Issue 
A&A
Volume 651, July 2021



Article Number  A51  
Number of page(s)  10  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/202140681  
Published online  09 July 2021 
Collisional effects in the blue wing of Lymanα
^{1}
Laboratoire de Physique et Chimie Quantique, Université de Toulouse (UPS) and CNRS,
118 route de Narbonne,
31400
Toulouse,
France
^{2}
GEPI, Observatoire de Paris, Université PSL, UMR 8111, CNRS, 61 avenue de l’Observatoire,
75014
Paris,
France
email: nicole.allard@obspm.fr
^{3}
Sorbonne Université, CNRS, UMR7095, Institut d’Astrophysique de Paris, 98bis boulevard Arago,
Paris,
France
^{4}
Department of Physics and Astronomy, University of Louisville,
Louisville,
Kentucky
40292,
USA
Received:
28
February
2021
Accepted:
12
April
2021
Spectral observations below Lymanα are now obtained with the Cosmic Origin Spectrograph (COS) on the Hubble Space Telescope. It is therefore necessary to provide an accurate treatment of the blue wing of the Lymanα line that enables correct calculations of radiative transport in DA and DBA white dwarf stars. On the theoretical front, we very recently developed very accurate HHe potential energies for the hydrogen 1s, 2s, and 2p states. Nevertheless, an uncertainty remained about the asymptotic correlation of the Σ states and the electronic dipole transition moments. A similar difficulty occurred in our first calculations for the resonance broadening of hydrogen perturbed by collisions with neutral H atoms. The aim of this paper is twofold. First, we clarify the question of the asymptotic correlation of the Σ states, and we show that relativistic contributions, even very tiny, may need to be accounted for a correct longrange and asymptotic description of the states because of the specific 2s 2p Coulomb degeneracy in hydrogen. This effect of relativistic corrections, inducing small splitting of the 2s and 2p states of H, is shown to be important for the ΣΣ transition dipole moments in HHe and is also discussed in HH. Second, we use existent (HH) and newly determined (HHe) accurate potentials and properties to provide a theoretical investigation of the collisional effects on the blue wing of the Lymanα line of H perturbed by He and H. We study the relative contributions in the blue wing of the H and He atoms according to their relative densities. We finally achieve a comparison with recent COS observations and propose an assignment for a feature centered at 1190 Å.
Key words: line: profiles / atomic data / white dwarfs / molecular data
© F. Spiegelman et al. 2021
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
In heliumdominated white dwarfs, the discrepancy in the hydrogen abundance between Balmerα from the optical data and Lymanα from the ultraviolet (UV) data is strong. In addition, the Lymanα line profile is asymmetric (see Xu et al. 2017, and references therein). The existence of a quasimolecular line satellite is crucial for understanding this asymmetrical shape of the Lymanα line observed with the Cosmic Origin Spectrograph (COS; see Fig. 1 in Allard et al. 2020). This absorption feature has been predicted by detailed collisional broadening profiles by Allard & Christova (2009). These authors made an exhaustive study of the red wing of the Lymanα line perturbed by HHe collisions. They considered high He densities met in cool DZ white dwarfs and examined the validity range of the oneperturber approximation that is widely used to calculate the line wings. HHe potentials were theoretically determined by several authors, namely Theodorakopoulos et al. (1984, 1987), Sarpal et al. (1991), Lo et al. (2006), Belyaev (2015), and Allard et al. (2020). Allard & Christova (2009) used the potentials and dipole moments of Theodorakopoulos et al. (1984, 1987), but were limited by a lack of accuracy of the molecular potential of the CΣ state. They noticed an unexpected well of about 150 cm^{−1} at R ~ 8 Å that is related to the choice of basis functions. Significant progress in the description of the HHe potential energies has been achieved in a recent work (Allard et al. 2020) using extensive basis sets and multireference configuration interaction (MRCI) calculations (Knowles & Werner 1992; Werner et al. 2015) to determine accurate ab initio potential energy curves. Nevertheless, because of the specific degeneracy of the hydrogen levels in the Coulomb model, the adiabatic correlation of the A and C states to either 2s (dipole forbidden from the ground state) or 2p (allowed) states is not fully characterized using the Coulomb Hamiltonian only. Relativistic effects that are smaller than 1 cm^{−1} for hydrogen are responsible for lifting the strict degeneracy of the hydrogen atomic levels in the Coulomb model (Kramida 2010). This level splitting is crucial for establishing the adiabatic correlation of the molecular states toward the asymptotic levels, and thus the specific asymptotic behavior of the dipole transition moments from the ground state. Thus, one aspect of the present paper is to redetermine and rediscuss the ground and lowest excited potential energy curves (PECs) of HHe and the electric transition dipole moments (Sect. 2.1), with a stronger focus on their longdistance behavior. A detailed correlation to the dissociated atomic states and its effect on the transition dipole moments is specifically discussed in Sect. 2.2. We consider spinorbit (SO) coupling in Sect. 2.3.
A similar asymmetry as in the Lymanα HHe line profile also exists in the resonance broadening of hydrogen perturbed by collisions with H atoms. In our first calculations for the resonance broadening of hydrogen perturbed by collisions with H and H^{+} (Allard et al. 1994), we were mainly interested in quasimolecular absorption of transient H_{2} and H molecules in the far red wing of Lymanα. Singlet states of H_{2} lead to line satellites from the freefree transitions → and C^{1} Π_{u} →. These states also are responsible for the boundbound Lyman and Werner H_{2} bands. Triplet states only lead to a blue asymmetry because of a close line satellite that appears as a shoulder in the blue wing. These improved theoretical calculations of the complete Lymanα profile including both red and blue wings were applied to the interpretation of International Ultraviolet Explorer (IUE) and Hubble Space Telescope (HST) spectra. They were shown to be fundamental in the interpretation of UV spectra of variable DA white dwarfs (ZZ Ceti stars; Koester et al. 1994). The analysis of the Lymanα satellites in the far red wing is not only a way to establish the location of the ZZ Ceti instability of variable DA white dwarfs, but also a test of the assumptions about convection efficiency (Bergeron et al. 1995).
