Collisional quenching of highly rotationally excited HF^{⋆}
^{1} Department of Physics and Astronomy and the Center for Simulational Physics, The University of Georgia, Athens, GA 30602, USA
email: yang@physast.uga.edu
^{2} Department of Physics, Penn State University, Berks Campus, Reading, PA 19610, USA
^{3} Department of Chemistry, University of Nevada, Las Vegas, NV 89154, USA
Received: 3 February 2015
Accepted: 23 April 2015
Context. Collisional excitation rate coefficients play an important role in the dynamics of energy transfer in the interstellar medium. In particular, accurate rotational excitation rates are needed to interpret microwave and infrared observations of the interstellar gas for nonlocal thermodynamic equilibrium line formation.
Aims. Theoretical cross sections and rate coefficients for collisional deexcitation of rotationally excited HF in the vibrational ground state are reported.
Methods. The quantummechanical closecoupling approach implemented in the nonreactive scattering code MOLSCAT was applied in the cross section and rate coefficient calculations on an accurate 2D HFHe potential energy surface. Estimates of rate coefficients for H and H_{2} colliders were obtained from the HFHe collisional data with a reducedpotential scaling approach.
Results. The calculation of statetostate rotational quenching cross sections for HF due to He with initial rotational levels up to j = 20 were performed for kinetic energies from 10^{5} to 15 000 cm^{1}. Statetostate rate coefficients for temperatures between 0.1 and 3000 K are also presented. The comparison of the present results with previous work for lowlyexcited rotational levels reveals significant differences. In estimating HFH_{2} rate coefficients, the reducedpotential method is found to be more reliable than the standard reducedmass approach.
Conclusions. The current statetostate rate coefficient calculations are the most comprehensive to date for HFHe collisions. We attribute the differences between previously reported data and our results to differences in the adopted interaction potential energy surfaces. The new He rate coefficients can be used in a variety of applications. The estimated H_{2} and H collision rates can also augment the smaller datasets previously developed for H_{2} and electrons.
Key words: molecular processes / molecular data / ISM: molecules
Rate coefficient tables are only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/578/A65
© ESO, 2015
1. Introduction
Molecular collisions, which are responsible for most of the excitation and reaction processes involving molecules, are important in the interstellar medium (ISM). Collisional excitation and deexcitation processes compete with radiative transitions in populating molecular levels. In cold environments, the important collision partners are H_{2} and He because of their large abundances, except in photodissociation regions (PDRs) and diffuse gas where collisions with electrons and H can become important. Precise laboratory data including collisional deexcitation rate coefficients are required for a range of temperatures to interpret the complicated interstellar spectra of molecular gas not in local thermodynamic equilibrium (LTE). Because of the complexity and difficulty of direct measurements, only limited statetostate collisional rate coefficients have been measured for systems of astrophysical interest (see, for example, Brunet et al. 2002). Therefore, astrophysical modeling heavily depends on theoretical prediction (e.g., Flower 2007; Faure & Lique 2012; Wiesenfeld & Faure 2013; Roueff & Lique 2013; Yang et al. 2013; Dubernet et al. 2013).
In this paper, we consider hydrogen fluoride (HF), an interstellar molecule containing a halogen element, which was first detected in the ISM by Neufeld et al. (1997). The reactivity of HF is weak, but it may be formed by the exoergic process F+H_{2}→HF+H. Experimental rate coefficients for this reaction were recently reported by Tizniti et al. (2014) at temperatures between 10 and 100 K. As a result of its structural stability and radiative properties, the HF molecule can be an alternative tracer to H_{2} in diffuse regions; HF is also the main reservoir of fluorine in the ISM (Monje et al. 2011; van der Tak 2012a). Monje et al. (2014) reported observations of HF in two luminous galaxies, NGC 253 and NGC 4945, using the Heterodyne Instrument for the FarInfrared (HIFI) on the Herschel Space Observatory. With Herschel/HIFI, Neufeld et al. (2010) detected HF in absorption from the ground rovibrational state. Phillips et al. (2010) presented a detection of the fundamental rotational transition of hydrogen fluoride in absorption toward Orion KL using Herschel/HIFI. The emission in the j = 1 → 0 rotational transition of HF has been observed in the carbon star envelope IRC +10216 by Agúndez et al. (2011). Monje et al. (2011) reported the first detection of HF toward a highredshift quasar at z = 2.56, while van der Tak et al. (2012b) presented observations of the HF j = 1 → 0 line in emission towards the Orion Bar.
The HFHe scattering system has been studied theoretically and experimentally (Lovejoy & Nesbitt 1990; Moszynski et al. 1994, 1996; Chapman et al. 1997; Stoecklin et al. 2003; Reese et al. 2005; Fajin et al. 2006). The availability of ab initio HFHe potential energy surfaces (PESs) has stimulated theoretical studies of HF excitation due to He impact. Lovejoy & Nesbitt (1990) reported the first experimental study of the nearinfrared vibrationalrotational spectra of the HeHF and HeDF complexes in a supersonic expansion. The HeHF (HeDF) spectra showed good agreement with the prediction obtained using the HartreeFock dispersion (HFD) type rigidrotor potential of Rodwell et al. (1981). The spectroscopic data were analyzed and used to probe the isotropic and anisotropic intermolecular potentials of the complexes. The twodimensional (2D) interaction potential of Moszynski et al. (1994) was developed from ab initio calculations using symmetryadapted perturbation theory (SAPT). This SAPT potential is in good agreement with the empirical PES of Lovejoy & Nesbitt (1990). All transition frequencies obtained from the boundstate calculations using the SAPT potential showed excellent agreement with the experimental spectra. The