Issue 
A&A
Volume 551, March 2013



Article Number  A140  
Number of page(s)  5  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/201219491  
Published online  11 March 2013 
On the X ^{1}Σ^{+} rovibrational spectrum of lithium hydride
^{1} Department of Physics and Astronomy and the Center for Simulational PhysicsUniversity of Georgia, Athens, GA 30602, USA
email: stancil@physast.uga.edu
^{2} Institute of Applied Physics and Computational Mathematics, 100094 Beijing, PR China
email: shi_yanbo@iapcm.ac.cn; wang_jianguo@iapcm.ac.cn
Received: 26 April 2012
Accepted: 21 January 2013
The complete line list of rovibrational transitions of the X ^{1}Σ^{+} state of LiH is computed. The line list includes all possible dipoleallowed transition energies and oscillator strengths that cover the transition energy range 3.27 − 19476 cm^{1}. The line list was obtained using an accurate potential and dipole moment function constructed from available experimental and theoretical data. This paper discusses the agreement of the current calculations with previous theoretical and experimental results. We also provide the radiative cooling function in the highdensity limit over a wide temperature range and compare them with previous results. A simulated collisionallybroadened LiH opacity relevant to cool dwarf stars is also presented.
Key words: molecular data / molecular processes
© ESO, 2013
1. Introduction
Lithium hydride has been the subject of theoretical and experimental molecular physics investigations for many years (see for example Partridge & Langhoff 1981). As the simplest neutral heteronuclear diatomic molecule, LiH is a favorite benchmark for various quantumchemical techniques (Tung et al. 2011; Holka et al. 2011). In astrophysics, LiH may play a role at temperatures T ≤ 5000 K in the cooling of primordial clouds (Stancil et al. 1996; Bougleux & Galli 1997; Galli & Palla 1998; Bovino et al. 2011) and is a means of monitoring the evolution of stars and interstellar clouds (Dulick et al. 1998). Furthermore, LiH, as well as other Libearing molecules (LiCl and LiOH), may be observable in cool (T_{eff} ≤ 2000 K) dwarf atmospheres (Lodders 1999; Weck et al. 2004). However, to date, searches for LiH at high redshift have proved unsuccessful (Friedel et al. 2011). Reviews of the lithium chemistry in the early Universe have been given by Lepp et al. (2002) and Bovino et al. (2011), while the status of laboratory infrared spectroscopy of LiH is outlined in Dulick et al. (1998). A complete and evaluated rovibrational line list for the ground electronic state, which is germane to its thermal evolution and needed for primordial spectral models, is needed^{1}.
In Sect. 2 we present the details of potential energy curve and dipole moment function for the electronic ground state of LiH. In Sect. 3, we briefly outline the relevant relations for computing transition energies, Einstein coefficients, oscillator strengths, and radiative cooling functions. Results of the current computation including the ground electronicstate rovibrational line list and cooling function are presented in Sect. 4 along with comparisons to previous work and astrophysical applications. Section 5 gives a summary of the current work. Atomic units are used throughout, unless otherwise noted.
2. Potential energy and dipole moment function
Within the wellknown BornOppenheimer approximation, the motion of a diatomic molecule may be partitioned into its electronic and nuclear components. As such, the nonrelativistic Hamiltonian of a molecule may be written as (1)with the timeindependent Schrödinger equation given by (2)By separation of variables, the total wave function may be written as (3)where R is the internuclear distance vector in bodyfixed coordinates, and r the electron coordinate vector with respect to the center of mass of the nuclei. The motion of electrons (e) and nuclei (N) can be described by the electronic Schrödinger equation (4)and the nuclear Schrödinger equation (5)where E_{e}(R) is the potential energy and E_{v,J} the rovibrational energy or eigenvalue.
The dipole moment function can be defined as (6)where Ψ_{e}(r;R) is the electronic eigenfunction. In this work we consider X^{1}Σ^{+} ← X^{1}Σ^{+} rovibrational transitions so that only the zcomponent of the r operator contributes.
For a diatomic molecule, the potential energy is only a function of nuclear distance and can be written as E_{e}(R). We applied an empirical potential (Coxon & Dickinson 2004), which was generated by directly fitting the spectroscopic LiH line positions, for 2 < R < 20 a_{0}. The longrange potential is given by for R > 20 a_{0}, where C_{6} = 66.536, C_{8} = 3279.99 and C_{10} = 223016.6 (Yan et al. 1996). For R < 2 a_{0}, the potential has been extrapolated with the shortrange form V^{SR}(R) = Aexp( − BR) + C.
In a similar way, the dipole moment function of the X state calculated by Partridge & Langhoff (1981) has been used over the range R = 1.75 to 17.5 a_{0}. For R > 17.5 a_{0}, the longrange form of Bottcher & Dalgarno (1974), D^{LR}(R) = d_{7}/R^{7}, was adopted. For R < 1.75 a_{0}, the shortrange form D^{SR}(R) = AR^{2} + BR was fit to the ab initio data. The potential and dipole moment function are displayed in Fig. 1.
Fig. 1 Potential energy (solid line) and the dipole moment function (dashed line) of the X state for LiH. 