The correlation diagram for H_{2} states contributing to Lymanα shown in Table 2 of Allard et al. (1994) was not correct because of an error in the preliminary ab initio calculations of the  transition moments. This error was noted in Allard et al. (1998a), who pointed out that the variation of the radiative dipole moment must be included in the line profile calculation. A new correlation diagram was presented in Allard & Kielkopf (2009) to correct Table 2 of Allard et al. (1994). This correlation diagram has been used in Allard et al. (1998a) and in our subsequent work. Electronic transition moments among singlets and triplets computed by Spielfiedel (2003) and Spielfiedel et al. (2004) were used in Allard & Kielkopf (2009) for an exhaustive study of the red wing of Lymanα in order to determine the contribution of the triplet transition that was not considered in Allard et al. (1994). Although we never clarified the correct contribution of triplet states to the blue wing of the Lymanα line, a blue line satellite was observed in experimental spectra (Kielkopf & Allard 1995, 1998).
We discuss the effect of the 2s2p degeneracy lifting on the longdistance behavior of H_{2} states in Sect. 2.4. In Sect. 3 we present a study of the blue wing of the Lymanα line perturbed by collisions with H and He atoms in order to examine their relative contributions in the Lymanα spectrum.
2 Diatomic potentials and electronic transition dipole moments
The transient interactions of atoms during radiative collisions are the main physical quantities needed for a good understanding of the effect of collisional processes on radiative transfer in stellar atmospheres and the spectra emitted by white dwarf stars. We consider herafter HHe with and without spinorbit coupling and HH.
2.1 HHe without SO coupling
The ab initio calculations of the potentials were carried out with the MOLPRO package (Werner et al. 2015) using a very large Gaussian basis set that was initially taken from the spd fgh augccpV6Z basis set of the MOLPRO library for both He and H atoms, complemented by additional diffuse functions. On both He and H, the augccpV6Z Gaussian basis set was complemented by diffuse functions in each l manifold. The added exponents on He were the same as in the previous work of Allard et al. (2020). A slightly larger complementary set of functions was determined and used on H in order to accurately describe the atomic spectrum of hydrogen up to n = 4, with both diffuse functions and intermediate exponents improving the nodal structure (Table A.2). Thus the basis set includes 239 Gaussian functions on He and 297 on H.
A unique set of molecular orbitals was obtained from a relativistic HartreeFock (RHF) calculation of HeH^{+}, incorporating the scalar relativistic effects (Darwin and massvelocity contributions) within the DouglasKrollHess (DKH) scheme (Reiher 2006; Nakajima & Hirao 2011) at second order. The virtual orbitals of HeH^{+} provide relevant excited molecular orbitals that properly dissociate into the exact orbitals of H (as represented within the present basis). All subsequent calculations include the DKH contributions. The configuration interaction (CI) was generated by a primary complete active space (CAS) including 14, 7, 7, and 4 molecular orbitals in each of the a_{1}, b_{1}, b_{2}, and a_{2} manifolds of the C_{2v} point group. This ensured that all Σ^{+}, Π, Δ, and Φ states dissociating up to n = 4 were properly described in the CAS space. Finally, an MRCI calculation (Knowles & Werner 1992) generated from this CAS space was conducted for the A_{1}, B_{1}, and A_{2} manifolds with 13, 7, and 3 eigenstates, respectively. Although we only discuss the Lymanα contribution here, the relevant electric dipole transition moments corresponding to Lyman and Balmer transitions up to n = 4 were determined in the same calculations and will be the object of future publication. Relativistic effects are obviously very tiny on hydrogen. Nevertheless, because all excited levels present degeneracy in the simple nonrelativistic Coulomb scheme, it may be important for a correct longrange description to account even for very tiny splitting.
The DKH contribution, which is essentially active in the inner parts of the wave functions close to the atoms, splits the degenerate components of a given n and stabilizes more significantly the low l levels. It also slightly increases the excitation energies from the 1s level. In the present work that is concerned with the 2s, 2p states, these scalar contributions were complemented by accounting for spinorbit coupling with similar magnitude. For all levels of H n = 1–4, the use of the extensive basis set and inclusion of the DKH correction provides transition energies obtained with an accuracy better than 0.5 cm^{−1} compared to the experimental atomic data (Kramida et al. 2020). Table 1 illustrates the achieved accuracy (in particular, the ordering of the 2s, 2p states) that is important to determine the configurational mixings that span the molecular states and the resulting transition probabilities that contribute to Lymanα.
The symmetry and labeling of the molecular states that dissociate into He(1s^{2}) + H(2s,2p) is shown in Table 2, and their potential energy curves are plotted in Fig. 1. States AΣ and BΠ are attractive with a minimum located around R = 0.75 Å. State C^{2}Σ^{+} presents a repulsive character at medium and long range, resulting in a barrier and a shallower shortdistance well depth. This mediumrange repulsion is associated with the repulsion between the electronic density of the 2s and 2p_{σ} hydrogen orbitals along the axis and the electrons of the helium atom.
