Terahertz spectroscopy of N{}^{18}O and isotopic invariant fit of several nitric oxide isotopologs

Terahertz spectroscopy of NO and isotopic invariant fit of several nitric oxide isotopologs

Holger S.P. Müller hspm@ph1.uni-koeln.de Kaori Kobayashi kaori@sci.u-toyama.ac.jp Kazumasa Takahashi Kazuko Tomaru Fusakazu Matsushima I. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany Department of Physics, Faculty of Science, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan
Abstract

A tunable far-infrared laser sideband spectrometer was used to investigate a nitric oxide sample enriched in O between 0.99 and 4.75 THz. Regular, electric dipole transitions were recorded between 0.99 and 2.52 THz, while magnetic dipole transitions between the and spin-ladders were recorded between 3.71 and 4.75 THz. These data were combined with lower frequency data of NO (unlabeled atoms refer to N and O, respectively), with rotational data of NO, NO, NO, and NO, and with heterodyne infrared data of NO to be subjected to one isotopic invariant fit. Rotational, fine and hyperfine structure parameters were determined along with vibrational, rotational, and Born-Oppenheimer breakdown corrections. The resulting spectroscopic parameters permit prediction of rotational spectra suitable for the identification of various nitric oxide isotopologs especially in the interstellar medium by means of rotational spectroscopy.

keywords:
nitric oxide, terahertz spectroscopy, electric dipole transitions, magnetic dipole transitions, fine structure, hyperfine structure
journal: Journal of Molecular Spectroscopy

1 Introduction

Nitric oxide, NO, is the only stable diatomic molecule with an odd number of electrons. It is, therefore, of great interest for fundamental sciences and in particular for molecular spectroscopy. This is an important reason for a large body of spectroscopic investigations into the ground electronic state of NO. Soon after an electron paramagnetic resonance study of NO in 1950 NO_ESR_1950 (), the first reports on its rotational spectrum in the ground vibrational state appeared NO_rot_1953 (); NO_rot_1956 (); NO_rot_1959 (). Further studies were carried out later on the main isotopolog NO_rot_1979 (); NO_rot_1980 (); NO_N-15-O_NO-18_rot_1991 (); NO_N-15-O_rot_1999 (); NO_N-15-O_NO-18_rot_1999 (), on NO NO_N-15-O_NO-18_rot_1991 (); NO_N-15-O_rot_1999 (); NO_N-15-O_NO-18_rot_1999 (), on NO NO_N-15-O_NO-18_rot_1991 (); NO_N-15-O_NO-18_rot_1999 (), and even on NO and NO NO-17_N-15-O-18_rot_1994 (); unlabeled atoms refer to N and O. The -doubling transitions in the radio-frequency (RF) and microwave (MW) regions were studied extensively for NO and NO in their ground vibrational states NO_N-15-O_RF_1972 () with additional data for NO NO_RF_1970 (); NO_RF_1976 (); NO_v=0+1_IR-RF-DR_1977 (); NO_v=0+1_IR-RF-DR_1981 (), even in its excited vibrational state NO_v=0+1_IR-RF-DR_1977 (); NO_v=0+1_IR-RF-DR_1981 (). The spin-orbit splitting in was determined directly from high-resolution observations of the    magnetic dipole spectrum of NO NO_N-15-O_rot_1999 (); NO_magnetic_1992 () and of NO NO_N-15-O_rot_1999 () near 123 cm. Numerous infrared (IR) studies have been carried out, mostly on the main isotopic species. Among those with experimental transitions frequencies we mention in particular heterodyne NO_heterdyne-IR_1986 () and Lamb-dip heterodyne studies NO_heterdyne-IR_1996 () of the fundamental vibrational band, which was also recorded with Fourier transform spectroscopy (FTS) NO_IR_1-0_1994 (); NO_IR_1-0-forbidden_1994 (); NO_IR_1-0_1995 (). Much higher vibrational levels were accessed through emission spectroscopy of the first NO_IR-emi_2-0_to_6-4_1979 (); NO_IR_emi_2-0_to_15-13+_1980 () and second overtone NO_IR-emi_10-7_to_22-19_1982 (). Isotopic data are also available, albeit to a lesser extent 565758_IR_1-0_+_1979 (); 5658_IR_1_2_3-0_1980 ().