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3. Rovibrational eigenvalues, transition probabilities, and related quantities
The radial nuclear Schrödinger equation of a diatomic molecule may be written as (7)which can be solved numerically by the NumerovCooley method (Johnson 1977), for example. Here μ is the reduced mass of ^{7}LiH, ν is vibrational quantum number, J the rotational quantum number, and χ_{vJ}(R) is the corresponding rovibrational eigenfunction. The Einstein A coefficient, or transition probability, in the dipole approximation, can be calculated given χ_{vJ}(R) and D(R), and is defined as (Weissbluth 1978) (8)where α = e^{2}/ħc, is the finestructure constant, a_{0} is the Bohr radius, and ΔE_{ν′J′,ν′′J′′} the transition energy. Here S_{J′J′′} is the line strength and defined as (9)Here ϕ_{J′J′′} is a HönlLondon factor (Herzberg 1989) given by (10)The oscillator strength may then be defined (Weissbluth 1978) as (11)In a variety of applications, it is useful to obtain the radiative cooling function in the highdensity or local thermodynamic equilibrium (LTE) limit (see for example Coppola et al. 2011). In LTE, the radiative cooling function only depends on the rovibrational energies, degeneracies, and transition probabilities. It is only a function of the gas kinetic temperature T and is given (in ergs/s) by (12)The sum is over all possible rovibrational deexcitation transitions within the electronic ground state and the denominator is the rovibrational partition function Q(T).
4. Results and discussion
The radial Schrödinger Eq. (7) was solved with a step size 1.0 × 10^{3}a_{0}, over a range of internuclear distances from R = 1.0 to 60.0 a_{0}. The reduced mass of ^{7}LiH was 0.881238162 u (Stwalley & Zemke 1993).
Vibrational binding energies G_{ν′} of the X^{1}Σ^{+} state of ^{7}LiH in units of cm^{1}.
The vibrational energies G_{ν′} and corresponding vibrational energy spacings ΔG_{ν′ + 1/2} of the X ^{1}Σ^{+} state obtained in the present study are listed in Tables 1 and 2, along with the theoretical values of Holka et al. (2011) and the measurements of Chan et al. (1986). The largest discrepancy for G_{ν′} between our calculations and the experiment is 2.47 cm^{1}, which occurs for v = 22, while the largest difference for ΔG_{ν′ + 1/2} is smaller at 1.85 cm^{1} for v = 22. Some select transition energies are listed in Table 3 along with the corresponding experimental results of Dulick et al. (1998) and the calculations of Coppola et al. (2011). It can be seen that the current results are in excellent agreement with experiments, the largest discrepancy being 0.027 cm^{1}.
Transition probabilities for some bandaveraged transitions in the X^{1}Σ^{+} state are compared in Table 4 with the previous calculation of Partridge & Langhoff (1981). The maximum difference is less than 6.79%. As a further illustration, the Jdependent dipole moment matrix elements for v = 1 to v = 0 are shown in Fig. 2. which are similar to those given in Gianturco et al. (1996). It can be seen that the transition probabilities for the Pbranch are much higher than those for the Rbranch. The line oscillator strengths as a function of wavelength are shown in Fig. 3 for the X X transition. The most intense line of the Rbranch is at 22.968 μm (band 00, R(60)). For the Pbranch, the strongest is at 16.822 μm (band 87, P(31)). The complete line list is available on the UGA Molecular Opacity Project website (http://www.physast.uga.edu/ugamop/) and in the format of the Leiden Atomic and Molecular Database (LAMDA, Schöier et al. 2005).
Utilizing the theoretical rovibrational energies and transition probabilities, the LTE radiative cooling function was calculated as a function of temperature as shown in Fig. 4. It is compared with the previous results of Dalgarno (1994, priv. comm.) and Coppola et al. (2011), along with the lowdensity limit cooling function (n_{H} ≤ 100 cm^{3}) due to H collisions (Galli & Palla 1998) (see the Appendix for a further discussion). All three LTE cooling functions are seen to be in excellent agreement over the considered temperature range.
Vibrational level spacings ΔG_{ν′ + 1/2} of the X state of ^{7}LiH where ΔG_{ν′ + 1/2} = G_{ν′ + 1} − G_{ν′} is in units of cm^{1}.
Rovibrational transition energies for in the X ^{1}Σ^{+} state of ^{7}LiH.
Avalues for transitions in the X ^{1}Σ^{+} state of ^{7}LiH.
Fig. 2 Dipole moment matrix elements for the Pbranch and Rbranch rovibrational transitions for ^{7}LiH for the v = 0 to v = 1 transition. 