Atomic energy levels of hydrogen (in cm^{−1}).
Symmetry and labeling of molecular states dissociating into H(2s,2p)+He and H(2s,2p)+H.
Fig. 1 MRCI adiabatic potential energy curves of HHe correlated with the 2s, 2p atomic levels. The zero energy corresponds to H(1s)+He(1s^{2}) at dissociation. 
2.2 HHe PECs and correlation to dissociated atomic states
For distances smaller than 8.1 Å, the results we show in Fig. 1 do not differ substantially from the recent calculation of Allard et al. (2020), which was essentially achieved with a similar methodology. We used the same augccpV6Z basis set complemented with diffuse functions (the same functions for helium, and slightly more diffuse functions on hydrogen), and a larger CAS space generating the MRCI (three electrons in 32 orbitals, instead of three electrons in 14 orbitals in the former work). None of these differences are expected to provide significant quantitative changes at short distance concerning the ground state and the states dissociating into H(n = 2)+He. However, no explicit mention of the relativistic effects was made in Allard et al. (2020), and the C state potential energy curve was found to converge to the unique Coulomb 2s/2p asymptote with a dipole moment for the X − C transition that vanished asymptotically, meaning an adiabatic correlation of the obtained C state with the dipoleforbidden 2s atomic state. Conversely, the transition dipole moment X − A was found to converge to that of the atomic 1s2p transition dipole moment value.
We here focus strongly on the medium and long distance of the potential curves, detailed in Fig. 2. In particular, at the asymptotic limit, the DKH relativistic correction lifts the hydrogen 2s/2p degeneracy, lowering the 2s state more than the 2p state. This results in a 2s2p splitting of 0.316 cm^{−1}. As a result, a longdistance avoided crossing around 8.1 Å occurs in the ^{2}Σ^{+} manifold between the lower adiabatic state correlated with 2s and the upper state correlated with 2p. The state correlated with 2s has a tiny well at 8.7 Å. This causes the adiabatic upper state (labeled C at short distance by spectroscopists) to correlate with the 2p asymptote, while state A is correlatedwith 2s. The examination of the dipole transition moments from the ground state, shown in Fig. 3, reveals a crucial implication. This longrange avoided crossing induces a kink at 8.1 Å in the X − C dipole moment and a sign change in the X − A dipole transition moment. Moreover, both adiabatic states have transition moments that stabilize around 0.4–0.6 au below R = 8.1 Å, which means that the upper state loses some 2p character while the lower gains it, regardless of the avoided crossing at 8.1 Å. Thus, the situation results from a gradual 2s/2p mixing increasing with short distance, superimposed with the sharp avoidance at 8.1 Å.
Fig. 2 Long range zoom of the MRCI adiabatic HHe potential energy curves of states A, B and C dissociating into H(n = 2)+He. The zero energy corresponds to H(1s)+He(1s^{2}) at dissociation. 
Fig. 3 Spinorbitless transition dipole moments of HHe from the X ground statetoward the adiabatic A, B, and C states of HHe. 
Fig. 4 Adiabatic H(n = 2)+He potential energy curves of molecular states including SO coupling. For better display, the red line of state (1)3∕2 state is shown with superimposed dots. Zero energy corresponds to H(1s)+He(1s^{2}) at dissociation. 
2.3 HHe with SO coupling
SO coupling was incorporated following the atominmoleculelike scheme introduced by Cohen & Schneider (1974). It relies on an effective monoelectronic spinorbit coupling operator, (1)
where and ŝ are the orbital and spinmoment operators, and ζ_{i} is the effective spin constant associated with a given atomic electronic shell.
The total Hamiltonian H_{el} + H_{SO} is expressedin the basis set of the eigenstates (here with total spin projection ) of the purely electrostatic Hamiltonian H_{el}. Because the core electrons of He define a closed shell, the spinorbit coupling between the molecular manyelectron doublet states Ψ_{kσ}, approximated at this step as single determinants with the same closed shell 1s σ^{2} as the He subpart, is isomorphic to that between the singly occupied molecular spinorbitals ϕ_{kσ}, which are asymptotically correlated with the six 2p spinorbitalsand two 2s spinorbitals of H. In its original formulation, the Cohen and Schneider approximation consists of assigning these matrix elements to their asymptotic (atomic) values. However, in the case of HeH, the 2s and 2p shells are asymptotically degenerate, which offers a favorable situation for electronic mixing. As previously mentioned, both adiabatic states Ψ_{2s Σ} and Ψ_{2p Σ} depart from their asymptotic character. Conversely, the Ψ_{2pΠ} state does not significantly mix with any other states and essentially conserves the asymptotic transition dipole moment from the ground state shown in Fig. 3.
In case of strong mixing, application of the Cohen and Schneider scheme is more relevant in a basis of diabatic or quasidiabatic states, as has been discussed in previous works (Allard et al. 2020). Thus we determined Φ_{2s Σ} and Φ_{2p Σ} diabatic states through a unitary transform of the adiabatic states Ψ_{2sΣ} and Ψ_{2pΣ} and the constraint that the transition dipole moments from the ground state X 1^{2} Σ^{+} to the quasidiabatic states at finite distance remain as close as possible to their asymptotic values, namely zero and 0.744 (asymptotic limit of the MRCI calculation), (2)
Using this procedure, the transition dipole moments to the quasidiabatic states are now exactly zero for state Φ_{2s Σ} and almost constant for state Φ_{2pΣ}. The two latter, and state Φ_{2pπ} = Ψ_{2pΠ} considered as diabatic, therefore span the Cohen and Schneider spinorbit matrix, the electronic Hamiltonian being nondiagonal in the Σ states manifold, (3)
