A theoretical room-temperature line list for Nh
A new room temperature line list for NH is presented. This line list comprised of transition frequencies and Einstein coefficients has been generated using the ‘spectroscopic’ potential energy surface NH3-Y2010 and an ab initio dipole moment surface. The NH line list is based on the same computational procedure used for the line list for NH BYTe reported recently and should be as accurate. Comparisons with experimental frequencies and intensities are presented. The synthetic spectra show excellent agreement with experimental spectra.
Although the abundance of N is 450 times lower than that of N Abbas et al. (2004), NH is an important astrophysical molecule. It is a maser source detected in interstellar molecular clouds (Schilke et al., 1991) and is also a tracer of the N/N isotopic ratio in interstellar medium (Charnley and Rodgers, 2002; Lis et al., 2010; Gerin et al., 2009), planetary (Fouchet et al., 2000, 2004) and Earth (Harper and Sharpe, 1998) atmospheres, meteorites (Pizzarello and Williams, 2012), comets (Mumma and Charnley, 2011), important as a probe of chemical processes in the astrophysical environment, of planetary atmospheric and formation processes etc. Very recently Fletcher et al. (2014) used the NH and NH spectral features to study the N/N ratio for Jupiter and Saturn.
Experimentally the ro-vibrational spectra of NH have been studied in a large number of works, including rotation-inversion spectrum (Shimzu and Shimizu, 1970; Carlotti et al., 1980; Sasada, 1980; Fusina et al., 1991; Schatz et al., 1994; Urban et al., 1994), fundamental bands (Cohen, 1980; Job et al., 1983; Urban et al., 1983, 1984, 1985; D’Cunha et al., 1985; Iwahori et al., 1986; Urban et al., 1986; Anders et al., 2000; Fusina et al., 2011), overtone bands (Dilonardo et al., 1982; Sasada et al., 1982; Fusina and Baldacchini, 1990; Moriwaki et al., 1991; Lees et al., 2006, 2008), hot bands (Karyakin et al., 1977; Urban et al., 1985; Sasada and Schwendeman, 1986; D’Cunha, 1987), and intensity measurements (Varanasi and Wyant, 1981; Devi et al., 1990; Siemsen et al., 1995; Lins et al., 2011). Some of these data are now collected in the HITRAN database Rothman et al. (2013). The electric dipole moment was experimentally studied by Orr and Oka (1977) and Dilonardo et al. (1981) using the Stark spectroscopy. The ground state energies were reported by Urban et al. (1985). Very recently a VECSEL laser source study of the 2.3 m region of NH was presented by Čermák et al. (2014) and a tentative assignment of new NH lines in the 1.51 m region was suggested by Földes et al. (2014).
Huang et al. (2011) presented theoretical ro-vibrational energies of NH computed variationally using an empirical PES HSL-2 for . These energies helped them to reassign and correct a number of transitions in HITRAN. An extensive hot line list BYTe for NH was recently generated (Yurchenko et al., 2011) using the TROVE approach Yurchenko et al. (2007). Containing 1.1 billion transitions BYTe was designed to be applicable for temperatures up to 1500 K. It has proven to be useful for astrophysical and spectroscopic applications (see, for example, Refs. Beaulieu et al. (2011); Lucas et al. (2010); Čermák et al. (2014)). In this work we build a room temperature line list for the N isotopologue of ammonia using the same computational approach based on the ‘spectroscopic’ potential energy surface (PES) NH3-Y2010 Yurchenko et al. (2011) and the ab initio dipole moment surface (DMS) from Ref. Yurchenko et al. (2009). The highest considered in this work is 18 defining the temperature limit of the current line list to be 300 K. It should be noted that TROVE was also used in the study of the thermal averaging properties of the spin-spin coupling constants of NH by Yachmenev et al. (2010) and a high-temperature partition function for NH Sousa-Silva et al. (2014).
The paper is structured as follows. In Section 2 we outline the theoretical approach used for the line list production. In Section 3 the structure of the line list and the description of the quantum numbers are presented, where some comparisons with experimental data are also given and the accuracy of the line list is discussed. In Section 4 some conclusions are offered.