SAPT potential has a global minimum for the linear HeHF arrangement and a secondary minimum for the linear HeFH geometry. The accuracy of the SAPT potential was also confirmed by agreement between calculated differential and integral cross sections on a slightly modified SAPT potential and experimental results (Moszynski et al. 1996). Another 2D HFHe potential was reported by Fajin et al. (2006) using the coupledcluster method with single and double excitations with perturbative triple excitation (CCSD(T)). More recently, a threedimensional (3D) PES was presented by Stoecklin et al. (2003). This PES was computed using the Brueckner coupledcluster method with perturbative triples excitations (BCCD(T)) in the supermolecular approach, and was fitted analytically using a kernel Hilbert space interpolation method. This 3D potential was also used in closecoupling (CC) calculations of pure rotational excitation of HF in collisions with He by Reese et al. (2005). Cross sections for transitions for rotational levels up to j = 9 of HF were calculated for collision energies up to 2000 cm^{1}. Rate coefficients were obtained from 0.1 to 300 K. However, the PES of Stoecklin et al. predicts global and local minima with well depths of of 43.70 and 25.88 cm^{1}, respectively, compared to 39.20 and 35.12 cm^{1} for the experimentally derived PES of Lovejoy & Nesbitt (1990).
In this work, explicit quantummechanical closecoupling scattering calculations of rotational quenching of HF in collisions with He at higher levels of rotational excitation were carried out using the SAPT potential of Moszynski et al. (1994). The statetostate rate coefficients are presented for a wide range of temperatures (0.1−3000 K), which will aid in modeling rotational spectra of HF in various astrophysical and atmospheric environments. The computational method is discussed in Sect. 2, and we compare the PESs of Moszynski et al. (1994) and Stoecklin et al. (2003) in Sect. 3. The results are presented in Sect. 4, while astrophysical applications and scaling approaches for estimating HF deexcitation by H_{2} and H collisions are described in Sect. 5.
Computed rotational excitation energies (cm^{1}) of HF.
2. Quantumscattering calculations
We adopted the timeindependent quantummechanical closecoupling (CC) theory developed by Arthurs & Dalgarno (1963) for the scattering of a linear rigidrotor by an Sstate atom. The statetostate integral cross section for a transition from an initial rotational state j to a final rotational state j′ can be expressed as (1)where j and l are the rotational angular momentum of the HF molecule and the orbital angular momentum of the collision complex, respectively. The total angular momentum J is given by J = l + j. is an element of the scattering matrix, which is obtained by solving coupledchannel equations and employing the usual boundary conditions. denotes the wave vector for the initial channel, E_{j} is the kinetic energy for the initial channel, and μ the reduced mass of the HFHe system. The total quenching cross section from an initial state j can be obtained by summing the statetostate cross sections σ_{j → j′}(E_{j}) over all final j′ states, where j′<j.
The quantumscattering code MOLSCAT (Hutson & Green 1994) was applied in the closecoupling calculations. The propagation in R was carried out to 50 Å with the coupledchannel equations solved using the modified logderivative Airy propagator (Alexander & Manolopoulos 1987). To ensure convergence in the crosssection calculations, at least five to ten closed channels in the basis and a sufficient number of partial waves were included. HF rotational energy levels are given in Table 1, which were obtained using the rotational constant B_{e} = 20.953 cm^{1} (Irikura 2007) and the centrifugal distortion constant D = 0.0021199 cm^{1} (Coxon & Hajigeorgiou 1990). The CC calculations were performed for collision energies ranging from 10^{5} to 15 000 cm^{1} with μ = 3.3353 u.
The rate coefficients for rotational transitions can be computed numerically by thermally averaging the corresponding cross sections over a Maxwellian kinetic energy distribution (2)where T is the temperature, β = (k_{B}T)^{1}, and k_{B} is Boltzmann’s constant.
3. Comparison of potential energy surfaces
In the rigidrotor scattering calculations, the interaction potential of HFHe, V(R,θ), was expanded in terms of Legendre polynomials, (3)where P_{λ} are Legendre polynomials of order λ, R is the distance between the HF center of mass and the He atom, and θ the angle between R and the HF molecular axis. The angular dependence of the interaction potential was expanded to λ_{max} = 20.
In Fig. 1, the first four components of v_{λ}(R), λ = 0, 1, 2, and 3 are plotted as a function of R for the PESs of Moszynski et al. (1994) and Stoecklin et al. (2003). To obtain a 2D rigidrotor PES, the optimal approach is to average the 3D PES over the groundstate vibrational function of the diatomic molecule (Faure et al. 2005; Kalugina et al. 2014). However, in this work, we mainly used the 2D PES of Moszynski et al. (1994), which was obtained at r_{e}, to calculate the rotational quenching cross sections and rate coefficients. For the 3D PES of Stoecklin et al., two values of the HF bond length, the equilibrium distance r_{e} = 1.7328a_{0} and the vibrationally averaged bond distance for the ground vibrational state r_{0} = 1.767a_{0} (Zhang & Zhang 1993), were used to compute v_{λ}(R). Comparing the plots, we can see that differences between the PESs of Moszynski et al. and Stoecklin et al. are apparent, particularly for components λ = 1 and 2. In the case of the PES of Stoecklin et al., except for λ = 0, some differences can be seen between the curves for R less than 6 a_{0} owing to the different values of the HF bond length. Therefore, one can expect discrepancies to arise in scattering calculations performed on the two different PESs. We recall that the Moszynski et al. PES agrees very well with the experimentally deduced surface of Lovejoy & Nesbitt (1990).
Fig. 1
Potential expansion terms v_{λ}, λ = 0,1,2, and 3, for the HFHe PESs of Moszynski et al. (1994) and Stoecklin et al. r_{e} and r_{0} are used to obtain 2D PESs for Stoecklin et al. (2003), while the Moszynski et al. PES was constructed for r_{e}. 