The effective spinorbit constant of the 2p shell of hydrogen is taken empirically as 2/3 times the spinorbit splitting 0.366 cm^{−1}, namely ζ_{2p} = 0.244 cm^{−1} (Kramida et al. 2020). The total Hamiltonian matrix H = H_{el} + H_{SO} for the n = 2 manifold is thus an 8 × 8 complex matrix that can decouple into real matrices according to different values of Ω =M_{l} + M_{s}, where M_{l} and M_{s} are the orbital and spin moment projections on the molecular axis. For Ω = ±1∕2 and Ω = ±3∕2, these matrices read as follows:
Here E_{2sΣ} and E_{2p Σ} are the energies of the quasidiabatic states and V is their electronic coupling. E_{2pΠ} is the energy of the Π state. The diagonalization of the above matrix at each internuclear distance provides the spinorbit eigenstates and energies. Despite being performed in an extensive basis, our calculation cannot reach sub cm^{−1} accuracy, which is required to investigate the molecular fine structure close to the asymptote because the SO splitting is only 0.366 cm^{−1}. In particular, the difference between the experimental atomic 2s state and the average of the 2p states is 0.209 cm^{−1}, while the calculation at separation yields 0.316 cm^{−1}. Thus we shifted the nonrelativistic diabatic potential E_{2sΣ} upward by 0.107 cm^{−1}. As a consequence of SO splitting, the 2p_{1∕2} level now lies closely below the 2s_{1∕2} state, by 0.034 cm^{−1} in the calculation versus 0.035 cm^{−1} experimentally.
The molecular eigenstates can be labeled in adiabatic order in each Ω manifold, namely (2)1/2, (3)1/2, (4)1/2, and (1)3/2 above the ground state (1)1/2 (also named herafter X_{1∕2}). Their potential curves are shown inf Figs. 4 and 5. They adiabatically correlate with atomic asymptotes 2p_{1∕2}, 2s_{1∕2}, 2p_{3∕2}, and 2p_{3∕2}, respectively. At short distance R < 1.2 Å, the lowest excited eigenstate (2)1/2 is essentiallyspanned by AΣ. However at intermediaterange 1.2–7.3 Å, (2)1/2 switches to BΠ which lies below AΣ. At R = 8.1 Å it undergoes another avoided crossing and is finally adiabatically correlated with the lowest asymptote 2p_{1∕2}. Consistently, state (3)1/2 has a BΠ character at short distance, switches to AΣ in the intermediate range and is adiabatically correlated with asymptote 2s_{1∕2}. Finally state (4)1/2 has essentially a CΣ character for R < 8.1 Å, however as a consequence of the avoided crossing between the Σ state at 8.1 Å, it becomes adiabatically correlated with asymptote 2p_{3∕2}. Only state (1)3∕2 remains identical to its parent state BΠ at all distances, except for an asymptotic shift. It should be noted that in a diabatic picture where the states may cross (but where the hamiltonian is no longer diagonal), three 1/2 states keeping the AΣ, BΠ and CΣ character almost up to dissociation could be defined, correlated to 2p_{1∕2}, 2p_{3∕2} and 2s_{1∕2} respectively after multiple crossings.
The transition dipole moments (Fig. 6) between the spinorbit states can easily be determined from those computed between the quasidiabatic CI wavefunctions, (4)
The dipole moments of transitions X_{1∕2} − (2)1∕2 and X_{1∕2} − (4)1∕2 show kinks at 8.1 Å, while that of the X_{1∕2} − (3)1∕2 transition changes sign. This feature results from the avoided or actual crossings between their three A, B, and C parents and the consecutive multiple crossings of the spinorbit states at this distance. The transition moment X_{1∕2} − (1)3∕2 remains clearly equal to that of X − B. The dipole moments of transitions X_{1∕2} − (2)1∕2 and X_{1∕2} − (3)1∕2 furthermore show a sudden exchange at 1.2 Å due to the crossing of their parent states AΣ and BΠ, respectively,at that distance.
The electronic structure of HeH involving fine structure has also been investigated with pseudopotentials (Kielkopf 2021), in excellent agreement with our calculation in the region R > 6 Å. Although we carried out an analysis of the effect of spinorbit coupling on the HHe potential curves and transition moments, spinorbit coupling is not taken into account in the collisional section below. The most important effect, namely the adiabatic correlation of the C state with the upper allowed 2p asymptote associated with a nonvanishing X − C dipole moment, is maintained when spinorbit coupling is accounted for: the C_{1∕2} state, spanned by the parent state CΣ at short distance, correlates with the dipoleallowed atomic state 2p_{3∕2}, and the X_{1∕2} − (4)1∕2 transition dipole remains finite at large distance.
Fig. 5 Longrange zoom of the adiabatic potential energy curves of H(n = 2)+He states including SO coupling. Zero energy corresponds to H(1s)+He(1s^{2}) at dissociation. 
Fig. 6 Transition dipole moments from the X_{1∕2} ground statetoward the adiabatic states dissociating into H(n = 2)+He including SO coupling. For better display, the red line of the X_{1∕2} − (1)3∕2 transition dipole moment is shown with superimposed dots. 
2.4 HH potentials
In Allard et al. (1994) the theoretical potentials for the binary interaction of one hydrogen atom with another hydrogen atom were taken from the calculations of Sharp (1971) and Wolniewicz & Dressler (1988). The dependence of the probabilities of the allowed molecular transitions on internuclear separation contributing to the Lymanα line were taken from Dressler & Wolniewicz (1985) for the singlet states and preliminary ab initio results for the transitions between the triplet states. The allowed transitions contributing to Lymanα were summarized in Table 4 of Allard et al. (1994), but the labels of the a and h triplet states were interchanged. The symmetry and labeling of the HH states are listed in Table 2. It might be wondered whether the inclusion of relativistic terms might seriously affect the longdistance behavior of the potentials and the transition dipole moments, as found in HHe. In Fig. 7 we show the longrange behavior of the theoretical quasifull CI HH potentials of Spielfiedel (priv. comm.) carried out in the Coulomb framework, and MRCI calculations (full CI) conducted on HH with the same basis set as was used on hydrogen in HHe (see above) and the DKH correction. The splitting of the 2s and 2p levels also induces avoided crossings in H_{2}. While states and join the asymptote around 15 Å states and remain significantly attractive up to very large separation because of the 1∕R^{3} contribution. They exhibit avoided crossings (within each symmetry manifold) with the former states (diabatically correlated with atomic forbidden asymptotes). However, in H_{2}, these avoided crossings take place at very long distance, namely around R = 45 Å. At this interatomic separation, the electronic coupling vanishes and we can consider that in a collisional situation the system will follow a diabatic behavior, namely it will move along diabatic potentials that cross. In the present case with vanishing coupling, diabatic potentials can easily be defined by reassigning the adiabatic potential values before and after the (avoided) crossing.
