Issue 
A&A
Volume 659, March 2022



Article Number  A157  
Number of page(s)  7  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/202142717  
Published online  22 March 2022 
Collisioninduced satellite in the blue wing of the Balmerβ line and consequences on the Balmer series
^{1}
Laboratoire de Chimie et de Physique Quantiques, Fédération FERMI, 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:
22
November
2021
Accepted:
3
January
2022
In this paper we emphasize the nonLorentzian behavior of the Balmer series in heliumdominated DBA white dwarf stars for which the decadesold problem exists for the determination of the hydrogen abundance. In a very recent work, we have shown that quasimolecular line satellites due to HHe and HH collisions are responsible for the asymmetrical shape of the Lymanα lines observed with the Cosmic Origin Spectrograph and that a similar asymmetry exists for the Balmerα line profiles. In continuation with very recent work, where the n = 2, 3 potential energies and transition dipole moments from the ground state were determined, here, we present accurate HHe potential energies and electronic transition dipole moments concerning the molecular states correlated with H(n = 4)+He and their transition dipole moments with the states correlated with H(n = 2)+He. Those new data are used to provide a theoretical investigation of the collisional effects in the blue wing of the Balmerβ line of H perturbed by He. Because of the general trend characterizing the repulsive Σ states of the potential energies involved in the Balmer series, the amplitude in the core of the line is decreasing very fast with the order of the series when the helium density gets as large as 10^{21} cm^{−3}. This study is undertaken by applying a unified theory of spectral line broadening that is valid at very high helium densities found in DZA white dwarf stars. The treatment includes collisioninduced line satellites due to asymptotically forbidden transitions, and it explains the asymmetry observed in their spectra.
Key words: line: profiles / white dwarfs / molecular data
© F. Spiegelman et al. 2022
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 the spectra of white dwarf stars exhibiting strong heliums lines with weaker hydrogen lines, that is to say DBA white dwarfs, there is a discrepancy between the values of the hydrogen abundances determined from Balmerα in the visible and those from Lymanα in the ultraviolet (see Xu et al. 2017, and references therein). The existence of close line satellites in the blue wing of these lines was found in detailed collisional broadening profiles computed for both HHe and HH (Spiegelman et al. 2021; Allard et al. 2022). These features are responsible for the asymmetrical shape and the decreasing strength of the core of the lines in those cases. In this paper we extend our study to the Balmerβ line by considering the H(n = 4) perturbed by He.
In recent previous works, we used new multireference configuration interaction (MRCI) calculations of the excited state potential energy curves dissociating into H(n = 1,2,3)+He, as well as the relevant electric dipole transition moments contributing to the Lymanα and Balmerα spectra. We illustrated how tiny relativistic effects affect the asymptotic correlation of the HHe adiabatic states and change the related transition dipole moments.
In the present work, we determine potential energy curves dissociating into H(n = 4)+He and transition dipole moments of HHe involving initial and final states correlated with H(n = 2)+He and H(n = 4)+He, respectively. Between these states, there are 23 HHe transitions which generate the complete Balmerβ line profile. The Σ − Σ transitions provide the essential contribution to the blue wing of the Balmer series lines, explain their asymmetrical behavior, and are relevant to the cool DBA white dwarf analyses. The potential energy curves and transition dipole moments are discussed in Sect. 2. Unified profile calculations are presented in Sect. 3. They show that in Herich white dwarf stars, the nonLorentzian shape of Balmerβ increases with He density and it will affect abundance analyses if not included in stellar atmosphere models. Consequently the strength and shape of the Balmer line series are also diagnostics of the atmospheric conditions of cool DZA white dwarf stars.
In Sect. 4 we consider the general trend of the repulsive Σ state of the excited states of HHe. They contribute through the Lyman and Balmer lines and lead to a nonLorentzian shape that is increasingly significant in higher series members that affect the visible spectrum of heliumrich white dwarfs.