The spectroscopy of NO is also important for diagnostic purposes. Nitric oxide is a minor constituent of Earth’s atmosphere with a prominent role in the catalytic decomposition of ozone in the stratosphere. However, it is not easily detected in the atmosphere employing rotational sepctroscopy because of its small dipole moment of 0.15872 (2) D NO_RF_1970 (), high-resolution IR spectroscopy of its fundamental band is commonly used instead. In fact, we are only aware of one report on microwave observations of atmospheric NO NO_atmo_rot_1992 (). The NRAO 11 m telescope on Kitt Peak was used to to record the transition of NO near 250.8 GHz. Filtering out emission with line widths larger than 1.5 MHz, they were sensitive only to NO at high altitudes of  km. Radio astronomy, on the other hand, was used frequently to observe NO in space. Nitric oxide was detected in the star-forming region Sagittarius B2(OH) NO_det_1978 () and in dark clouds NO_dark-cloud_1990 (). Its abundance in these dense molecular clouds is rather high, around that of CO NO_abundance_1992 (), so it was hardly surprising that it was also detected in external galaxies such as the star-burst galaxy NGC 253 NO_extragal_2003 (). Higher rotationally excited transitions of NO have been observed with the high-resolution instrument HIFI on board of the Herschel space observatory in the frame work of molecular line surveys of the prolific star-forming regions Orion KL Orion-KL_HIFI_2014 () and Sagittarius B2(N) Sgr-B2(N)_HIFI_2014 (). The Atacama Large Millimeter Array (ALMA), which is currently under construction, will provide not only very high spatial resolution, but also very high sensitivity and spectral resolution, which should facilitate the detection of minor isotopic species of NO.

We recorded magnetic dipole transitions of NO around 4 THz to determine the spin-orbit splitting directly. In addition, we recorded electric dipole transitions around 2 THz for better prediction of higher rotational states. The resulting data were combined with other NO rotational data to determine its spectroscopic parameters. Ultimately, the data were also combined with rotational data of other NO isotopologs and with heterodyne IR data of the main species for an isotopic invariant fit along with Born-Oppenheimer breakdown (BOB) corrections to derive, in turn, predictions of the rotational spectra of NO isotopic species for radio astronomy.

2 Experimental details and observed spectrum

The terahertz spectrometer used at the University of Toyama in the present study is a so-called Evenson-type tunable far-infrared spectrometer (TuFIR) based on a frequency synthesizing technique developed by Evenson and co-workers exp_1 (). Details of the spectrometer can be found elsewhere exp_2 (). The basic principle is the stable far-infrared genaration by the difference frequency generated from two frequency-stabilized CO lasers. The difference frequency is mixed with the microwave radiation from a synthesized sweeper on a metal-insulator-metal (MIM) diode to achieve tunability. Two side bands (upper and lower) are generated. The frequency of the absorption can be determined by the phase of the signal. A liquid-helium-cooled Si bolometer is used to detect the terahertz radiation. The 1 detection signal from the lock-in amplifier was recorded with a computer. A path length modulator was inserted into the terahertz path in order to eliminate standing waves.

Two glass cells, 250 cm or 40 cm long, were used for most of the measurements. The isotopically enriched NO (Shoko Co. Ltd., 97 % O) was used without further purification. The sample pressure was maintained at about 712 Pa for the pure rotational (electronic dipole) transitions and at 120 Pa for the weak magnetic transitions. All measurements were carried out at room temperature.

Figure 1: Terahertz spectrum of NO in the region of the   , fine structure transition with resolved N hyperfine splitting.
Frequency OC
998525 .402 0 .037
998824 .023 0 .061
1024876 .524 0 .074
1093603 .395 0 .012
1093894 .752 0 .061
1122177 .958 0 .027
1122224 .596 0 .065
1188668 .423 0 .017
1188952 .059 0 .026
1219450 .321 0 .074
1219504 .465 0 .012
1283718 .135 0 .012
1283993 .739 0 .024
1316649 .397 0 .027
1316711 .601 0 .044
1378750 .049 0 .040
1379017 .279 0 .026
1413770 .721 0 .029
1413841 .225 0 .019
1473761 .367 0 .031
1474020 .025 0 .065
1510810 .030 0 .008
1510889 .084 0 .039
1568749 .227 0 .051
1568998 .950 0 .001
1853539 .862 0 .038
1853761 .992 0 .033
1898181 .516 0 .036
2516576 .862 0 .009
2516732 .584 0 .046

Uncertainty 50 kHz for each rotational line.