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Fig. 2
Statetostate rotational quenching cross sections from initial state j = 1 of HF in collisions with He. 

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4. Results and discussion
4.1. Statetostate and total quenching cross sections
We performed calculations of the statetostate quenching cross sections for initial HF rotational states of j = 1, 2,..., 20 using the PES of Moszynski et al. (1994)^{1}. To evaluate the accuracy of the current computed cross section and to compare with the results obtained using the PES of Stoecklin et al. (2003), calculations of statetostate quenching cross sections from j = 1 and 3 were also performed using the PES of Stoecklin et al.^{2}. In the rigidrotor approximation calculations carried out on the PES of Stoecklin et al., the 2D PESs are obtained by fixing the HF bond distance at r_{e} and r_{0}. Correspondingly, rotational constants of HF B_{e} = 20.9537 cm^{1} (Irikura 2007) and B_{0} = B_{e} −1/2α_{e} = 20.5570 cm^{1} were used in the crosssection calculations, where the vibrationrotation interaction constant α_{e} = 0.7934 cm^{1} (Irikura 2007). As examples, the statetostate quenching cross sections from initial states j = 1 and 3 are presented in Figs. 2 and 3, respectively. In the case of quenching j = 1 → 0, Fig. 2 illustrates that there are significant differences between the cross sections obtained using the PESs of Moszynski et al. and Stoecklin et al. that are due to the different structures of the two PESs, as shown in Fig. 1. For the PES of Stoecklin et al., the cross sections calculated using (B_{0}, r_{0}) agree better with the results obtained using their 3D potential. A number of resonances in the cross section, which occur for low collision energies associated with the van der Waals wells, demonstrate their sensitivity to the adopted PES.
Fig. 3
Statetostate rotational quenching cross sections from initial state j = 3 of HF in collisions with He. 