2. Diatomic HHe potentials and electronic transition dipole moments
The calculation scheme for the potentials dissociating into H(n = 2,3,4)+He is the same as described in our previous publications concerned with the Lymanα (Spiegelman et al. 2021) and the Balmerα (Allard et al. 2022) broadenings. Briefly, a MRCI (Knowles & Werner 1992; Werner et al. 2015) calculation was run within an extensive Gaussiantype orbital (GTO) basis set. Scalar relativistic corrections (namely the Darwin and massvelocity terms) according to the DouglasKrollHess (DKH) scheme (Reiher 2006; Nakajima & Hirao 2011), breaking the specific degeneracy of the atomic hydrogen levels determined with the Coulomb Hamiltonian only, were added. In order to improve the description of the adiabatic molecular states of HHe toward the n = 4 atomic limit, we refined the basis set used in our previous works (Spiegelman et al. 2021; Allard et al. 2022), consisting of an augccpv6z quality description complemented by diffuse exponents for all s, p, d, f, g, and h functions for H and He, reoptimizing the exponents of the external d and f GTO functions of hydrogen (see Table A.1). Table A.2 compares the calculated transitions from 1s to n = 4 with the javeraged experimental ones. The calculated 4s − 4f splitting is 0.14 cm^{−1} larger, but of the same order of magnitude as the experimental value 0.061 cm^{−1}.
In the following, we use the spectroscopist’s notations X, A, B, and C to label the lower adiabatic molecular states of HHe, namely the ground state 1 ^{2}Σ^{+}, the lowest excited states 2 ^{2}Σ^{+}, 1 ^{2}Π, and 3 ^{2}Σ^{+} correlated with H(n = 2)+He, while we label the upper states correlated with H(n = 4)+He according to their adiabatic ranking in their respective symmetries. For a detailed characterization of states n = 2, 3, we refer readers to our previous papers (Spiegelman et al. 2021; Allard et al. 2022) and to earlier literature (Theodorakopoulos et al. 1987; Ketterle et al. 1985, 1988; Ketterle 1989; Sarpal et al. 1991; Lo et al. 2006; van Hemert & Peyerimhoff 1991; Ketterle 1990a,b; Allard et al. 2020). The molecular states dissociating into 4s, 4p, 4d, and 4f are (7–10) ^{2}Σ^{+}, (4–6) ^{2}Π, (2,3) ^{2}Δ, and 1 ^{2}Φ. Figure 1 shows the potential energy curves of the states dissociating into H(n = 2)+He and H(n = 4)+He. All states are bound with equilibrium distances close to 0.77 Å, most of them have similar dissociation energies around 16 500 cm^{−1} and vibrational constants ω_{exe}, and they can hardly be individually identified at the scale of the figure. Those dissociation energies are also quite similar to those of the HeH^{+} ion (R_{e} = 0.7742 Å, D_{e} = 16 455.64 cm^{−1}, and ω_{e} = 3220 cm^{−1}) calculated by Koos & Peek (1976), which are consistent with the Rydberg nature of those states (Ketterle 1990c,d). The detailed spectroscopic constants are given in the Table A.3. The most stable state is 7 ^{2}Σ^{+} with a dissociation energy of D_{e} = 16 882 cm^{−1}. Conversely, state 10 ^{2}Σ^{+} is the least stable (D_{e} = 15 157 cm^{−1}), and it has a barrier to dissociation. This feature is analog to those observed for the respective highest states dissociating into the n = 2 and n = 3 configurations, namely C^{2}Σ^{+} and 6 ^{2}Σ^{+}. To our knowledge, no previous theoretical data exist for the molecular states of HHe correlated with n = 4. A detailed experimental investigation was published by Ketterle (1990c) who could characterize states 4 ^{2}Π, 5 ^{2}Π, and 2 ^{2}Δ. The present calculated v′ = 0 to v″ = 0 vibrational transitions (T_{00} values) are in agreement with Ketterle’s data with a rms deviation of 16 cm^{−1} (see Table A.3). At long distance in the range 17–28 Å, the states undergo multiple avoided crossings, which determine their correlation with the atomic asymptotes, as illustrated in Fig. 2.
Fig. 1. MRCI adiabatic potential energy curves of molecular states dissociating into H(n = 2,4)+He. The R scale is a logarithmic graduation. The dashed lines correspond to potential energy curves dissociating into H(n = 3)+He. 