Table 1: Rotational (electric dipole) transitions of NO, frequency (MHz) and residuals OC (MHz) between observed frequency and that calculated from the isotopic invariant fit.
Frequency OC
10.5 10.5 3719850 .891 0 .225
9.5 9.5 3719910 .708 0 .097
8.5 8.5 3719968 .880 0 .152
8.5 8.5 3722980 .772 0 .047
9.5 9.5 3723030 .929 0 .052
10.5 10.5 3723081 .461 0 .027
11.5 11.5 3745903 .024 0 .302
10.5 10.5 3745963 .357 0 .121
9.5 9.5 3746023 .090 0 .049
9.5 9.5 3749294 .640 0 .089
10.5 10.5 3749343 .231 0 .055
11.5 11.5 3749391 .245 0 .028
5.5 4.5 4053636 .418 0 .057
4.5 3.5 4053694 .601 0 .000
3.5 2.5 4053738 .502 0 .054
3.5 2.5 4054966 .173 0 .226
4.5 3.5 4055021 .910 0 .696
5.5 4.5 4055093 .835 0 .279
7.5 6.5 4274260 .709 0 .446
6.5 5.5 4274321 .120 0 .416
5.5 4.5 4274371 .189 0 .056
5.5 4.5 4276288 .069 0 .396
6.5 5.5 4276341 .083 0 .024
7.5 6.5 4276404 .386 0 .417
8.5 7.5 4388268 .502 0 .117
7.5 6.5 4388330 .480 0 .058
6.5 5.5 4388383 .531 0 .004
6.5 5.5 4390643 .905 0 .072
7.5 6.5 4390695 .621 0 .274
8.5 7.5 4390756 .112 0 .257
10.5 9.5 4623467 .645 0 .168
9.5 8.5 4623532 .062 0 .214
8.5 7.5 4623589 .493 0 .149
8.5 7.5 4626538 .285 0 .095
9.5 8.5 4626588 .085 0 .253
10.5 9.5 4626642 .944 0 .211
11.5 10.5 4744561 .295 0 .439
10.5 9.5 4744627 .399 0 .063
9.5 8.5 4744686 .692 0 .032
9.5 8.5 4747980 .294 0 .194
10.5 9.5 4748028 .065 0 .254
11.5 10.5 4748081 .730 0 .024

[2pt]

Uncertainty 250 kHz for each fine structure line.

Table 2:    magnetic dipole (fine structure) transitions of NO, frequency (MHz) and residuals OC (MHz) between observed frequency and that calculated from the isotopic invariant fit.

The NO rotational transitions were found easily based on predictions generated from previous work NO_N-15-O_NO-18_rot_1991 (); NO_N-15-O_NO-18_rot_1999 (). Hyperfine structure (HFS) was not resolved in these transitions, and good signal-to-noise ratios (SNR) were obtained. Uncertainties of 50 kHz were assigned to these data which are gathered in Table 1. Combining our new data with the previous ones and taking into account the fine structure (FS) splitting in NO and NO NO_N-15-O_rot_1999 (), the weaker magnetic dipole transition were observed readily. As can bee seen in Fig. 1, HFS was resolved in these spectral recordings, and the SNR were reasonable. We assigned uniformly 250 kHz as uncertainties to these transition frequencies mainly because of the lower SNR, but also because of the larger line width caused by pressure broadening. The magnetic dipole transitions are summarized in Table 2.

3 Spectroscopic analysis

NO is a stable radical with a regular ground electronic state, i.e., the spin ladder is at lower energies than the spin ladder. It has a fairly small electric dipole moment of 0.15872 (2) D NO_RF_1970 (). As a diatomic consisting of two fairly light atoms, its spin-orbit splitting is comparatively small (3.7 THz) while its rotational constant is with 51 GHz fairly large. As a consequence, its spectrum is close to Hund’s case (a) at lower rotational quantum numbers, but closer to Hund’s case (b) at intermediate and higher rotational quantum numbers.