2 Theoretical approach
We use the same computational procedure and the associated program TROVE Yurchenko et al. (2007) as was employed to generate the hot ammonia line list BYTe Yurchenko et al. (2011), therefore the reader should refer to this paper for a detailed description. Here we present only a short outline of this approach.
In order to obtain energies and associated wavefunctions required for building the line list of NH we solve the Schrödinger equation for the nuclear motion variationally. Both the kinetic and potential energy terms of the Hamiltonian were expanded to 6th and 8th orders, respectively, in terms of five linearized coordinates around the reference geometry, defined as a non-rigid reference configuration associated with the inversion motion characterized by a relatively low barrier to the planarity. The linearized coordinates are chosen to be close to the three stretching modes associated with the N-H vibrations and two asymmetric bending modes combined from the three bending vibrations of the interbond angles H–N–H. Our vibrational basis set is a product of six one-dimensional (1D) basis functions. The stretching, bending, and inversion 1D basis sets are obtained by solving the corresponding reduced 1D Schrödinger equations using the Numerov-Cooley approach Noumeroff (1923); Cooley (1961) for each degree of freedom independently. This so-called primitive basis set is then improved through a number of pre-diagonalizations and consecutive contractions. The latter is controlled by the polyad number
where are the quantum numbers associated with the three stretching modes, are associated with the asymmetric bending modes, and counts the inversion mode functions. As in Ref. Yurchenko et al. (2011), we define the size of the basis set using the condition . We use the so-called representation, where the final contracted ro-vibrational basis functions are represented by direct symmetrized products of the vibrational eigenfunctions and the rigid rotor wavefunctions , where is the rotational angular momentum, is the projection of the rotational angular momentum to the molecular axis , and is the rotational parity (see Yurchenko et al. (2005) for further details). The eigenfunctions are the eigensolutions of the pure vibrational problem. The highest rotational excitation presently considered is . We only compute and store the energy term values and wavefunctions below 14 000 cm above the zero point energy (ZPE) obtained as 7414.08 cm. These thresholds are chosen to get a reasonable population at room temperature according with the Boltzmann distribution.
As in Yurchenko et al. (2009) here we employ the EBSC (empirical basis set correction) scheme, where some of the band centers are substituted with the corresponding experimental values, where available. For NH however there are only very few band centers known experimentally with high enough accuracy, namely for , , , , , , , , , , , , , as well as the ground state inversion splitting Cohen (1980); Sasada et al. (1982); Dilonardo et al. (1982); Urban et al. (1983); Shojachaghervand and Schwendeman (1983); Job et al. (1983); Urban et al. (1984, 1985); D’Cunha et al. (1985); Urban et al. (1985); Sasada and Schwendeman (1986); Urban et al. (1986); D’Cunha (1987); Fusina and Baldacchini (1990); Lees et al. (2006, 2008); Li et al. (2007); Fusina et al. (2011); Čermák et al. (2014). Therefore the effect of this otherwise very efficient procedure is rather limited. With this approach the energies are reproduced exactly, while the ro-vibrational coupling leads to a gradual ‘de-focus’ of the energies. In Table 1 we compare the original theoretical term values with the experimental band centres used in our EBSC approach. The Obs.-Calc. residuals in this table illustrate the deficiency of our model based on the NH PES applied for the 15th isotopologue. Although the accuracy of these particular bands is recovered through the EBSC approach, the error of other band centers can be expected to be as large as up to about 0.2 cm at least, as illustrated in Table 1.
The same PES and DMS as in Yurchenko et al. (2011) were used. The potential energy surface NH3-Y2010 was obtained by Yurchenko et al. (2011) by fitting to the experimentally derived term values of the main isotopologues only, with covering term values up to cm. Because of the approximations used in the fitting, this ‘spectroscopic’ PES is an effective object. Therefore it does not guarantee, at least in principle, the same accuracy for NH as was reached for NH. We make a comparison with the experiment in the next section. The ab initio dipole moment surface ATZfc DMS used here was developed by Yurchenko et al. (2005a) which should be capable of accurate modelling of NH spectra. For the description of the intensity calculations see Refs. Yurchenko et al. (2005b, 2011).