Fig. 2. Long range zoom on the adiabatic potential energy curves of H(n = 4)+He. 
All transition dipole moments D(R) between the n = 2 and n = 4 adiabatic states were calculated using the MRCI wavefunctions, namely transitions from A^{2}Σ^{+}, C^{2}Σ^{+} to 7,8,9,10 ^{2}Σ^{+} and to 4,5,6 ^{2}Π, from B^{2}Π to 7,8,9,10 ^{2}Σ^{+}, 4,5,6 ^{2}Π and 2,3 ^{2}Δ. The A^{2}Σ^{+}, C^{2}Σ^{+}, B^{2}Π − 10^{2}Σ^{+} asymptotically forbidden transitions are involved in the blue wing contribution, as discussed below. The A^{2}Σ^{+}, C^{2}Σ^{+} − 7,8,9,10 ^{2}Σ^{+} transition moments display a rather complex picture shown in Fig. 3. Their evolution can be rationalized considering the following three key features: (i) the avoided crossing between the lower states A^{2}Σ^{+} and C^{2}Σ^{+} around R = 8.1 Å (Spiegelman et al. 2021), (ii) the multiple avoided crossing between the upper states (7 − 10)^{2}Σ^{+} in the range R = 17–28 Å illustrated in Fig. 2, and (iii) other avoided crossings between the upper states at short distance of R ≤ 5 Å.
Fig. 3. Transition dipole moments D(R) of HHe between ^{2}Σ^{+} states n = 2 and n = 4. 
The situation is thus similar to what was observed for the Lymanα and Balmerα molecular transitions: while relativistic effects, partly taken into account here, obviously play a negligible role in the energies of the H(4f)+He states (less than ≈0.1 cm^{−1}) and in the transition dipole moments at short distance, they break the Coulomb asymptotic degeneracy and induce cascade avoided crossings which determine the asymptotic correlation of the adiabatic states and the variation of the dipole transition moments at medium and long distance.
3. Analysis of the Balmerβ profile perturbed by He atoms
The highest molecular states correlated to n = 2 and n = 3 have a prominent role in the appearance of blue satellite features in Lymanα (Spiegelman et al. 2021) and Balmerα (Allard et al. 2022) line profiles. If we define the potential energy V(R) for a state e as
the prediction of a line satellite in the blue wing is related to the maximum of V(R) of these repulsive states. The upper box of Fig. 4 shows a comparison of the shortrange part of the potential curve V(R) of the following repulsive states: C^{2}Σ^{+}, 6 ^{2}Σ^{+}, and 10 ^{2}Σ^{+}, corresponding to the highest states correlated to the n = 2, n = 3, and n = 4 levels, respectively. With increasing n, the barrier height decreases (5386 cm^{−1}, 1390 cm^{−1}, and 574 cm^{−1} for n = 2, 3, and 4, respectively) and the repulsion of the state above the asymptote rises at greater distances, namely R = 8, 15, and 25 Å for n = 2, 3, and 4, as can be seen in Figs. 1 and 2.
Fig. 4. Repulsive potentials and their difference. Top: potential energy V of the repulsive states of Lymanα (dashed black curve), Balmerα (dashed red curve), and Balmerβ (blue curve). Bottom: difference potential ΔV(R) for the Lymanα transition X → C (dashed black curve), for the Balmerα transition C → 6^{2}Σ^{+} (dashed red curve), and for the Balmerβ transitions C → 10^{2}Σ^{+} (blue curve). 