The effective Hamiltonian suitable to fit the rotational spectrum of NO has been described rather often, and a rather detailed description can be found in Ref. NO_N-15-O_rot_1999 (). Further discussion on the Hamiltonian of a molecule in terms of Hund’s cases (a) and (b) can be found elsewhere radi-Hamiltonian (). Pickett’s SPCAT and SPFIT programs spfit_1991 () were used for prediction and fitting of the NO spectra. The programs were intended to be rather general, thus being able to fit asymmetric top rotors with spin and vibration-rotation interaction in support of the spectral line lists of the Jet Propulsion Laboratory (JPL) JPL-catalog_1998 () and Cologne Database for Molecular Spectroscopy (CDMS) CDMS_1 (); CDMS_2 (). Hund’s case (b) quantum numbers are employed in SPCAT and SPFIT whereas Hund’s case (a) quantum numbers are more common for NO. We follow the latter labeling in Fig. 1 and Tables 1 and 2. Conversion of Hund’s case (b) quanta to case (a) or vice versa depends on the magnitude of the rotational energy relative to the magnitude of the spin-orbit splitting. For , levels with correlate with and levels with correlate with ; for larger values of , the correlation is reversed. In the case of the NO isotopologs, the reversal occurs between and .

During the fitting process, we contained a spectroscopic parameter in the fit if it reduced the rms error of the fit, as measure of the quality of the fit, by an appreciable amount. This meant in most instances that the parameter was determined with great significance, meaning its uncertainty in the fit was about one fifth of the magnitude of its value or less. Care was also taken to evaluate which parameter reduced the rms error by the greatest amount.

Among the available data of one isotopic species and within one vibrational state, we used those, which were most accurate because data with larger uncertainties have considerably lower weights in the fit; the weight of a datum in the fit scales inversely to the square of the uncertainty. In a few cases, multiple data were used if the uncertainties were similar. We scrutinized the reported uncertainties in all instances. For the great majority of the data, the reported values were employed in the fit. Few transition frequencies were omitted from the fit if their residuals in the fits were much larger than the reported uncertainties. In few other cases with large residuals, the uncertainties were increased. Some uncertainties appeared to be conservative, and they were reduced somewhat. Details will be given below.

In order to evaluate NO spectroscopic parameters, we combined our data with the lower frequency data from Saleck et al. NO_N-15-O_NO-18_rot_1991 (). Uncertainties assigned to the transition frequencies pertaining to the lowest quantum numbers of the ladder ( = 1.50.5 and 2.51.5) appeared to be too conservative, not only for NO NO_N-15-O_NO-18_rot_1991 () in the single isotopolog fit, but also in the combined fit and for NO NO_N-15-O_NO-18_rot_1991 (), and for NO and NO NO-17_N-15-O-18_rot_1994 (); therefore, we reduced the uncertainties somewhat for these transitions. We omitted the data from Ref. NO_N-15-O_NO-18_rot_1999 () because they had slightly to considerably larger uncertainties, and the data with only slightly larger uncertainties had residuals frequently much larger than the quoted uncertainties.

The initial spectroscopic parameter set consisted of those employed for NO and NO NO_N-15-O_rot_1999 (). It is worthwhile mentioning that was used there and in the present fits whereas most other NO parameter sets employed instead. and make essentially indistinguishable contributions in a radical, and the same holds for their vibrational or rotational corrections Veseth_2Pi_1971 (); only one of the two parameters can be determined usually. One way to resolve the indeterminacy is an isotopic invariant fit AD_gamma_ii_1977 (), which will be described for NO in the following part of this section. If at least one of the atoms of the molecule has a non-zero spin, the combination of HFS and Zeeman effects may allow to disentangle and AD_gamma_Zeeman_2002 (); it turned out that in NO the contributions come almost entirely from , whereas dominated the contributions in FO AD_gamma_Zeeman_2002 (). Several of the initial parameters in the NO fit were poorly determined and were omitted successively from the fit without increasing the rms error much. The nuclear spin-rotation parameter was retained in the fit despite being not determined significantly because its omission increased the rms error by more than 5 % and because its value was correct within its uncertainty. The final set of NO spectroscopic parameters is given in Table 3.

Parameter Value
48 211 .775 59 (114)
147 .517 3 (35)
32 .7 (30)
3 691 991 .767 (53)
184 .205 (41)
1 .420 (110)
332 .201 8 (70)
2 .536 0 (18)
37 .3 (25)
84 .214 (39)
22 .425 (142)
58 .871 (200)
112 .585 (11)
1 .837 (27)
23 .76 (74)
12 .8 (47)

[2pt]

Numbers in parentheses are 1  uncertainties in units of the least significant figures.

