# On the 1${}^{1}\Sigma^{+}_{\mathrm{u}}$ electronic state in strontium dimer

Jacek Szczepkowski<EMAIL_ADDRESS>Wlodzimierz Jastrzebski
<EMAIL_ADDRESS>Anna Grochola  Institute of Physics, Polish Academy of
Sciences, al. Lotników 32/46, 02-668 Warsaw, Poland  Pawel Kowalczyk
<EMAIL_ADDRESS>Institute of Experimental Physics, Faculty of
Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland

###### Abstract

The laser technique of polarisation labelling of levels has been used to study
the excited 1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state of 88Sr2, in most part
heavily perturbed by the neighbouring electronic states. By observation of
completely resolved rovibronic spectra we have determined positions of 2556
spectral lines originating from the labelled levels in the ground
X${}^{1}\Sigma^{+}_{\mathrm{g}}$ state and terminating on levels of the
perturbed complex. The experimental results are confronted with the existing
theoretical predictions.

laser spectroscopy; alkali earth dimers; electronic states; perturbations

##  I Introduction

Despite recent interest in ultracold strontium dimers [1, 2, 3, 4, 5, 6]
relatively little is known about the structure of excited electronic states of
this molecule. Experimental studies are hindered by congestion of spectra due
to the presence of several isotopologues of Sr2 at relatively high abundance
as well as strong interaction between singlet and triplet states, which makes
both manifolds observable and causes formation of irregular spectral patterns.
As a result, beside the well known ground X${}^{1}\Sigma^{+}_{\mathrm{g}}$
state, only lower parts of the 1${}^{1}\Sigma^{+}_{\mathrm{u}}$,
1${}^{1}\Pi_{\mathrm{u}}$ and 2${}^{1}\Sigma^{+}_{\mathrm{u}}$ states are well
characterised by state of art spectroscopic methods [7, 8, 9]. Theoretical
investigation on electronic structure of Sr2 also meets with difficulties
related in particular to a large number of electrons (76, four of them being
valence electrons) with highly correlated behaviour. The theoretical papers
known to us [10, 12, 13, 14, 15] provide often contradictory results and
comparison with experiments shows that the accuracy of these calculations is
questionable.

In the present paper we investigate experimentally the lowest excited state of
88Sr2 available by optical transition from the ground state, i.e. the
(A)1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state. This state was observed before by
Stein et al. [9] but only the lowest vibrational levels up to $v’=10$ were
characterised. The gross properties of the electronic states of strontium
dimer, located in the studied energy range, can be inferred from theoretical
calculations. All of them predict a presence of four states there, besides the
1${}^{1}\Sigma^{+}_{\mathrm{u}}$ also 2${}^{3}\Sigma^{+}_{\mathrm{u}}$,
1${}^{3}\Pi_{\mathrm{u}}$ and 2${}^{3}\Pi_{\mathrm{u}}$, although their mutual
positions vary depending on different approaches (see Figure 1 for an example
of the relevant states calculated by Boutassetta et al. [10, 11]). It is
important that all four states can be mutually coupled via spin-orbit or
rotational interactions. In our experiment we employed the optical-optical
double resonance polarisation labelling spectroscopy technique which proved
successful in studies of many excited states in diatomic molecules. In the
following sections we present a principle of the method together with
experimental details, analysis of the recorded spectra and discussion of our
findings in view of the existing theoretical predictions.

Figure 1: Potential energy curves of Sr2 relevant to this paper, calculated in
Hund’s case (a) by Boutassetta et al. [10, 11].

##  II Experimental method and apparatus

The V-type double resonance polarisation labelling spectroscopy (PLS) has been
described in the literature (see for example [16, 17]) and extensively used
for studying excited electronic states of simple molecules [18, 19, 20, 21,
22, 23, 24, 25, 26]. In our version of the method we use beams of two lasers.
A fixed frequency, weak probe laser beam is tuned to resonance with a known
rovibronic transition in the investigated molecule, thus selecting
(’labelling’) the lower level in the involved transition. A tuneable,
polarised strong beam of the second laser establishes by means of optical
pumping a difference in population among $M$ sublevels of the initial (and
final) levels of the pump transition. The molecular sample is placed between
two crossed polarisers inserted in the path of the probe beam. A linearly
polarised probe beam traversing the prepared sample is therefore stopped by
the second, crossed polariser at all frequencies except those at which the
sample is anisotropic. Accordingly, the signal recorded behind the second
polariser corresponds only to frequencies of transitions originating from the
labelled level in the ground state. (Possible transitions from or to the upper
labelled level can be distinguished by their spectral features and actually
have not been observed in the present experiment.)