It is essential to use a general unified theory in which the electric dipole moment varies during a collision because all transitions of Balmerβ from the 2s and 2p states to the 4f state are asymptotically forbidden. Most of the problems in collisioninduced radiative transitions have been solved within the oneperturber approximation. At low densities, the binary model for an optically active atom in collision with one perturber is valid for the whole profile, except for the central part of the line. In dense plasmas, as in very cool DZ white dwarfs (Allard et al. 2016; Kawka et al. 2021) and liquid helium clusters (Allard et al. 2013), the possibility of several atoms interacting strongly is high, and the effects play a role in the wavelength of the line center, for example in the shift of the line as well as the general shape of the line profile. For such a high perturber density, the collisional effects should be treated by using the autocorrelation formalism in order to take simultaneous collisions with more than one perturbing atom into account. A pairwise additive assumption allows us to calculate the total profile I(Δω), when all the perturbers interact through the Fourier transform (FT) of the Nth power of the autocorrelation function ϕ(s) of a unique atomperturber pair. Therefore
That is to say, we neglect the interperturber correlations. The radiator can interact with several perturbers simultaneously, but the perturbers do not interact with each other. It is what Royer (1980) calls the "totally uncorrelated perturbers approximation”. The fundamental result expressing the autocorrelation function for many perturbers in terms of a single perturber quantity g(s) was first obtained by Anderson (1952) and Baranger (1958b) in the classical and quantum cases, respectively. From the point of view of a general classical theory, the solution to the Anderson (1952) model corresponds to the first order approximation in the gas density obtained by the cumulant expansion method (Royer 1972). The higher order terms representing correlations between the perturbers are neglected since they are extremely complicated (Royer 1972; Kubo 1962a,b; Kampen 1974). In Allard et al. (1999), we derived a classical path expression for a pressurebroadened atomic spectral line shape that allows for an electric dipole moment that is dependent on the position of perturbers, which is not included in the approximations of Anderson (1952) and Baranger (1958a,b).
The spectrum I(Δω) can be written as the FT of the dipole autocorrelation function Φ(s) ,
where s is time. The FT in Eq. (3) is taken such that I(Δω) is normalized to unity when integrated over all frequencies, and Δω is measured relative to the unperturbed line. We have provided an overview of the unified theory in Sect. 3 of Spiegelman et al. (2021).
The unified theory predicts that line satellites will be centered periodically at frequencies corresponding to integer multiples of the extrema of ΔV(R), the difference between the energies of the quasimolecular transition, defined as follows:
The bottom of Fig. 4 shows ΔV(R) for the transitions X → C related to Lymanα, C → 6 ^{2}Σ^{+} related to Balmerα compared to ΔV(R) for the transition C → 10 ^{2}Σ^{+} related to Balmerβ. The main contribution to the blue wing is due to the forbidden C 2p^{2}Σ^{+} →4f 10^{2}Σ^{+} transition. The transition dipole moment D(R) between the C 2p electronic state and the 4f state asymptotically vanishes as the radiative transition between the corresponding atomic states is forbidden (Fig. 3). Although this transition should not contribute to the unperturbed line profile, radiative transitions can be induced by close collisions because D(R) differs from zero when a He atom passes close to the H atom. To point out the importance of the variation of the dipole moment on the formation of collisioninduced (CI) satellite, we have displayed D(R) together with the corresponding ΔV(R) for the C 2p − 4f transition in Fig. 5. The satellite amplitude depends on the value of D(R) through the region of the potential maximum responsible for the satellite and on the position of this extremum (Allard et al. 1998). The presence of quasimolecular satellites due to collision effects is unimportant when the He density is low; however, in the physical conditions found in the cool atmospheres of Herich white dwarfs, their presence leads to an asymmetrical shape of the Balmer profiles which is a signature of a high helium density. Figure 6 shows a very broad blue wing with a close CI line satellite at about 200 cm^{−1} from the line center corresponding to the maximum of ΔV(R). The line profile obtained when taking only the allowed transitions into account is overplotted. The main objective of this figure is to illustrate the contribution of the forbidden transitions to the Balmerβ profile. Figure 7 shows that the presence of a line satellite in the near line wing leads to a complex behavior of the dependence of the line shape on He density. As in the case of the DZA white dwarf L74546A obtained at the ESO La Silla 3.6m telescope, the observed spectrum of Ross 640 obtained at Calar Alto (Fig. 7 in Koester & Wolff 2000) shows an observed Balmerβ line with a very broad blue wing due to high He densities which can reach 10^{21} cm^{−3} in the Balmerα and β formation region (Vennes, priv. comm.). Figure 8 shows a very broad line profile at this He density for both Balmer lines. In particular, we have found that the Balmerβ line profile is totally asymmetric in this range of He density. This effect is already present at the lower density, 2 × 10^{20} cm^{−3}, and it has an increasing importance with He density (Fig. 7). The development of the blue wing leads to the center of the main line being overwhelmed by the blend of line satellites. As a result, the full treatment of the Balmerβ line reveals strong effects on its opacity outside the impact approximation. The Lorentzian approximation is overplotted in Figs. 6 and 7. Consequently, we conclude that these collisioninduced effects are not always negligible when models of stellar spectra are compared with observations to determine abundances, structural properties, or ages.