Table 3: Spectroscopic parameters (MHz) of NO from a single isotopolog fit.

The ro-vibrational energy levels of a diatomic molecule AB, such as NO, can be represented by the Dunham expression Dunham_1932 ()

(1)

where the are the Dunham parameters. In electronic states different from states, i.e. in states with orbital angular momentum , the expansion in is often replaced by an expansion in , see, e.g., Ref. Dunham_BO_Watson2 (). The ground electronic state of NO is (), and the expansion is often carried out in , and this expansion was used here. The expansion in is quite common also, see, e.g., the case of the radical BrO BrO_rot_2001 ().

Several isotopic species of AB can be fit jointly by constraining the to Dunham_BO_Watson2 (); Dunham_BO_Watson1 ()

(2)

where is isotopic invariant, is the mass of the electron, is the reduced mass of AB, is the mass of atom A, and is a BOB term assiciated with atom A. The abbreviation is sometimes used for . We need to point out that both and are defined negatively in some publications. Obviously, and are defined equivalently.

Rotational and vibrational corrections to the -doubling, FS, and HFS parameters have been expressed analogously as in Eqs. (1) and (2), the isotopic dependences were given explicitly, e.g., in Refs. radi-Hamiltonian (); BrO_rot_2001 (). Briefly, the lowest order fine structure parameters and scale with 1 and , respectively. The -doubling parameters and scale with and , respectively. The electron spin-nuclear spin coupling parameters , , , and all scale with the respective nuclear factor . In the case of nitrogen, both N and N have non-zero spins ( and 0.5, respectively). The N/N factor ratio is 1.4027548 moments_1989 (); magnetic_moments_2012 (). There is only one oxygen nucleus, O, with non-zero spin of 5/2, so no factor ratios needed to be considered for oxygen substitution. The lowest order quadrupole parameters, , , and , all scale with the quadrupole moment , but there is only one nucleaus for each atom with , N and O. The lowest order nuclear spin-rotation parameters and scale with .

Isotopic invariant fits were carried out for numerous diatomics, among them BrO BrO_rot_2001 (), CdH CdH_ii_2004 (); ZnH_CdH_ii_2006 (), ZnH ZnH_CdH_ii_2006 (), CH CH+_fitting_2010 (), and O O2_1Delta_2012 (); O2_3states_2012 ().

The atomic masses were taken from a recent compilation AME_2012 (). It includes recent improvements for N Mass-14N_2004 (), O Mass-18O_2009 (), and O Mass-17O_2010 (). Among these, the updated O value is the most relevant one for high resolution spectroscopy.

The aim of the present study was modeling of the ground state rotational spectra of NO isotopic species. However, in order to separate contributions of the breakdown of the Born-Oppenheimer approximation to a certain spectroscopic parameter from the frequently larger vibrational correction to this parameter, see e.g., Refs. CdH_ii_2004 (); ZnH_CdH_ii_2006 (), we needed to consider some information on vibrationally excited NO. These were NO -doubling data NO_v=0+1_IR-RF-DR_1977 (); NO_v=0+1_IR-RF-DR_1981 () along with heterodyne = 10 IR transition frequencies NO_heterdyne-IR_1986 (); NO_heterdyne-IR_1996 (). A subsequent study will consider the extensive available IR data. The study should include not only experimental transition frequencies with appropriate uncertainties, but also intensity information from experimental measurements as well as from empirical and theoretical modeling NO_EDMF_2014 ().