Because only the difference potentials are meaningful for the treatment of collisional broadening, we can infer that scalar relativistic corrections are not needed and the potentials in the Coulomb description can be used simply as diabatic potentials. We therefore used the theoretical HH potentials of Spielfiedel (priv. comm.) together with the electronic transition moments among singlets and triplets taken from Spielfiedel (2003) and Spielfiedel et al. (2004). As with HHe, spinorbit coupling is not considered in the collisional section that follows.
3 Collisional profiles perturbed by neutral H and He atoms
Quasimolecular lines in the red wing of Lymanα, Lymanβ, and Lymanγ arise from radiative collisions of excited atomic hydrogen with unexcited neutral hydrogen atoms or protons Allard & Kielkopf (1991); Allard et al. (1998b,a, 2000, 2004b,a, 2009). A general unified theory in which the electric dipole moment varies during a collision (Allard et al. 1999) is essential as the blue HH quasimolecular line satellite is due to an asymptotically forbidden transition b ^{3}Σ_{u} → h ^{3}Σ_{g}.
Fig. 7 Longrange zoom of the adiabatic potential energy curves of states , , , and of H+H(n = 2). Top: data of Spielfiedel et al. (priv. comm.) without scalar relativistic terms. Bottom: our CI results including the DKH correction. Zero energy corresponds to H(1s)+H(1s) at dissociation. The dotted line corresponds to the C_{3} ∕R^{3} asymptotic extrapolation (hidden within the calculated curves lines in the bottom plot). 
3.1 Unified theory
Starting with the Anderson (1952) theory suitably generalized to include degeneracy, a unified theory of spectral line broadening (Allard et al. 1999) was developed to calculate neutral atom spectra, given the interaction and the radiative transition moments of relevant states of the radiating atom with other atoms in its environment. The unified profiles are the Fourier transforms (FT) of the autocorrelation functions as given by Eq. (121) of Allard et al. (1999), in which the contributions from the different components of a transition enter with their statistical weights. A pairwise additive assumption allows us to calculate I(Δω), when N perturbers interact as the FT of the Nth power of the autocorrelation function ϕ(s) of a unique atomperturber pair. For a perturber density n_{p}, we obtain (5)
where the decay of the autocorrelation function with time leads to atomic line broadening.
For a transition α = (i, f) from an initial state i to a final state f, we have (6)
In Eq. (6) the symbols e and e′ label the energy surfaces on which the interacting atoms approach the initial and final atomic states of the transition. The sum is over all pairs (e, e′) such that as R → ∞. In the equations that follow, we review the meaning of these terms in Eq. (6). In our context, the perturbation of the frequency of the atomic transition during the collision results in a phase shift, η(s), which is calculated along a classical path R(t) that is assumed to be rectilinear. At time t from the point of closest approach, the atoms are separated by (7)
where ρ is the impact parameter of the perturber trajectory, is the relative velocity, and x the position of the perturber along its trajectory at time t = 0. We have for the phase term in Eq. (6) (8)
where ΔV (R), the difference potential, is given by (9)
and represents the difference between the energies of the quasimolecular transition. The potential energy for a state e is (10)
3.2 Satellite bands due to HH and HHe
Within the assumption of additive superposition of interactions, we can write the total profile as the convolution of the individual profiles corresponding to perturbations by H and He, (11)
The unified theory (Allard & Kielkopf 1982) predicts that line satellites are centered periodically at frequencies corresponding to integer multiples of the extrema of ΔV (R) (Eq. (9)). However, their appearance depends on the value of the electronic dipole moments in the region of the maximum of ΔV (R) (Allard et al. 1998a).
The prediction of the shape of the blue wing required us to study the potential energies of the two triplet transitions b →h and b →i ^{3}Π_{g} that contribute to the blue wing of the Lymanα line (Fig. 8). For comparison we overplot the potential energies of the X Σ and C Σ states of the HHe system. The prediction of a line satellite in the blue wing of the HH and HHe line profiles is related to the potential maximum at short distance R = 2−3 Å of the C, h, and i states. This leads to a maximum of the potential energy difference ΔV (R) for these transitions shown in Fig. 9. The electronic states h and b of the isolated radiator are not connected by the dipole moment operator: d_{hb} (R →∞) = 0. As reported in Spielfiedel et al. (2004), the 2, 3 ^{3}Σ_{g} states (labeled h, a) undergo an avoided crossing at equilibrium distance and thus exchange their character. The radiative dipole moment varies dramatically with R (bottom of Fig. 9). Allowed radiative transitions cannot occur between these two states, but d_{hb} (R) differs from zero when a perturber passes close to the radiator. Our theoretical approach allows us to take this asymptotically forbidden transition of quasimolecular hydrogen that dissociates into (1s, 2s) atoms into account. An other important factor is the variation of the dipole moment during the collision once modulated by the Boltzmann factor e^{−βVe(r)} (Eq. (117) of Allard et al. 1999), (12)