Fig. 5. Difference potential energy ΔV(R) in cm^{−1} (black line) and dipole moment D(R) (black dashed line) for the forbidden C 2p → 4f 10^{2}Σ^{+} transition. 
Fig. 6. Comparison of the total profile (blue line) to the profile restrictly obtained with the allowed transitions (red dashed line). The Balmerβ line profile is also compared to the Lorentzian approximation (black dashed curve). The He density is 5 × 10^{20} cm^{−3}, and the temperature is 8000 K. 
Fig. 7. Variation with the He density of the Balmerβ line profile for 2 × 10^{20} (black line), 3 × 10^{20} (blue line), and 5 × 10^{20} cm^{−3} (red line); the temperature is 8000 K. The Balmerβ line profiles are also compared to the Lorentzian approximation for the He densities 2 × 10^{20} cm^{−3} (dashed black line) and 3 × 10^{20} cm^{−3} (dashed blue line). 
4. Consequences on the Balmer series
Figure 1 illustrates the potential energies involved in Lymanα, Balmerα, and Balmerβ. Due to the more diffuse character of the 4s, 4p, 4d, and 4f orbitals with respect to those with smaller n, the barrier height of the highest state extends to a wider distance range and is lower than that of states C^{2}Σ^{+} and 6^{2}Σ^{+}. The characteristics of the highest states must be stressed in view of the study of the asymmetrical shape dependence on the order of the Balmer series. The top of Fig. 4 summarizes the potential energies of the highest states involved in Lymanα, Balmerα, and Balmerβ, and the bottom of it shows the corresponding ΔV(R). The essential characteristics to note are that the maximum in ΔV occurs at larger internuclear distances (R_{max} ∼ 10 Å) for Balmerβ than for Balmerα (R_{max} ∼ 5.4 Å) and Lymanα (R_{max} ∼ 2 Å). ΔV(R) presents a flat maximum at a larger distance for Balmerβ. The average number of perturbers in the interaction volume at R_{max} is the most determining parameter for the amplitude of the satellites in the spectral line (Allard 1978; Royer 1978; Allard & Kielkopf 1982). This dependence on the average number of perturbers in the collision volume is expected on the basis of the Poisson distribution, which indicates the probability of finding a given number of uncorrelated perturbers in the collision volume. It was decisively identified in the theoretical analysis of experimental Cs spectra by Kielkopf & Allard (1979). The other crucial point is that the maximum ΔV_{max} is smaller with 350 cm^{−1} for Balmerβ, versus 1000 cm^{−1} for the Balmerα line and 5000 cm^{−1} for the Lymanα line. The Balmerβ line satellite gets closer to the main line than the Balmerα and Lyman satellites, as shown in Fig. 8, Fig. 6 of Allard et al. (2022), and Fig. 11 of Spiegelman et al. (2021). Its effect on the asymmetrical shape of the Balmer profile and on its amplitude becomes more important and appears at a lower density.
Fig. 8. Comparison of the Balmerα (black curves) and Balmerβ (red curves) line profiles. The He densities are 10^{21} cm^{−3} (full curves) and 6 × 10^{20} cm^{−3} (dashed curves); the temperature is 8000 K. 