In Tables 2–5 we compare our theoretical term values of NH with their ‘experimental’ counterparts for the vibrational () and pure rotational () states available in the literature (see Introduction). As far as the accuracy of these term values is concerned it is comparable to the accuracy of the vibrational term values for the main isotopologue using the same PES. The pure rotational and rotation-inversional term values also show a very good agreement with experiment. This is reassuring especially if the underlying PES was generated using the main isotopologue only, although the effect from the isotopic substitution 14 15 is not expected to be large.
In Table 6 some vibrational term values () of NH from this work are compared to the theoretical values computed by Huang et al. (2011) using their empirical PES HSL-II. The agreement at lower energies is very good but deteriorates at about 5000 cm. It is difficult to claim the better accuracy for any of these two approaches based on this comparison only. We believe that at least some of our band centers above 6000 cm should be more accurate, see e.g. Table 5. However according to Čermák et al. (2014) the line positions of NH reported by Huang et al. (2011) are more precise at least for the 2.3 m region.
We have compared our intensities to the HITRAN data Rothman et al. (2013) as well as to those reported by Devi et al. (1990) () and Lins et al. (2011) (near infrared). In Figs. 1 and 2 we show a generated absorption spectrum of NH at K compared to the HITRAN intensities. The agreement is similar to that achieved by BYTe for NH Yurchenko et al. (2011). The NH experimental data is rather sparse compared to the data available for the main isotopologue. A number of obvious outliers (5014.4776, 5084.8734, 5104.2963, and possibly 5103.8909 cm) in the experimental spectra indicate problems with the assignment of the NH transitions in HITRAN. Similar problems have recently been studied and resolved for the NH data Down et al. (2013). Another outlier is at 6586.747 cm from the recent work by Lins et al. (2011) which also appears to be too strong, see Figs. 1 and 2.
3 The line list
Our room temperature NH line list contains 80 515 767 transitions representing all non-zero ( K) intensities covering the wavenumber range up to 8 000 cm constructed from 270 646 upper state term values below 14 000 cm and 9772 lower state term values below 6 000 cm with rotational excitations up to . Following Refs. Barber et al. (2006); Yurchenko et al. (2011) we use the two-files ExoMol format Tennyson et al. (2013) to organize the line list for NH. The Energy file (see an extract in Table 7) contains the energy term values (cm), quantum numbers both in the local and normal mode representations. Each energy record is indexed with a running number . These indexes are then used in the Transition file (see extract in Table 8) to refer to a pair of states and participating in the transition . Apart from these indexes, only the Einstein coefficient is needed to complete the transition record. With this format the size of the line list is significantly reduced. The line list can be also found via www.exomol.com as a part of the ExoMol project Tennyson and Yurchenko (2012). We also supply a sample Fortran code to be used together with our line list to simulate intensities or cross sections (Hill et al., 2013). In fact the unified ExoMol-format of the present NH line list makes this code useful with any line lists stored in this format.
The largest expansion coefficients of the ro-vibrational eigenfunctions were used to assign the corresponding final eigenvalues with the vibrational quantum numbers , the rotational quantum numbers and , the total symmetry as well as the symmetry of the vibrational basis function . Here and are represented by six irreducible representations , , , , , in the molecular symmetry group (M) Bunker and Jensen (1998). In this case our ‘local’ mode basis functions functions are used as reference and provide approximate labels for the eigenstates. The problem with this approach (as well as many other assigning approaches) is the strong mixing of basis set functions at high excitations which gives rise to the ambiguity in assignment. As a manifestation of the quality of the assignment we also provide values of the corresponding largest expansion coefficients, see the -column in Table 7: small numbers (less than 0.5) indicate strong mixing of reference states and show that that the suggested quantum numbers can be ambiguous.
Recognizing the importance of the conventional ‘normal’ mode quantum numbers, we map our ‘local’ modes to the ‘normal’ mode quantum numbers using the same procedure as in Ref. Yurchenko et al. (2011). It should be noted however that there is no direct transformation between these two labelling schemes. Furthermore due to the approximate nature of the assignment in some cases we obtain ambiguous normal mode quantum numbers, which do not always correspond to the experimental normal mode labels. Again, the value can be used as measure of this ambiguity.