The Boltzmann factor in Eq. (12) appears because the perturbing atoms are in thermal equilibrium with the radiating atom, which affects the probability of finding them initially at a given R. In this case, where we consider absorption profiles due to triplet transitions, V_{e} is the b groundstate potential. In Fig. 10 we show D(R) together with the corresponding ΔV (R) for the hb transition. The dipole moment D(R) and the energy difference determining the transition wavelength ΔV (R) are maximum at R_{ext}= 2.8 Å. In thisinstance, a radiative transition is induced by collisions. Figure 10 shows that D(R) is not dependent on temperature throughout the region where the collisioninduced satellite is formed, and therefore the blue wing of Lymanα will not change with increasing temperature in the range of temperatures 10 000–15 000 K.
An examination of Fig. 9 leads us to expect a farther blue satellite for HHe that arises from the extremum of 5000 cm^{−1} when the two atoms are separated by about 2 Å compared to 2100–2500 cm^{−1} for i − b and h − b. Figure 11 shows the distinct wide satellite at about 4200 cm^{−1} owing to the XC transition, whereas the b →h transition yields a blue shoulder centered approximately at 1900 cm^{−1} in the blue wing of Lymanα of HH. Although ΔV (R) for the allowed transition b ^{3}Σ_{u} → i ^{3}Σ_{u} has a maximum (ΔV = 2100 cm^{−1}), it simply contributes to the blue asymmetry as it is blended in the near wing.
The wavelength of the theoretical collisioninduced satellite is largely dependent on the accuracy of the difference potential of the two contributing states to the transition, while the strength of the absorption as a function of wavelength is dependent on the radiative dipole moment shown in Fig. 10 and on the accuracy of the spectral line shape theory. This line satellite has been observed experimentally in the spectrum of a laserproduced plasma source, see Fig. 2 of Kielkopf & Allard (1995) and Fig. 7 of Kielkopf & Allard (1998).
Fig. 8 Top: shortrange part of the repulsive potential curve CΣ (black line) of the HHe molecule compared to the h (dashed blueline) and i ^{3} Π_{g} (dashed red line) states of the H_{2} molecule. Bottom: X (black line) and b (blue line). 
Fig. 9 Top: difference potential for the triplet transitions h − b (dashed blueline), i − b (dotted red line) of HH compared to the X − C transition of HHe. Bottom: electric dipole transition moments between the triplet states of HH. The h − b transition is forbidden asymptotically as it is a transition between the 2s and 1s states of the free hydrogen atom at large R. 
Fig. 10 Variation with temperature in the modulated dipole and the difference potential of the triplet state hb of the HH molecule. T = 15 000 K (dashed red line), T = 12 000 K (black line), and T = 10 000 K (dotted blue line). 
Fig. 11 Blue wings of Lymanα due to HH collisions (blue curve), HHe collisions (dashed red curve), and simultaneous collisions by H and He (black curve). The H and He densities are 10^{18} cm^{−3}, and the temperature is 12 000 K. 
3.3 Collisional profiles simultaneaously perturbed by He and H atoms
The spectra of heliumdominated white dwarf stars with hydrogen in their atmosphere present a distinctive broad feature centered around 1160 Å in the blue wing of the Lymanα line (see Fig. 1 in Allard et al. 2020). Figure 11 shows that this line satellite is quite close to the one due to HH collisions centered at 1190 Å. We caused the ratio n_{He} /n_{H} of their densities to vary. Line profiles that are simultaneously perturbed by H and He are computed for a ratio varying from 10^{−2} to 10^{3}, and the H density n_{H} remains equal to 10^{18} cm^{−3}. When the ratio is 10^{3}, the HHe line profile is identical to a pure helium profile, whereas for 10^{−2}, the HHe line satellite at 1160 Å is not seen. This is illustrated in Fig. 12. The blue wings of Lymanα perturbed by He or H atoms are compared in Fig. 13. The line profile calculations were made at a temperature of 12 000 K for a perturber density of 10^{18} cm^{−3} of He or neutral H. An additional feature is shown in the blue wing of HH Lymanα at 1150 Å. This feature is a line satellite of Lymanβ quite far from the unperturbed Lymanβ line center; it is even closer to the Lymanα line. Figure 13 shows that it is therefore necessary to take the total contribution of the Lymanα and Lymanβ wings of neutral H throughout this region into account.
Fig. 12 Blue wing of Lymanα simultaneously perturbed by He and H atoms for a different ratio of helium and hydrogen densities. From top to bottom, n_{He} /n_{H} is 10^{3}, 10^{2}, 10, 1, 0.1, and 10^{−2}. The Lymanα profiles resulting from perturbation by either HHe or HH collisions separately are overplotted (dotted lines) for ratios of 10^{3} and 10^{−2}, respectively.The temperature is 12 000 K. 
Fig. 13 Comparison of the blue wings of Lymanα perturbed by HHe collisions (black curve) with the blue wing in the Lymanα range (red curve) due to HH collisions. The contribution of the red wing of Lymanβ leads to a large feature at 1150 Å. The H and He densities are 10^{18} cm^{−3}, and the temperature is 12 000 K. 
3.4 Observation of the 1150 Å satellite
This absorption feature due to the  X transition was predicted by Allard et al. (2000). The ab initio calculations of Spielfiedel (2003) have shown that for the isolated radiating atom (R →∞), this transition is not asymptotically forbidden, as was explicitly stated in Allard et al. (2000).
We reported a theoretical study of the variation of the Lymanβ profile with the relative density of ionized and neutral atoms and demonstrated that a ratio of 5 of the neutral and proton density is enough to make this line satellite appear in the far wing (Fig. 5 of Allard et al. 2000). The line satellite appearance is then very sensitive to the degree of ionization and may be used as a temperature diagnostic. In Allard et al. (2004a) we reported its first detection in farUV (FUV) observations of the pulsating DA white dwarf G226−29 obtained with the Far Ultraviolet Spectroscopic Explorer (FUSE). This broad feature was also detected in the laboratory by Kielkopf et al. (2004) and observed in another variable DA star, G18532, by Dupuis et al. (2006).