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Fig. 3 Oscillator strengths for the X X transition of ^{7}LiH versus wavelength. Rbranch (red) and Pbranch (black). 

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Fig. 4 Radiative cooling functions of LiH in the LTE (upper three lines) and the lowdensity limit (n_{H} ≤ 100 cm^{3}, lower solid red and black lines) considering only H collisions. 

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Fig. 5 Opacity of the LiH (red) rovibrational transitions in the infrared compared to the LiCl (blue) opacity for a temperature of 1800 K and a pressure of 100 atms. 

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To simulate the opacity of LiH in a cool dwarf atmosphere, we computed the absorption cross section using Eq. (6) of Dulick et al. (2003) for a temperature of 1800 K, as an example. The line absorption cross section, which is a function of the transition probability, line frequency, partition function, and temperature, is multiplied by a Lorentzian line profile (Bernath 1995) with width corresponding to collisional broadening. Assuming a canonical collisional broadening cross section of 10^{16} cm^{2} with a line width proportional to the pressure, Figure 5 displays the opacity for 100 atms. There is a series of strong individual pure rotational lines for λ > 20 μm, while the fundamental vibrational band occurs near 8 μm. Comparison is made to the LiCl opacity (Weck et al. 2004) for the same temperature and pressure. The LiCl fundamental vibrational band falls in the gap between the pure LiH rotational lines and the LiH 01 band, but is typically one to two orders of magnitude smaller. According to the cool dwarf atmosphere models of Lodders (1999), the abundances of LiH and LiCl, as well as LiOH, are comparable at 1800 K, but for higher temperatures, LiH becomes the dominant Libearing molecule. Weck et al. (2004) suggest that it would be difficult to detect LiCl in cool dwarfs due to strong absorption features from other more abundant molecules (e.g., water) in the same wavelength range. LiH, with its somewhat higher opacity,might be a better candidate for detecting lithium molecules.
5. Conclusions
The rovibrational energy levels of ^{7}LiH were computed with an empirical potential generated from a direct fit of available spectroscopic line positions. The computed vibrational energies and transition energies are found to be in very good agreement with experiments. Oscillator strengths and transition probabilities were also computed and used to construct a complete rovibrational line list. The transition energies were evaluated against available experimental data and found to be reliable to better than 0.03 cm^{1}. The line list was used to compute an LTE radiative cooling function for LiH and to explore its infrared opacity. Among all dominant Libearing molecules, LiH is a better candidate for detecting lithium in cool dwarf atmospheres.
Appendix: Collisional excitation of LiH
While in this work, we present an LTE radiative cooling function (i.e., valid within the highdensity limit), Bougleux & Galli (1997) have approximated a densitydependent nonLTE (NLTE) cooling function (see also Galli & Palla 1998). In this approach, the NLTE cooling function is parameterized in terms of the LTE radiative cooling function and a collisional cooling function in the zerodensity limit. For applications to the early Universe, they considered only collisions due to atomic hydrogen. Unfortunately, rotational or vibrational excitation rate coefficients of LiH due to H are not available, requiring Bougleux & Galli (1997) to adopt the LiHHe calculations of Jendrek & Alexander (1980), but adjusted to emulate LiHH collisions via massscaling of the collisional rate coefficients. The massscaling approach is typically adopted when rate coefficients due to He impact are used to approximate paraH_{2} impact. Recent work on other molecules suggests this approach is questionable at best (e.g., Cernicharo et al. 2011) even though the PESs for He and H_{2}complexes are expected to be similar, being characterized by weakly bound van der Waals minima. Collision complexes involving H, on the other hand, typically involve very deep wells leading to very fast exothermic reactive channels. As a consequence, massscaling to deduce Himpact excitation rate coefficients from He or H_{2}impact data is likely to be a poor approximation.
For general astrophysical applications, collisions due to the most abundant species, H, H_{2}, He, e^{−}, and H^{+}, should be included in an NLTE cooling function, though H_{2} dominates in most interstellar environments. LiHHe collisional excitation calculations have been more recently given by Taylor & Hinde (1999, 2005) and Bodo et al. (2001a), and discussed in a review of lithium chemistry by Bodo et al. (2003). The reactive process (13)and the Hexchange process (14)have been considered in a number of recent studies (Prudente et al. 2009; Bovino et al. 2009), but to date, inelastic collisional excitation of LiH due to H has not been investigated because it is a very computationally demanding problem that would require simultaneous treatment with reactions (13) and (14). We are unaware of any inelastic data due to H_{2}, e^{−}, or H^{+} impacts, though an excited state PES for the LiH complex is available (Bodo et al. 2001b), which could be used later. It is therefore, not currently possible to construct an accurate, comprehensive NLTE cooling function for LiH. Figure 4 gives an indication of the expected range of the NLTE cooling function.
A previous, though unevaluated, line list is available on the ExoMol database: http://www.exomol.com
Acknowledgments
The work of P.C.S. was partially supported by NSF grant AST0607733 and NASA grant NNX07AP12G. Y.B.S. acknowledges travel support by the International Cooperation and Exchange Foundation of CAEP. We thank the referee for helpful comments that improved the manuscript.
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All Tables
Vibrational binding energies G_{ν′} of the X^{1}Σ^{+} state of ^{7}LiH in units of cm^{1}.
Vibrational level spacings ΔG_{ν′ + 1/2} of the X state of ^{7}LiH where ΔG_{ν′ + 1/2} = G_{ν′ + 1} − G_{ν′} is in units of cm^{1}.
All Figures
Fig. 1 Potential energy (solid line) and the dipole moment function (dashed line) of the X state for LiH. 

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In the text 
Fig. 2 Dipole moment matrix elements for the Pbranch and Rbranch rovibrational transitions for ^{7}LiH for the v = 0 to v = 1 transition. 

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In the text 
Fig. 3 Oscillator strengths for the X X transition of ^{7}LiH versus wavelength. Rbranch (red) and Pbranch (black). 

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In the text 
Fig. 4 Radiative cooling functions of LiH in the LTE (upper three lines) and the lowdensity limit (n_{H} ≤ 100 cm^{3}, lower solid red and black lines) considering only H collisions. 

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In the text 
Fig. 5 Opacity of the LiH (red) rovibrational transitions in the infrared compared to the LiCl (blue) opacity for a temperature of 1800 K and a pressure of 100 atms. 

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In the text 
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