We started the combined analysis by determining spectroscopic parameters for the ground vibrational state of the main isotopolog. As in Ref. NO_N-15-O_rot_1999 (), electric and magnetic dipole transitions in the THz region were taken from that work, and RF and MW -doubling data were taken from Refs. NO_N-15-O_RF_1972 (); NO_RF_1970 (); NO_RF_1976 (); NO_v=0+1_IR-RF-DR_1977 (); NO_v=0+1_IR-RF-DR_1981 (). We used also unpublished data from Pickett et al. NO_rot_1979 () which had been used in prior analyses NO_rot_1980 (); NO_N-15-O_NO-18_rot_1991 (); NO_heterdyne-IR_1996 (). These data were not only of similar accuracy as the THz data NO_N-15-O_rot_1999 (), but also extended from the lowest up to . In contrast, the THz data started at and for the and spin components, respectively NO_N-15-O_rot_1999 (). One transition frequency from Ref. NO_rot_1979 (), () kHz, was omitted from the fit because the residual was about five times the uncertainty. The three transition frequencies from Ref. NO_RF_1976 () with reported uncertaities of 1, 1, and 2 kHz, respectively, were assigned uncertainties of 3 kHz in accordance with the residuals. Ground and excited state -doubling data from Ref. NO_v=0+1_IR-RF-DR_1977 () appeared to be judged too conservatively with uncertainties of 20 or 25 kHz. They were reproduced to within 5 kHz for the most part even with the reported uncertainties. Thus we used 5 kHz as uncertainty for each of these lines. Approximately 30 kHz were reported as uncertainties for the ground and excited state -doubling data from Ref. NO_v=0+1_IR-RF-DR_1981 (). We assigned 20 kHz to all of the data and to most of the data; 50 kHz were attributed to the = 20.520.5 data which appeared to require a larger uncertainty of 30 or 50 kHz.

The choice of spectroscopic parameters for NO in its ground vibrational state was straightforward for the most part based on previous analyses NO_N-15-O_rot_1999 (); NO_heterdyne-IR_1996 (). The distortion parameter had uncertainties in the fits slightly smaller than in Ref. NO_N-15-O_rot_1999 (), but its value was much smaller in magnitude and not determined with significance; the value obtained with the final line list was ( MHz. Hence, it was omitted from the final fit, as was done in Ref. NO_heterdyne-IR_1996 (). No distortion correction was needed for any of the HFS parameters with the exception of . It was necessary to include (N) in the fit to reproduce the -doubling data from Ref. NO_N-15-O_RF_1972 () well. This parameter describes the difference in between the and spin components. It was used, e.g., in a previous study of BrO BrO_rot_2001 (). The approach in the original -doubling study NO_N-15-O_RF_1972 () was equivalent, because two independent parameters and were used to determine the N quadrupole coupling within the and substates, respectively.

Parameter Value
56 240 .216 66 (14)
51 119 .680 7 (42)
4 .469 2 (29)
4 .027 2 (27)
526 .763 3 (22)
163 .944 1 (30)
0 .044 7 (24)
0 .484 2 (55)
37 .940 (114)
3 695 104 .22 (65)
204 .98 (26)
167 .83 (38)
7 335 .247 (55)
0 .122 8 (59)
193 .40 (21)
7 .476 3 (55)
1 .611 0 (56)
350 .623 40 (91)
17 .11 (93)
403 .50 (32)
34 .1 (12)
2 .844 711 (39)
44 .282 (65)
42 .319 (112)
84 .304 2 (106)
202 .3 (211)
22 .271 (21)
249 . (43)
58 .890 4 (14)
112 .619 47 (132)
30 .3 (27)
105 .6 (145)
1 .898 6 (32)
77 .4 (64)
23 .112 6 (62)
6 .89 (83)
12 .293 (27)
7 .141 (123)
173 .058 3 (101)
35 .460 (109)
92 .871 (171)
206 .121 6 (70)
1 .425 (47)
30 .02 (163)
32 .7 (23)

[2pt]

Numbers in parentheses are 1  uncertainties in units of the least significant figures.

Table 4: Spectroscopic parameters (MHz) for NO determined from the isotopic invariant fit.

Vibrational corrections were evaluated next by including heterodyne IR measurements of the NO fundamental band NO_heterdyne-IR_1986 (); NO_heterdyne-IR_1996 () with reported uncertainties in the fit and subsequently the -doubling transition frequencies of NO in its excited vibrational state NO_v=0+1_IR-RF-DR_1977 (); NO_v=0+1_IR-RF-DR_1981 () with uncertainties as described above. The choice of parameters to be included in the fit was straightforward for the most part. After inclusion of in the fit at most two of the three vibrational corrections to , , and could be determined. The best result was obtained in the final fits with vibrational corrections to and . Each of these parameters led to a modest reduction of the rms error. In case of the quadrupole parameters, a vibrational correction was only needed for .