In Allard et al. (2004c) we discussed in detail how important it is to take the Boltzmann factor in absorption into accountin Eq. (12), especially in stellar atmospheres for temperatures lower than 15 000 K. We considered local thermal equilibrium model atmospheres with a pure hydrogen composition that explicitly include the Lymanα and Lymanβ quasimolecular opacities. For the case of G22629 shown in Fig. 14, we used a very high signaltonoise ratio (S/N) spectrum obtained using timeresolved HST spectra presented by Kepler et al. (2000). This comparison allowed us to make a temperature and gravity determination that is compatible with a fit to the FUSE observation of this object. Figure 4 of Allard et al. (2004a) showed our fit to the HST spectrum using our adopted values for T_{eff}= 12 040 K and log g = 7.93. We extracted data from Fig. 5 of Allard et al. (2004c), where synthetic spectra in the Lymanβ range are compared with a FUSE spectrum of G22629, to show in Fig. 14 the synthetic spectrum obtained for T = 12 040 K and log g = 7.93. G22629 was more recently observed with HST COS under program 14 076, and we overplot the spectrum of the G130M_{2} grating thatcovers 1130–1270 Å. The HST COS observation is noisy but consistent with the one obtained by FUSE and fills the gap above 1180 Å where the shoulder in the blue wing of Lymanα appears. This part of the spectrum 1180–1200 Å could not be obtained with FUSE or HST. However, we should point out that we need to divide the COS flux by a factor of ~1.3 to bring it to the FUSE spectrum, we had a similar problem in Allard et al. (2004a) to fit the FUSE and IUE flux. We considered that this difference by a factor of ~1.3 in the flux calibration of observations performed with two different instruments of a faint target was acceptable, but obtaining the same factor with COS would mean that the error is likely due to FUSE.
Fig. 14 Comparison of the synthetic spectrum obtained with T_{eff} = 12 040 K and log g = 7.93 (green line) in the Lymanβ range with a FUSE spectrum of G22629 (red line; extracted from Fig. 5 of Allard et al. 2004c). We overplot the COS observation of G22629 (Gaensicke 2016; orange line). 
4 Conclusions
The effect of collision broadening by atomic H and He on spectral lines is central for understanding the opacity of stellar atmospheres. A correct determination of the Lymanα line requires the determination of the ground and first excited potential energy curves and the electric transition dipole moments with high accuracy. We showed for HHe how important it is to account for relativistic effects to characterize the longdistance behavior of the potentials in detail and the avoided crossing situations. These effects, although tiny, become crucial in determining the character and the adiabatic correlation of the states, and in particular, the behavior of the associated dipole transition moments from the ground state, because of the specific degeneracy of hydrogenexcited states in the Coulomb model. This problem does not seem so stringent for H+H (n = 2) collisions. Due to the 1∕R^{3} asymptotic behavior of the states that also undergo avoided crossings, but at very large separation, in this case, the behavior of the system is expected to be fully diabatic, except perhaps in ultracold and ultraslow systems.
Our study was conducted assuming classical motion for the nuclei, as well as an adiabatic picture for the electronic states during the collisional process. The socalled diagonal adiabatic corrections (Kolos & Wolniewicz 1968; Pachucki & Komasa 2014; Komasa et al. 1999; Gherib et al. 2016), which mostly contribute at short distance, might also be added for an improved accuracy. Moreover, it should be mentioned that in the regions in which the potential curves of different states are very close, such as at large separation especially in the HHe case or in shortrange avoided crossing regions such as in HH, the BornOppenheimer approximation defining the adiabatic states is likely to break down, and it might be necessary to take offdiagonal nonadiabatic couplings and collisional branchings between the adiabatic states into account. Lique et al. (2004) took the rotational coupling between states and C^{1} Π_{u} in HH collisions into account and concluded that the nonadiabatic effects with respect to the adiabatic treatment were very weak in this case. The effect of nonadiabatic couplings remains an open question in the triplet case, which presents avoided crossings at short distance.
The HST observations have motivated this theoretical work, in which we used accurate molecular data for both the HH (Spielfiedel 2003; Spielfiedel et al. 2004) and HHe (this work) and extended that of Allard et al. (2020). This allowed a thorough study of the atomic underlying atomic physics and accurate line profile calculations of Lyman lines perturbed by collisions with H and He atoms given here. Furthermore, it is also very gratifying to observe features that have been predicted theoretically. This was the case of the 1150 Å broad feature in the Lymanβ wing of the FUSE spectrum of the DA white dwarf G22629 and now this one at 1160 Å in the blue wing of COS spectra of DBA white dwarfs. The COS observation of G22629 was also an opportunity to reconsider the blue wing of Lymanα. Finally, ourstudy is a first step toward obtaining the accurate data for both Lymanα and for Balmerα that are essential for determining the hydrogen abundance correctly.
Appendix A Additional tables
Exponents of Gaussiantype functions on hydrogen added to the augccpV6Z basis set.
Spectroscopic constants of the molecular states dissociating into H(2s,2p)+He(1s^{2}).
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All Tables
Symmetry and labeling of molecular states dissociating into H(2s,2p)+He and H(2s,2p)+H.
Exponents of Gaussiantype functions on hydrogen added to the augccpV6Z basis set.
Spectroscopic constants of the molecular states dissociating into H(2s,2p)+He(1s^{2}).
All Figures
Fig. 1 MRCI adiabatic potential energy curves of HHe correlated with the 2s, 2p atomic levels. The zero energy corresponds to H(1s)+He(1s^{2}) at dissociation. 

