncertainties in units of the lastquoted digit. The relative weight is given only for blended transitions. able 3
Summary of the data used for the multi-isotopologue fit of imidogen
Pure rotational Ro-vibrationalno. of lines no of. vib states Refs. no. of lines no of. bands Refs.NH 96 2 Flores-Mijangos et al. , Lewen et al. , TW 451 6 Bernath ,Geller et al. , Ram and Bernath ND 144 5 Saito and Goto , Takano et al. , Dore et al. , TW 406 6 Ram NH 61 2 Bailleux et al. , Bizzocchi et al. – – ND 43 2 This work (TW) – –
Table 4
Ro-vibrational Dunham Y lm constants and isotopically invariant U lm parameters determined in the multi-isotopologue fit for imidogen radical Y lm U lm l m units value units value1 0 / cm − . ( ) cm − u / . ( ) − − . ( ) cm − u − . ( ) − . ( ) cm − u / . ( ) − − . ( ) cm − u − . ( ) − − . ( ) cm − u / − . ( ) − − . ( ) cm − u / − . ( ) . ( ) cm − u . ( ) − . ( ) cm − u / − . ( ) . ( ) cm − u . ( ) − . ( ) cm − u / − . ( ) × − − . ( ) cm − u − . ( ) × − . ( ) cm − u / . ( ) × − − . ( ) cm − u / − . ( ) × − − . ( ) cm − u − . ( ) . ( ) cm − u / . ( ) × − − . ( ) cm − u − . ( ) × − . ( ) cm − u / . ( ) × − − . ( ) cm − u − . ( ) × − . ( ) cm − u . ( ) × − − . ( ) × − cm − u / − . ( ) × − − . ( ) × − cm − u − . ( ) × − − . ( ) × − cm − u / − . ( ) × − − . ( ) × − cm − u − . ( ) × − − . ( ) × − cm − u / − . ( ) × − . ( ) × − cm − u / . ( ) × − X l m δ X lm ∆ X lm N 0 1 / MHz . ( ) − . ( ) N 1 1 / MHz − . ( ) − . ( ) H 1 0 / cm − . ( ) − . ( ) H 2 0 / cm − − . ( ) − . ( ) H 0 1 / MHz . ( ) − . ( ) H 1 1 / MHz − . ( ) − . ( ) H 0 2 / MHz − . ( ) − . ( ) H 0 3 / MHz . ( ) × − − . ( ) Notes.
The Dunham constants Y lm are referred to the most abundant NH isotopologue. The BOB coefficients ∆ X lm are adimensional.Number in parentheses are the σ statistical errors in units of the last quoted digit. able 5 Fine and hyperfine Dunham Y lm constants and isotopically invariant U ylm parameters determined in the multi-isotopologue fit for NH Dunham type Isotopically invariantFine structure parameters λ / MHz . ( ) U λ / MHz u . ( ) λ / MHz . ( ) U λ / MHz u / . ( ) × − λ / MHz − . ( ) U λ / MHz u − . ( ) × − λ / MHz . ( ) U λ / MHz u / . ( ) × − γ / MHz − . ( ) U γ / MHz u − . ( ) γ / MHz . ( ) U γ / MHz u / . ( ) γ / MHz − . ( ) U γ / MHz u / − . ( ) × − γ / MHz . ( ) U γ / MHz u / . ( ) × − γ / MHz . ( ) U γ / MHz u . ( ) × − γ / MHz − . ( ) U γ / MHz u / − . ( ) × − γ / MHz . ( ) U γ / MHz u . ( ) × − γ / MHz − . ( ) U γ / MHz u − . ( ) × − δ λ , N00 / MHz − . ( ) ∆ λ , N00 . ( ) δ λ , H00 / MHz − . ( ) ∆ λ , H00 . ( ) δ λ , H10 / MHz . ( ) ∆ λ , H10 − . ( ) δ γ , H00 / MHz . ( ) ∆ γ , H00 . ( ) δ γ , H10 / MHz − . ( ) ∆ γ , H10 . ( ) Hyperfine structure parameters b F , ( H ) MHz − . ( ) U b F ( H ) cm − − . ( ) b F , ( H ) MHz − . ( ) U b F ( H ) cm − − . ( ) × − c ( H ) MHz . ( ) U c ( H ) cm − . ( ) c ( H ) MHz − . ( ) U c ( H ) cm − − . ( ) × − C ( H ) MHz − . ( ) U C ( H ) cm − − . ( ) × − b F , ( D ) MHz − . ( ) U b F ( D ) cm − − . ( ) × − b F , ( D ) MHz − . ( ) U b F ( D ) cm − u / − . ( ) × − c ( D ) MHz . ( ) U c ( D ) cm − . ( ) × − c ( D ) MHz − . ( ) U c ( D ) cm − u / − . ( ) × − c ( D ) MHz − . ( ) U c ( D ) cm − u / − . ( ) × − eQe ( D ) MHz . ( ) U eQq ( D ) cm − . ( ) × − b F , ( N ) MHz . ( ) U b F ( N ) cm − . ( ) × − b F , ( N ) MHz − . ( ) U b F ( N ) cm − u / − . ( ) × − b F , ( N ) MHz − . ( ) U b F ( N ) cm − u / − . ( ) × − c ( N ) MHz − . ( ) U c ( N ) cm − u / − . ( ) c ( N ) MHz . ( ) U c ( N ) cm − u / . ( ) × − eQq ( N ) MHz − . ( ) U eQq ( N ) cm − − . ( ) × − eQq ( N ) MHz . ( ) U eQq ( N ) cm − u / . ( ) × − C ( N ) MHz . ( ) U C ( N ) cm − u / . ( ) × − C ( N ) kHz − . ( ) U C ( N ) cm − u / − . ( ) × − b F , ( N ) MHz − . ( ) U b F ( N ) cm − − . ( ) × − b F , ( N ) MHz . ( ) U b F ( N ) cm − u / . ( ) × − c ( N ) MHz . ( ) U c ( N ) cm − . ( ) c ( N ) MHz − . ( ) U c ( N ) cm − − . ( ) × − C ( N ) MHz − . ( ) U C ( N ) cm − u / − . ( ) × − C ( N ) MHz . ( ) U C ( N ) cm − u / . ( ) × − Notes.
The Dunham constants Y lm are referred to the most abundant NH isotopologue. The BOB coefficients ∆ X lm are adimensional.Number in parentheses are the σ statistical errors in units of the last quoted digit.
10 | 1–11 able 6
Born–Oppenheimer and equilibrium bond distances (in pm) fromthe individual isotopologues (see text).
Species r e r e − r BO e NH 103.716377(16) 0.109656 NH 103.715864(16) 0.109143ND 103.665420(10) 0.058699 ND 103.664908(10) 0.058187 r BO e = . ( ) r theor e = . Table 7
Contributions of the Born–Oppenheimer Breakdown coefficientsto the U constant Atom ∆ (exp) adiabatic non-adiabatic DunhamN -3.8592 -0.6515 -3.1326 -0.0751H -3.6874 -1.0379 -2.5744 -0.0751 Table 8
Zero Point Energies (in cm − ) of imidogen isotopologues. Species This work Ref. 53 Ref. 54NH 1623.5359(17) 1623.6(6) 1621.5 a NH 1619.9485(17) 1617.9 b ND 1190.0859(11) 1190.13(5) 1189.5 c ND 1185.1413(11) 1183.6 b Notes.
Number in parentheses are the σ statistical errors in unit of the lastquoted digit. ( a ) From Ref. . ( b ) Computed. ( c ) From Ref. ..