Inclusion of the NO -doubling transitions NO_N-15-O_RF_1972 () into the fit turned out to be challenging. In fact, whereas the data were fit very satisfactorily in the original study NO_N-15-O_RF_1972 (), none of the transition frequencies was reproduced there within the uncertainties. They showed deviations between more than 5 times to almost 70 times the quoted uncertainties NO_N-15-O_RF_1972 (). Interestingly, our initial trial fits of the NO data lead only to a rejection of the , transition of the ladder at () MHz because of a residual of 13 kHz. In the combined fit, two -doubling transition frequencies with and were omitted in addition because each one deviated from the calculated frequency by about 40 times the reported uncertainty; furthermore, two corresponding frequencies with deviated each by about 8 times the reported uncertainty and were omitted also. All other transitions were retained in the fit with the uncertainties as reported. The rotational NO_N-15-O_NO-18_rot_1991 (); NO_N-15-O_rot_1999 () and FS transition frequencies NO_N-15-O_rot_1999 () were included with the reported uncertainties, except for the modifications in the low- data NO_N-15-O_NO-18_rot_1991 () as mentioned above.

The inclusion of the NO data called for BOB parameters for and to be included in the fit, as was expected. In addition, BOB parameters were required for and because transition frequencies with very high rotational quantum numbers were determined for NO and for NO NO_N-15-O_rot_1999 ().

Subsequently, NO data were used in the fit as described above. Only BOB parameters for and were needed because the NO data did not reach as high quantum numbers as the NO and NO data. Inclusion of NO data did not afford any additional parameters. Finally, the NO data were used in the fit. Obviously, new parameters were necessary to account for the O HFS splitting; no other parameters were introduced to the fit. The final set of spectroscopic parameters determined in the fit is given in Table 4, derived parameters are presented in Table 5.

All input data were reproduced on average to within the uncertainties employed in the fit; the rms error of the fit is 0.924. There is some scatter among the various subdata sets, but none has residuals on average much larger than 1.0. Among the smallest values are -doubling transitions from Refs. NO_v=0+1_IR-RF-DR_1977 (); NO_v=0+1_IR-RF-DR_1981 (). The rms error of our NO data is 0.871, slightly better for the pure rotational data and slightly worse for the FS data.

Parameter Value
51 111 .184 2 (11)
2 .231 66 (147)
2 .296 99 (156)
163 .899 4 (27)
6 .96 (37)
3 695 477 .03 (21)
1 .416 0 (18)
1 .324 3 (30)
350 .606 29 (17)
1 .246 (68)

[2pt]

Numbers in parentheses are 1  uncertainties in units of the least significant figures.
Unitless.

Table 5: Derived spectroscopic parameters (MHz) of NO from the isotopic invariant fit.

4 Discussion and conclusion

We have reproduced extensive rotational data of several NO isotopologs along with heterodyne IR data in one isotopic invariant fit. The later inclusion of extensive rovibrational data may affect some parameters outside the present uncertainties. Moreover, additional vibrational and possibly BOB corrections will be required for some parameters. The NO energy difference, in particular, is merely a fitting parameter at present.

NO NS CO CS SiO SiS O SO
2 .2317 (15) 3 .424 (68) 2 .05603 (24) 2 .5434 (49) 1 .2976 (44) 1 .3935 (42) 1 .7353 (31) 1 .830 (56)
2 .2970 (16) 2 .856 (96) 2 .09934 (24) 2 .3945 (34) 2 .0507 (16) 1 .8728 (55) 1 .7353 (31) 2 .700 (24)
6 .96 (37) 6 .3978 (20) 4 .9 (57)
11 .0 (66)

[2pt]

Numbers in parentheses are 1  uncertainties in units of the least significant figures.
This work.
Ref. NS_isos_rot_1995 ().
Ref. CO_fit_2012 ().
Ref. CDMS_2 ().
Ref. SiO_rot_2013 ().
Ref. SiS_rot_2007 ().
Ref. O2_3states_2012 (), value for ; : 1.9144 (56), : 2.1333 (74).
Ref. SO_isos_rot_1982 ().

Table 6: Comparison of Born-Oppenheimer breakdown parameters and of NO with those of related molecules.

The hyperfine parameters, however, will not be affected by additional IR data because of the lower accuracy of the data and because HFS is not resolved in the FTS data. Contributions to the interpretation of the NO HFS parameters have been provided numerous times, e.g., in Refs. NO_rot_1956 (); NO_N-15-O_NO-18_rot_1999 (); NO_HFS_interpretation_1955 (); NS_rot_1969 (). The N HFS parameter of NO and NS are quite similar NS_rot_1969 () with the ones of NS being consistently smaller NS_rot_1969 (); NS_isos_rot_1995 (), hence the spin density is smaller at N in NS compared to NO.