In the text 
Fig. 2 Long range zoom of the MRCI adiabatic HHe potential energy curves of states A, B and C dissociating into H(n = 2)+He. The zero energy corresponds to H(1s)+He(1s^{2}) at dissociation. 

In the text 
Fig. 3 Spinorbitless transition dipole moments of HHe from the X ground statetoward the adiabatic A, B, and C states of HHe. 

In the text 
Fig. 4 Adiabatic H(n = 2)+He potential energy curves of molecular states including SO coupling. For better display, the red line of state (1)3∕2 state is shown with superimposed dots. Zero energy corresponds to H(1s)+He(1s^{2}) at dissociation. 

In the text 
Fig. 5 Longrange zoom of the adiabatic potential energy curves of H(n = 2)+He states including SO coupling. Zero energy corresponds to H(1s)+He(1s^{2}) at dissociation. 

In the text 
Fig. 6 Transition dipole moments from the X_{1∕2} ground statetoward the adiabatic states dissociating into H(n = 2)+He including SO coupling. For better display, the red line of the X_{1∕2} − (1)3∕2 transition dipole moment is shown with superimposed dots. 

In the text 
Fig. 7 Longrange zoom of the adiabatic potential energy curves of states , , , and of H+H(n = 2). Top: data of Spielfiedel et al. (priv. comm.) without scalar relativistic terms. Bottom: our CI results including the DKH correction. Zero energy corresponds to H(1s)+H(1s) at dissociation. The dotted line corresponds to the C_{3} ∕R^{3} asymptotic extrapolation (hidden within the calculated curves lines in the bottom plot). 

In the text 
Fig. 8 Top: shortrange part of the repulsive potential curve CΣ (black line) of the HHe molecule compared to the h (dashed blueline) and i ^{3} Π_{g} (dashed red line) states of the H_{2} molecule. Bottom: X (black line) and b (blue line). 

In the text 
Fig. 9 Top: difference potential for the triplet transitions h − b (dashed blueline), i − b (dotted red line) of HH compared to the X − C transition of HHe. Bottom: electric dipole transition moments between the triplet states of HH. The h − b transition is forbidden asymptotically as it is a transition between the 2s and 1s states of the free hydrogen atom at large R. 

In the text 
Fig. 10 Variation with temperature in the modulated dipole and the difference potential of the triplet state hb of the HH molecule. T = 15 000 K (dashed red line), T = 12 000 K (black line), and T = 10 000 K (dotted blue line). 

In the text 
Fig. 11 Blue wings of Lymanα due to HH collisions (blue curve), HHe collisions (dashed red curve), and simultaneous collisions by H and He (black curve). The H and He densities are 10^{18} cm^{−3}, and the temperature is 12 000 K. 

In the text 
Fig. 12 Blue wing of Lymanα simultaneously perturbed by He and H atoms for a different ratio of helium and hydrogen densities. From top to bottom, n_{He} /n_{H} is 10^{3}, 10^{2}, 10, 1, 0.1, and 10^{−2}. The Lymanα profiles resulting from perturbation by either HHe or HH collisions separately are overplotted (dotted lines) for ratios of 10^{3} and 10^{−2}, respectively.The temperature is 12 000 K. 

In the text 
Fig. 13 Comparison of the blue wings of Lymanα perturbed by HHe collisions (black curve) with the blue wing in the Lymanα range (red curve) due to HH collisions. The contribution of the red wing of Lymanβ leads to a large feature at 1150 Å. The H and He densities are 10^{18} cm^{−3}, and the temperature is 12 000 K. 

In the text 
Fig. 14 Comparison of the synthetic spectrum obtained with T_{eff} = 12 040 K and log g = 7.93 (green line) in the Lymanβ range with a FUSE spectrum of G22629 (red line; extracted from Fig. 5 of Allard et al. 2004c). We overplot the COS observation of G22629 (Gaensicke 2016; orange line). 

In the text 
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