Because of this general trend characterizing the repulsive Σ states, the average number of perturbers in the interaction volume will get larger for higher series of Balmer lines, leading to a higher probability of multiplepertuber effects. The core of the main line due to allowed transitions will disappear, being replaced by the blend of multiple satellites (Kielkopf 1983, 1985). This is what is already happening in the Balmerβ line when the He density gets as large as 10^{21} cm^{−3}. Figure 8 shows a broad line profile with a decreasing asymmetry and a change in the sign of its shift resulting from the disappearance of the main line due to the allowed transitions replaced by the blend of multiple line satellites. Increasing the He density has the same effect as considering a higher member of the Balmer series as it increases the probability of multiple pertuber effects. This behavior is well known in line broadening theory and has been thoroughly studied in the 1950s by the Ch’en group (Allard & Kielkopf 1982). The spectra of members of the Balmer series corresponding to a transition to upper level n are then progressively more blueshifted and shallower as n increases. The observed spectra of PG 1157+004 (Fig. 17, 11598+007 of Limoges et al. 2015) or of WDJ01030522 (Fig. 2 of Tremblay et al. 2020), which are both massive white dwarfs with a very high log g, exhibit a strong asymmetry and a blue shift, providing further evidence that we are dealing with an unresolved blend of quasimolecular line satellites due to high He densities. For PG 1157+004, the helium density is 3 × 10^{20} cm^{−3} in the region of the formation of the Balmerα and β lines (Vennes, priv. comm.). Figure 7 shows that even at this rather low density, the Balmerβ line profile shows an asymmetrical shape. For cooler atmospheres, as in the case of WDJ01030522, the He density is as high as 10^{21} (Vennes, priv. comm.). Many explanations have tried to solve that puzzle of such white dwarf spectra, such as two sources contributing to the Balmer features (Limoges et al. 2015) or magnetic fields (Tremblay et al. 2020).
The resonance broadening of H perturbed by collisions with H atoms produces asymmetry in the Balmerα line profiles similar to that due to HHe (Allard et al. 2022), but the H density in DA white dwarfs is not as large as the He density in cool Herich DBA to have such an effect in their observed spectra. This a major difference between Balmer spectra of a DA or of a cool DBA white dwarf. More generally, the important sensitivity of the Balmer line shapes to the nature of the perturber and its density suggests that they could be used as a diagnostic tool given precision lownoise observed spectra and appropriate line shape theory based on the high accuracy now provided by firstprinciples theoretical atomic and molecular physics.
In conclusion, an accurate determination of neutralcollisionbroadened lines is required to interpret the strength, broadening, and shift of the resulting Balmer line profiles correctly. Calculations reported in Allard et al. (2022) for Balmerα and in this paper for Balmerβ, especially for HH collisions, involve a huge number of transitions, and they are dependent on very accurate molecular data. The past laboratory experimental work and analysis of observations of multipleperturber effects on the broadening of atomic spectral lines such as by Kielkopf (1983, 1985) is also fundamental, as we have pointed out because it confirms that the higher the excitation of the atomic state, the larger the effects of multiple perturbers are at a lower density in the laboratory (Kielkopf & Allard 1979), as shown in Fig. 8 for stellar atmospheres.
Acknowledgments
We would like to thank Adela Kawka and Stephane Vennes who initiated this study. We also acknowledge with appreciation Stephane Vennes, for his determinations of He density in the Balmerα and β formation regions of the DZ white dwarfs which are cited in the paper. We thank the anonymous referee for helpful comments that improved the manuscript.