We follow Ref. Down et al. (2013) and define the normal mode quantum numbers as given by
where . Here , are the vibrational normal mode quantum numbers, , , and are the vibrational angular momentum labels; is the total angular momentum quantum number, is the projection of the total angular momentum on the molecule fixed axis ; is the inversion symmetry of the vibrational motion; and , and are the symmetry species of the rotational, vibrational, and total internal wave-functions in the molecular symmetry group (M), respectively, spanning , , , , , and . As was argued by Down et al. (2013), the definition of the signs of the vibrational angular momentum quantum numbers and is ambiguous (as ambiguous the sign of ). Therefore we follow the suggestion of Down et al. (2013) and use the absolute values (1) instead.
The symmetries of the initial and final states are important for the line intensities, which is manifested by the selection rules and the nuclear statistical weights . Similar to the main isotopologue of ammonia, the ro-vibrational states with the symmetries of and do not exist (i.e. the corresponding ). The non-zero nuclear statistical weights factors are 8, 4, 8, 4 for , , , , respectively, which are different from those of NH owning to the different nuclear spin of N, 1 against of N. The TROVE approach uses the symmetrically adapted basis set, which gives the symmetry labels of the eigenstates automatically. The selection rules are the following
The non-existing pure vibrational () and term values are also included into the line list with the total statistical weight , which can be useful as band centers.
With our computed energies of NH we obtain the partition function of 1165.4 which can be compared to the room-temperature partition function supplied by HITRAN of 1152.7 by Fischer et al. (2003).
|0||0||0||0||0||0||0||0.761||0.758||Urban et al. (1985)|
|0||1||0||0||0||0||0||928.509||928.457||Urban et al. (1985)|
|0||1||0||0||0||0||0||962.912||962.894||Urban et al. (1985)|
|0||2||0||0||0||0||0||1591.236||1591.185||Dilonardo et al. (1982)|
|0||0||0||0||1||1||1||1623.130||1623.149||Dilonardo et al. (1982)|
|0||0||0||0||1||1||1||1624.190||1624.202||Dilonardo et al. (1982)|
|0||2||0||0||0||0||0||1870.823||1870.853||Fusina and Baldacchini (1990)|
|0||3||0||0||0||0||0||2369.274||2369.314||Dilonardo et al. (1982)|
|0||1||0||0||1||1||1||2533.382||2533.380||Dilonardo et al. (1982)|
|0||1||0||0||1||1||1||2577.571||2577.590||Dilonardo et al. (1982)|
|0||3||0||0||0||0||0||2876.144||2876.130||Dilonardo et al. (1982)|
|0||0||0||0||2||0||0||3210.614||3210.430||Fusina et al. (2011)|
|0||0||0||0||2||0||0||3212.335||3212.120||Fusina et al. (2011)|
|0||0||0||0||2||2||2||3234.107||3233.925||Fusina et al. (2011)|
|0||0||0||0||2||2||2||3235.504||3235.338||Fusina et al. (2011)|
|1||0||0||0||0||0||0||3333.306||3333.220||Fusina et al. (2011)|
|1||0||0||0||0||0||0||3334.252||3334.160||Fusina et al. (2011)|
|0||0||1||1||0||0||1||3435.167||3435.143||Fusina et al. (2011)|
|0||0||1||1||0||0||1||3435.540||3435.475||Fusina et al. (2011)|
|1||1||0||0||0||0||0||4288.186||4288.024||Urban et al. (1985)|
|1||1||0||0||0||0||0||4312.345||4312.304||Urban et al. (1985)|
|1||0||0||0||2||2||2||6546.951||6546.987||Li et al. (2007)|
|1||0||0||0||2||2||2||6548.560||6548.449||Li et al. (2007)|
|1||0||1||1||0||0||1||6596.569||6596.605||Lees et al. (2006)|
|1||0||1||1||0||0||1||6597.607||6597.498||Lees et al. (2006)|
|1||0||1||1||0||0||1||6664.486||6664.627||Lees et al. (2008)|
|1||0||1||1||0||0||1||6665.480||6665.303||Lees et al. (2008)|
Estimated from the band centers.
Estimated from the corresponding transition frequencies.