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The PES of Moszynski et al., used in our calculations, was constructed by fixing the HF bond distance at its equilibrium value r_{e}. We used the rotational constant B_{e} to evaluate the HF rotational energy levels (see Table 1). However, to study the effect of the rotational constant on the cross sections, we carried out crosssection calculations from initial states j = 1, 3, and 10 using the rotational constant B_{0}. Figures 2 and 3 show that for the PES of Moszynski et al., the statetostate quenching cross sections from initial j = 1 and 3 using B_{e} and B_{0} are nearly identical. For statetostate quenching cross sections from initial j = 10 (not shown), the differences between the results obtained using B_{e} and B_{0} are generally lower than 10%.
In Fig. 3 we compare the statetostate quenching cross section from initial state j = 3. As observed in the case of j = 1, the cross sections display resonances in the intermediate energy region from ~0.01 cm^{1} to ~10 cm^{1} due to quasibound levels supported by the attractive part of the interaction potential. Importantly, for astrophysical applications, the properties of the resonances influence the quenching rate coefficients at low temperatures. In contrast to initial j = 1, the difference between the cross sections obtained on the two PESs is smaller. In particular, for the PES of Stoecklin et al. (2003) the results using (B_{0}, r_{0}) are similar to the results using the 3D potential for collision energies higher than 1.0 cm^{1}. As can be seen, the deexcitation process from j = 3 is dominated by the Δj = −1 transition. Furthermore, the computed cross sections show that the Δj = j′ − j = −1 transition dominates the deexcitation for all j, and the cross sections generally increase with increasing j′, with that for j′ = 0 being the smallest.
Fig. 4
Total deexcitation cross sections from initial states j = 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20 of HF in collisions with He with the PES of Moszynski et al. (1994). 

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The total quenching cross section from a given initial level j can be computed by summing over all final states j′. In Fig. 4 the total quenching cross sections from selected initial levels j = 2, 4, 6, ..., 18, and 20 are displayed. Generally, the total quenching cross sections have similar behavior, decreasing with j for E_{j} ≤ 50 cm^{1}, but differences result for small j at high energy that are due to a limited number of final exit channels. Each of the cross sections exhibit the behavior predicted by the Wigner (1948) threshold law at ultralow collision energies below ~10^{3} cm^{1}, where only swave scattering contributes and the cross sections vary inversely with the relative velocity. In the intermediate energy region, between 0.1 and 10 cm^{1}, the cross sections display scattering resonances, but they reveal somewhat different structures depending on the initial rotational state j between 0.05 and 1 cm^{1}. Except for j = 2, the total deexcitation cross sections decrease to a global minimum near 50 cm^{1}.
Fig. 5
Statetostate rotational quenching rate coefficients from initial states j = 1 and 3 of HF that are due to collisions with He. a) j = 1; b)j = 3. Lines: present results with the PES of Moszynski et al. (1994); symbols: Reese et al. (2005) with the PES of Stoecklin et al. (2003). 