An early RF study yielded a value  kHz with rather small uncertainty of less than 2 kHz NO_N-15-O_RF_1972 (), not in agreement with our value of  kHz, whereas an extended update of that RF study yielded  kHz NO_RF_1976 (). The parameter was also determined, e.g., for BrO BrO_rot_2001 (). If we scale that value of  MHz with the NO/BrO ratios of and , we obtain a value of  kHz, rather close to our NO value. The agreement should not be overinterpreted because deriving values of ClO and IO analogously from the BrO value yields 0.96 MHz and 141 MHz compared to experimental values of () MHz ClO_rot_2001 () and () MHz IO_rot_2001 (), respectively.

The values  MHz and  MHz, derived from Zeeman spectroscopy of NO AD_gamma_Zeeman_2002 (), and our values of  MHz or  MHz in and  MHz are in reasonable accordance. No uncertainties were quoted in the Zeeman study. But even if uncertainties were determined, it would be difficult to judge which values are more reliable. A combination of both methods should yield improved values. Both results demonstrate that fitting in rotational or rovibrational spectra of NO isotopologs is more appropriate than fitting even though the ratio is small with respect to if is omitted from the fit.

An isotopic invariant fit of rotational data of three NO isotopologs was carried out earlier NO_N-15-O_NO-18_rot_1991 (). Their values, and , compare very favorably with ours, and . Our uncertainties are considerably smaller because of extensive additional data for NO and NO NO_N-15-O_rot_1999 (), for NO from the present study, and data for NO and NO NO-17_N-15-O-18_rot_1994 ().

The available BOB parameters and of NO are compared in Table 6 with data of related diatomics. There are three contributions to Dunham_BO_Watson1 (): (i) a higher-order semiclassical term that originates in the Dunham formalism and is usually very small, (ii) a diabatic (or nonadiabatic) term that is proportional to the molecular -value , and finally, (iii) an adiabatic term that is derived from the experimental value by subtracting the two former contributions. The latter contribution appears to depend more on the two atoms in a given diatomic molecule than on specifics of this molecule BOB_4-6_1982 (). The second contribution is usually the dominant one, and its magnitude is particularly large if there exist low-lying electronic states of the same spin-multiplicity in the molecule Dunham_BO_Watson1 ().

The similarity of the NO and CO values should not be overinterpreted because values for, e.g., CO are quite different:  (14) and  (33) CO+_rot_2013 (). The values in Table 6 cover a considerable part of the normal values, and trends are hard to detect. The magnitudes of the NS values are comparatively large for a diatomic consisting of fairly light atoms, indicative of at least one fairly low lying electronic doublet state Dunham_BO_Watson1 (); NS_isos_rot_1995 (). However, these values are still much smaller in magnitude than those of CH,  (105) and  (76) CH+_fitting_2010 ().

The BOB parameters appear to be usually negative and increasing in magnitude with , as in the case of AlH AlH_IR_1993 () or, with more limited values, CH CH+_fitting_2010 (). As can be see in Table 6, this is also the case for of CO and NO. In addition, it appears as if the poorly determined values of CS are of correct order of magnitude. The remaining BOB parameters determined for NO, , , and are all of fairly small magnitude.

Predictions of the rotational spectra of several NO isotopologs will be available in the catalog section111http://www.astro.uni-koeln.de/cdms/ of the CDMS CDMS_1 (); CDMS_2 (). The line, parameter, and fit files from the isotopic invariant fit are deposited as supplementary material. In addition, these files, along with other auxiliary files, will be available in the fitting spectra section222http://www.astro.uni-koeln.de/site/vorhersagen/pickett/beispiele/NO/ of the CDMS.

Acknowledgements

This study was partly supported by a Grant-in-Aid for Scientific Research on Innovative Areas by the Ministry of Education, Culture, Sports, Science, and Technology of Japan (grant no. 26108507). K.K. is grateful for support of her stay in Cologne by the collaborative research grant SFB 956.

Appendix A. Supplementary Material

Supplementary data for this article are available on ScienceDirect (www.sciencedirect.com) and as part of the Ohio State University Molecular Spectroscopy Archives (http://library.osu.edu/sites/msa/jmsa_hp.htm). Supplementary data associated with this article can be found, in the online version, at doi: .

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