References
 Allard, N. F. 1978, J. Phys. B: At. Mol. Opt. Phys., 11, 1383 [NASA ADS] [CrossRef] [Google Scholar]
 Allard, N. F., & Kielkopf, J. F. 1982, Rev. Mod. Phys., 54, 1103 [Google Scholar]
 Allard, N. F., Drira, I., Gerbaldi, M., Kielkopf, J. F., & Spielfiedel, A. 1998, A&A, 335, 1124 [NASA ADS] [Google Scholar]
 Allard, N. F., Royer, A., Kielkopf, J. F., & Feautrier, N. 1999, Phys. Rev. A, 60, 1021 [Google Scholar]
 Allard, N. F., Nakayama, A., Spiegelman, F., Kielkopf, J. F., & Stienkemeier, F. 2013, Eur. Phys. J. D, 67, 52 [NASA ADS] [CrossRef] [Google Scholar]
 Allard, N. F., Leininger, T., Gadéa, F. X., BrousseauCouture, V., & Dufour, P. 2016, A&A, 588, A142 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Allard, N. F., Kielkopf, J. F., Xu, S., et al. 2020, MNRAS, 494, 868 [Google Scholar]
 Allard, N. F., Spiegelman, F., Kielkopf, J. F., & Bourdreux, S. 2022, A&A, 657, A121 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Anderson, P. W. 1952, Phys. Rev., 86, 809 [Google Scholar]
 Baranger, M. 1958a, Phys. Rev., 111, 481 [NASA ADS] [CrossRef] [Google Scholar]
 Baranger, M. 1958b, Phys. Rev., 111, 494 [NASA ADS] [CrossRef] [Google Scholar]
 Kampen, N. G. V. 1974, Physica, 74, 215 [NASA ADS] [CrossRef] [Google Scholar]
 Kawka, A., Vennes, S., Allard, N. F., Leininger, T., & Gadéa, F. X. 2021, MNRAS, 500, 2732 [Google Scholar]
 Ketterle, W. 1989, Phys. Rev. Lett., 62, 1480 [CrossRef] [Google Scholar]
 Ketterle, W. 1990a, J. Chem. Phys., 93, 3752 [NASA ADS] [CrossRef] [Google Scholar]
 Ketterle, W. 1990b, J. Chem. Phys., 93, 3760 [NASA ADS] [CrossRef] [Google Scholar]
 Ketterle, W. 1990c, J. Chem. Phys., 93, 6929 [NASA ADS] [CrossRef] [Google Scholar]
 Ketterle, W. 1990d, J. Chem. Phys., 93, 6935 [NASA ADS] [CrossRef] [Google Scholar]
 Ketterle, W., Figger, H., & Walther, H. 1985, Phys. Rev. Lett., 55, 2941 [Google Scholar]
 Ketterle, W., Dodhy, A., & Walther, H. 1988, J. Chem. Phys., 89, 3442 [Google Scholar]
 Kielkopf, J. 1983, J. Phys. B At. Mol. Phys., 16, 3149 [NASA ADS] [CrossRef] [Google Scholar]
 Kielkopf, J. 1985, J. Quant. Spectr. Rad. Transf., 33, 267 [NASA ADS] [CrossRef] [Google Scholar]
 Kielkopf, J. F., & Allard, N. F. 1979, Phys. Rev. Lett., 43, 196 [NASA ADS] [CrossRef] [Google Scholar]
 Knowles, P. J., & Werner, H.J. 1992, Theor. Chim. Acta, 84, 95 [CrossRef] [Google Scholar]
 Koester, D., & Wolff, B. 2000, A&A, 357, 587 [NASA ADS] [Google Scholar]
 Koos, W., & Peek, J. M. 1976, Chem. Phys., 12, 381 [NASA ADS] [CrossRef] [Google Scholar]
 Kramida, A., Ralchenko, Yu., Reader, J., & NIST ASD Team 2020, NIST Atomic Spectra Database (ver. 5.8) (Gaithersburg, MD: National Institute of Standards and Technology), https://physics.nist.gov/asd [Google Scholar]
 Kubo, R. 1962a, J. Phys. Soc. Jpn., 17, 1100 [NASA ADS] [CrossRef] [Google Scholar]
 Kubo, R. 1962b, J. Math. Phys., 4, 174 [Google Scholar]
 Limoges, M. M., Bergeron, P., & Lépine, S. 2015, ApJS, 219, 19 [NASA ADS] [CrossRef] [Google Scholar]
 Lo, J. M. H., Klobukowski, M., BielińskaWaz, D., Schreiner, E. W. S., & Diercksen, G. H. F. 2006, J. Phys. B: At. Mol. Opt. Phys., 39, 2385 [NASA ADS] [CrossRef] [Google Scholar]
 Nakajima, T., & Hirao, K. 2011, Chem. Rev., 112, 385 [Google Scholar]
 Reiher, M. 2006, Theor. Chem. Acc., 116, 241 [Google Scholar]