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4.2. Statetostate quenching rate coefficients
The quenching rate coefficients can be computed by averaging the appropriate cross sections over a MaxwellBoltzmann distribution of collision energy E_{j}, as given by Eq. (2). The statetostate quenching rate coefficients for initial HF rotational states of j = 1,2,..., 20 were calculated using B_{e} and the PES of Moszynski et al. (1994). However, to our knowledge, there are no published experimental rate coefficients available. Our rate coefficients, computed using the PES of Moszynski et al., are only compared with the theoretical results of Reese et al. (2005), which were obtained over the limited temperature range of 0.1 to 300 K. As examples, Figs. 5–7 present selected statetostate quenching rate coefficients from initial rotational levels j = 1,3,5, and 9. Except for deexcitation from j = 1, the current results generally follow similar trends with that of Reese et al., which were computed using the 3D potential of Stoecklin et al. (2003). However, at 0.1 K, the rate coefficients of Reese et al. are always significantly larger than the current results for all transitions, except for j = 1.
Fig. 6
Statetostate rotational quenching rate coefficients from initial states j = 5 of HF that are due to collisions with He. Lines: present results with the PES of Moszynski et al. (1994); symbols: Reese et al. (2005) with the PES of Stoecklin et al. (2003). 

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Fig. 7
Same as Fig. 6, but for initial rotational state j = 9. 

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For initial state j = 1, Fig. 5a shows that the current rate coefficients are larger than the results of Reese et al. As for initial state j = 3 shown in Fig. 5b, for temperatures above 1 K, the current statetostate rate coefficients agree reasonably well with that of Reese et al., although our results are somewhat larger for the j = 3 → 1 transition. Statetostate quenching rate coefficients from initial states j = 5 and 9 are compared in Figs. 6 and 7, respectively, with the results of Reese et al. Except for the deexcitation to the final state j′ = 0 and for 0.1 K, the current rate coefficients are smaller than those of Reese et al., similar to what is found for j = 3. For each dominant deexcitation transition, Δj = j′ − j = −1, j = 1,2,..., 9, at a temperature of 50 K, we compare the percent differences between our rate coefficients and the results of Reese et al. (2005). In Fig. 8, the percent differences are displayed as a function of initial rotational state j. The percent difference is near zero for j = 3, but the differences vary from 20% to 75% for all other j values.
For illustration, in Fig. 9 we present the statetostate deexcitation rate coefficients for temperatures ranging from 0.1 K to 3000 K for initial states j = 10 and 20. Over the whole temperature range considered, the rate coefficients generally increase with increasing temperature for all transitions. Furthermore, the rate coefficients clearly decrease with increasing  Δj  =  j′ − j  with the Δj = −1 transitions dominant.
Fig. 8
Percent difference of HFHe rate coefficients for the deexcitation of level j for the dominant transition Δj = −1 at 50 K between current results using the PES of Moszynski et al. (1994) and the results of Reese et al. (2005) with the PES of Stoecklin et al. (2003). 

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Fig. 9
Statetostate rotational deexcitation rate coefficients from initial state j = 10 and j = 20 of HF in collisions with He using the PES of Moszynski et al. (1994). a) j = 10; b) j = 20. 

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5. Applications
As highlighted in the Introduction, HF has been observed in both emission and absorption in a variety of astronomical environments that may be characterized by diverse physical conditions. As a consequence, the rotational levels of HF may be populated by different, or multiple, mechanisms leading to spectra differing from LTE. For the UVirradiated environment of the Orion Bar, van der Tak et al. (2012b) considered electrons and H_{2} as possible impactors for inelastic collisional excitation. However, given that the Orion Bar is a prototypical photodissociation region (PDR), the abundances of both H and He are most likely higher than or comparable to that of H_{2}. In many PDR enviroments, collisions due to all four colliders may need to be considered.
For a variety of reasons, it is often the case that excitation rate coefficients for a molecular species may only be available for He collisions, as performed in the current work. In such instances, a common practice is to estimate H_{2}, and occasionally H, collisional excitation rate coefficients using He data by scaling by the square root of the ratio of the collision systems’ reduced masses, here a factor of 1.4 for H_{2}. Walker et al. (2014) demonstrated on both theoretical and numerical grounds that this standard reducedmass scaling approach is typically invalid. We therfore do not recommend here that such an approach be adopted with the present HFHe rate coefficients. Fortunately, explicit HFH_{2} rate coefficients for some rotational quenching transitions of the HF ground vibrational state have been computed by Guillon & Stoecklin (2012).
Fig. 10
HF–paraH_{2} (Guillon & Stoecklin 2012) and HFHe rate coefficients (current results) compared to HF–paraH_{2} rate coefficients deduced by various scaling approaches for the j = 1 → 0 transition. See text for discussion. 

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5.1. Prediction of HF rate coefficients by scaling
Given the availability of HFH_{2} and HFHe collisional data, we can test various scaling methods, including the reducedpotential approach introduced in Walker et al. (2014). In the reducedpotential method, the collisional data are scaled by the reduced potentials μ_{X}ε_{X} according to (4)where is the reduced mass of the HFX system, ε_{X} is the van der Waals minimum of the HFX system, and C is a phenomenological exponent. X = Y is usually He, with X = Z typically paraH_{2}. The reducedpotential and standard reducedmass approaches are compared in Fig. 10 for the HF rotational deexcitation transition j = 1 → 0. Standard reducedmass scaling results in an estimate for paraH_{2} collisions that is a factor of 10 too small, while the reducedpotential method with an exponent of C = 1.7 agrees well.
Given its better performance, the reducedpotential scaling approach of Walker et al. (2014) was therefore used to predict unknown rate coefficients for paraH_{2} and H colliders with HF. We adopted the He collision data computed here and the smaller set of paraH_{2} rate coefficients for j ≤ 5 calculated by Guillon & Stoecklin (2012) for T = 10 − 150 K. The rate coefficients are very sensitive to the presence of quasibound resonances over this temperature range, which may partially be accounted for with the reducedpotential approach since it takes into the account the different interaction welldepths. Compared to the reducedmass scaling technique, where the rate coefficients simply scale as the square root of the ratio of the reduced masses, the reducedpotential scaling approach offers an improvement for predicted HF rate coefficients, much like the improved CO and H_{2}O rate coefficient predictions of Walker et al. (2014). While the value of C for CO rate coefficients ranged from −0.2 to 1.3 and from 0.5 to 1.2 for H_{2}O, it was noted that symmetries were involved in obtaining the best value of C.
The accuracy of the reducedpotential scaling approach can be further tested here with the heteronuclear molecule HF. The normalized rootmeansquare deviation (NRMSD), σ_{norm}, quantifies the residual variance between the calculated H_{2} rate coefficients, k_{calc}, and those scaled from He, k_{scale}, and is given by (5)where N is the number of temperature data points and k_{max} and k_{min} are the values of the maximum and minimum rate coefficients. The resulting NRMSD percentages for the H_{2} rate coefficient predictions for both standard reducedmass scaling and reducedpotential scaling is given in Table 2 and Fig. 11 for the first fifteen transitions of HF at 50 K. When comparing the NRMSD for both methods for each transition of HF, the reducedpotential scaling predictions exhibit less residual variance in all fifteen transitions with a mean of 14% in its ability to reproduce the explicit HFH_{2} calculations as computed by Guillon & Stoecklin (2012). In many cases, the NRMSD for standard reducedmass scaling exceeds 100%, while the reducedpotential approach gives NRMSD <35%. A linear leastsquares analysis was then performed, using the first fifteen transitions of HF, for each j′ and the resulting linear functions are plotted in Fig. 12. Except for j′ = 2, the lines converge around  Δj  = 6 and C = −3. For  Δj  = 1, the values of C decrease linearly with j′ and are listed in Table 3 for j′ = 0 − 4. The optimal values for C, valid for all temperatures, were forced to exactly reproduce the reducedpotential scaling result for the dominant Δj = −1 transitions.
Optimized values of C and their respective normalized rootmeansquare deviations (NRMSD, in %) for collisional deexcitation transitions of HF with H_{2} scaled via the standard reducedmass (rm) and reducedpotential (rp) methods from HFHe collisional rate coefficients.
Fig. 11
Normalized rootmeansquare deviation (NRMSD) in standard reducedmass scaling and reducedpotential scaling for 15 transitions of HFparaH_{2}, truncated at 100%. 