 Royer, A. 1972, Phys. Rev. A, 6, 1741 [NASA ADS] [CrossRef] [Google Scholar]
 Royer, A. 1978, Acta Phys. Pol. A, 54, 805 [Google Scholar]
 Royer, A. 1980, Phys. Rev. A, 22, 1625 [NASA ADS] [CrossRef] [Google Scholar]
 Sarpal, B. K., Branchett, S. E., Tennyson, J., & Morgan, L. A. 1991, J. Phys. B: At. Mol. Opt. Phys., 24, 3685 [NASA ADS] [CrossRef] [Google Scholar]
 Spiegelman, F., Allard, N., & Kielkopf, J. 2021, A&A, 651, A51 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Theodorakopoulos, G., Petsalakis, I. D., Nicolaides, C. A., & Buenker, R. J. 1987, J. Phys. B, 20, 2339 [Google Scholar]
 Tremblay, P. E., Hollands, M. A., Gentile Fusillo, N. P., et al. 2020, MNRAS, 497, 130 [NASA ADS] [CrossRef] [Google Scholar]
 van Hemert, M. C., & Peyerimhoff, S. D. 1991, J. Chem. Phys., 94, 4369 [NASA ADS] [CrossRef] [Google Scholar]
 Werner, H. J., Knowles, P. J., Knizia, G., et al. 2015, MOLPRO, version 2015.1, a package of ab initio programs [Google Scholar]
 Xu, S., Zuckerman, B., Dufour, P., et al. 2017, ApJ, 836, L7 [Google Scholar]
Appendix A: Complementary HHe molecular data
Complementary basis set information
Calculated atomic energy levels of Hydrogen (n = 4) vs experiment
Spectroscopic constants of HHe molecular states correlated with H(n = 4)+He
All Tables
All Figures
Fig. 1. MRCI adiabatic potential energy curves of molecular states dissociating into H(n = 2,4)+He. The R scale is a logarithmic graduation. The dashed lines correspond to potential energy curves dissociating into H(n = 3)+He. 

In the text 
Fig. 2. Long range zoom on the adiabatic potential energy curves of H(n = 4)+He. 

In the text 
Fig. 3. Transition dipole moments D(R) of HHe between ^{2}Σ^{+} states n = 2 and n = 4. 

In the text 
Fig. 4. Repulsive potentials and their difference. Top: potential energy V of the repulsive states of Lymanα (dashed black curve), Balmerα (dashed red curve), and Balmerβ (blue curve). Bottom: difference potential ΔV(R) for the Lymanα transition X → C (dashed black curve), for the Balmerα transition C → 6^{2}Σ^{+} (dashed red curve), and for the Balmerβ transitions C → 10^{2}Σ^{+} (blue curve). 

In the text 
Fig. 5. Difference potential energy ΔV(R) in cm^{−1} (black line) and dipole moment D(R) (black dashed line) for the forbidden C 2p → 4f 10^{2}Σ^{+} transition. 

In the text 
Fig. 6. Comparison of the total profile (blue line) to the profile restrictly obtained with the allowed transitions (red dashed line). The Balmerβ line profile is also compared to the Lorentzian approximation (black dashed curve). The He density is 5 × 10^{20} cm^{−3}, and the temperature is 8000 K. 

In the text 
Fig. 7. Variation with the He density of the Balmerβ line profile for 2 × 10^{20} (black line), 3 × 10^{20} (blue line), and 5 × 10^{20} cm^{−3} (red line); the temperature is 8000 K. The Balmerβ line profiles are also compared to the Lorentzian approximation for the He densities 2 × 10^{20} cm^{−3} (dashed black line) and 3 × 10^{20} cm^{−3} (dashed blue line). 

In the text 
Fig. 8. Comparison of the Balmerα (black curves) and Balmerβ (red curves) line profiles. The He densities are 10^{21} cm^{−3} (full curves) and 6 × 10^{20} cm^{−3} (dashed curves); the temperature is 8000 K. 

In the text 
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