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Fig. 12
Phenomological constant C as a function of  Δj . A linear leastsquares analysis was performed for each j′ and the resulting linear functions are plotted. Note the convergence of these functions (except for j′ = 2) around  Δj  = 6 and C = −3. Symbols correspond to C determined for actual HFparaH_{2} and HFHe data. 

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HF reducedpotential fitting parameters.
Fig. 13
Phenomological constant C as a function of  Δj . As a result of the linear decrease of C with j′ and the convergence around  Δj  = 6 and C = −3, the slope and yintercept can be obtained for each transition and the value of C can be predicted. Symbols and fit lines are for j′ = 0 − 4, same as in Fig. 12. 

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Fig. 14
Same as Fig. 10, but for the j = 5 → 1 transition. 

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To use the reducedpotential approach in astrophysical applications, however, one needs estimates of C when data for the impactor of interest Z, e.g. paraH_{2}, are unknown. Using the information from Fig. 12, the slope and yintercept can be obtained for each transition and the value of C predicted as given in Fig. 13 and Table 3 for j′ = 5−20. The best prediction for transitions with  Δj  ≥ 6 is obtained with C = −3. Rate coefficients for the transitions of H_{2} with j< 5 are reproduced reasonably well by the reducedpotential approach with C decreasing with increasing  Δj  and increasing j′. Figures 10 and 14 compare rate coefficient estimates using the fit prediction of C to direct reducedpotential and reducedmass values for select transitions. Weak transitions are scaled with C = −3, and since these rate coefficients are several orders of magnitude smaller than the dominant transitions, larger error in the predicted values is acceptable.
Given that the available HFH_{2} rate coefficients are limited to j ≤ 5 (Guillon & Stoecklin 2012), we used the reducedpotential scaling method predictions for C and the current HFHe data to estimate HF rotational deexcitation for j = 6−20 for paraH_{2} collisions. Furthermore, as there is a complete lack of HF rotational excitation data for H impact, we extended the reducedpotential scaling approach to estimate HFH deexcitation rate coefficients from the current HFHe rate coefficients with the trend in C taken from Fig. 13. In using the reducedpotential approach, the following parameters were adopted for HF collisions with He, H_{2}, and H: μ_{He} = 3.3353 u, μ_{H2} = 1.818 u, μ_{H} = 0.9596 u, ε_{He} = 39.68 cm^{1} (Moszynski et al. 1994), ε_{H2} = 359.0 cm^{1} (Guillon et al. 2008), and ε_{H} = 100.0 cm^{1} (Stark & Werner 1996). All rate coefficient data are available in the LAMDA format, as mentioned in footnote 1. The availability of a complete set of HF rotational quenching rate coefficients due to collisions with H_{2}, He, H, and e^{−} will allow for detailed modeling of HF rotational emission lines in a variety of environments with a varying molecular fraction. However, we caution that the data obtained via reducedpotential scaling are approximate, but are reasonable estimates until explicit calculations become available.
6. Conclusion
Rate coefficients for the deexcitation of rotational excited HF due to He collisions were computed using the closecoupling method and an accurate potential energy surface. The adopted ab initio PES agrees well with an experimentally deduced empirical PES. Differences of 75% and larger were found with previous HFHe scattering calculations that used a less reliable PES. New rate coefficients were obtained for HF rotational levels j = 1−20 for 0.1−3000 K due to He. A recently introduced scaling approach was used to estimate rate coefficients for HFH and missing HFH_{2} collisional data.
All statetostate deexcitation cross sections and rate coefficients for HFHe are available on the UGA Excitation Database website (www.physast.uga.edu/amdbs/excitation). The rate coefficients are also available in the BASECOL (Dubernet et al. 2013) and the Leiden Atomic and Molecular Database (LAMDA; Schöier et al. 2005) formats. In addition, estimates for HFH_{2} and HFH rate coefficients obtained by the reducedpotential scaling approach, described below, are included in the new LAMDA file along with the original data from the LAMDA website.
Acknowledgments
Work at UGA was supported by NASA grant NNX12AF42G, at Penn State by NSF Grant No. PHY1203228, and at UNLV by NSF Grant No. PHY1205838. We thank Ad van der Avoird and T. Stoecklin for sending their potential subroutines and for helpful correspondence.
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All Tables
Optimized values of C and their respective normalized rootmeansquare deviations (NRMSD, in %) for collisional deexcitation transitions of HF with H_{2} scaled via the standard reducedmass (rm) and reducedpotential (rp) methods from HFHe collisional rate coefficients.
All Figures
Fig. 1
Potential expansion terms v_{λ}, λ = 0,1,2, and 3, for the HFHe PESs of Moszynski et al. (1994) and Stoecklin et al. r_{e} and r_{0} are used to obtain 2D PESs for Stoecklin et al. (2003), while the Moszynski et al. PES was constructed for r_{e}. 

Open with DEXTER  
In the text 
Fig. 2
Statetostate rotational quenching cross sections from initial state j = 1 of HF in collisions with He. 

Open with DEXTER  
In the text 
Fig. 3
Statetostate rotational quenching cross sections from initial state j = 3 of HF in collisions with He. 

Open with DEXTER  
In the text 
Fig. 4
Total deexcitation cross sections from initial states j = 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20 of HF in collisions with He with the PES of Moszynski et al. (1994). 

Open with DEXTER  
In the text 
Fig. 5
Statetostate rotational quenching rate coefficients from initial states j = 1 and 3 of HF that are due to collisions with He. a) j = 1; b)j = 3. Lines: present results with the PES of Moszynski et al. (1994); symbols: Reese et al. (2005) with the PES of Stoecklin et al. (2003). 

Open with DEXTER  
In the text 
Fig. 6
Statetostate rotational quenching rate coefficients from initial states j = 5 of HF that are due to collisions with He. Lines: present results with the PES of Moszynski et al. (1994); symbols: Reese et al. (2005) with the PES of Stoecklin et al. (2003). 

Open with DEXTER  
In the text 
Fig. 7
Same as Fig. 6, but for initial rotational state j = 9. 

Open with DEXTER  
In the text 
Fig. 8
Percent difference of HFHe rate coefficients for the deexcitation of level j for the dominant transition Δj = −1 at 50 K between current results using the PES of Moszynski et al. (1994) and the results of Reese et al. (2005) with the PES of Stoecklin et al. (2003). 

Open with DEXTER  
In the text 
Fig. 9
Statetostate rotational deexcitation rate coefficients from initial state j = 10 and j = 20 of HF in collisions with He using the PES of Moszynski et al. (1994). a) j = 10; b) j = 20. 

Open with DEXTER  
In the text 
Fig. 10
HF–paraH_{2} (Guillon & Stoecklin 2012) and HFHe rate coefficients (current results) compared to HF–paraH_{2} rate coefficients deduced by various scaling approaches for the j = 1 → 0 transition. See text for discussion. 

Open with DEXTER  
In the text 
Fig. 11
Normalized rootmeansquare deviation (NRMSD) in standard reducedmass scaling and reducedpotential scaling for 15 transitions of HFparaH_{2}, truncated at 100%. 

Open with DEXTER  
In the text 
Fig. 12
Phenomological constant C as a function of  Δj . A linear leastsquares analysis was performed for each j′ and the resulting linear functions are plotted. Note the convergence of these functions (except for j′ = 2) around  Δj  = 6 and C = −3. Symbols correspond to C determined for actual HFparaH_{2} and HFHe data. 

Open with DEXTER  
In the text 
Fig. 13
Phenomological constant C as a function of  Δj . As a result of the linear decrease of C with j′ and the convergence around  Δj  = 6 and C = −3, the slope and yintercept can be obtained for each transition and the value of C can be predicted. Symbols and fit lines are for j′ = 0 − 4, same as in Fig. 12. 

Open with DEXTER  
In the text 
Fig. 14
Same as Fig. 10, but for the j = 5 → 1 transition. 

Open with DEXTER  
In the text 