In practical realisation of the above idea in our experiment the pump light
was generated by a parametric oscillator/amplifier system (OPO/OPA, Sunlite
Ex, Continuum) pumped with the third harmonic of an injection seeded Nd:YAG
laser (Powerlite 8000). Depending on the spectral region of interest either
the idler or the signal beam from the OPO/OPA system was used, for the purpose
of the present experiment tuneable between 645 and 770 nm. The system
delivered pulses of 10 ns duration, spectral width 0.1 cm-1 and energy of few
tens of mJ, however, to avoid saturation of molecular signals, the laser
intensity was attenuated with neutral filters of variable optical density.
Frequency measurements were referenced to the optogalvanic lines of an Ar
hollow cathode lamp and fringe markers of a 0.5 cm long Fabry-Pérot
interferometer, providing accuracy of measured laser wavenumbers estimated as
0.1 cm-1. As a source of the probe (labelling) light we used a cw single-mode
ring dye laser (Coherent 699-21 operated on Rhodamine 6G dye), with a
wavelength controlled using a HighFinesse WS-7 wavemeter, set at fixed
frequencies of selected transitions from the X${}^{1}\Sigma^{+}_{\mathrm{g}}$
ground state to low levels of the 2${}^{1}\Sigma^{+}_{\mathrm{u}}$ state,
taken from the list published in the supplementary materials to Ref. [9].
Unfortunately, according to the selection rules specific to PLS [27], when
labelling levels via P or R lines (the only ones allowed in the
${}^{1}\Sigma-^{1}\Sigma$ transition), the polarisation spectra were also
limited to P and R lines only, irrespectively of symmetry of the upper state.
This was an unfortunate drawback which prevented us from direct distinguishing
between $\Sigma\leftarrow\Sigma$ and $\Pi\leftarrow\Sigma$ transitions in the
recorded spectra. However, in the frequency range covered by our ring dye
laser and at the same time by the available spectral data there was no other
choice of labelling transitions.

The Sr2 molecules were produced in a three section heat-pipe oven [28] of 1 m
length, with the inner surface of the pipe lined with three separate pieces of
steel mesh. The oven was filled with 15 g of strontium and 1.5 g of metallic
magnesium. As strontium and magnesium form an alloy with substantially lower
melting point than its constituents [29], this experimental trick ensured a
proper circulation of the metal inside the heat-pipe and, in particular,
prevented condensation of solid strontium in the outer zones of the oven. The
central part of the heat-pipe (about 20 cm in length) was heated to 1020∘C,
while external parts were maintained at 720∘C and it was operated with argon
as a buffer gas at a pressure of 15 Torr.

##  III Experimental results

The present experiment provided frequencies of 2556 spectral lines, all of
them originating from the ground state X${}^{1}\Sigma^{+}_{\mathrm{g}}$ of the
88Sr2 isotopologue, measured in the range $13000-15500$ cm-1. The line
frequencies were converted to energies of the corresponding upper levels using
highly precise molecular constants of the ground state [8]. In addition, we
had to our disposal 287 levels of 88Sr2 listed in the supplementary materials
to the paper by Stein et al. [9]. The data by Stein cover the range of
vibrational levels $v^{\prime}=1-20$ but the majority of levels corresponds to
$v^{\prime}\leq 10$. The energies of rovibrational levels in the range $1\leq
v^{\prime}\leq 10$ and $J^{\prime}<220$ were represented in Ref. [9] by a set
of as many as 17 Dunham coefficients, most of them being merely fitting
parameters and having no physical meaning. Stein et al. claimed that attempts
to incorporate higher vibrational levels into the fit failed, probably due to
perturbation of these levels. However a scarcity of data in this range made
further analysis futile. As the observations from the present experiment,
providing a vast majority of the combined data, cover energy levels with
rotational quantum numbers $J^{\prime}$ below 150, for further discussion we
have discarded higher rotational levels from the database of Ref. [9]. Our
analysis of the available data consists in two parts.

For the energy levels $1\leq v^{\prime}\leq 10$ we confirm that they can be
described by a Dunham series in a form

$T(v^{\prime},J^{\prime})=T_{e}+\sum_{m,n}Y_{mn}(v^{\prime}+0.5)^{m}[J^{\prime}(J^{\prime}+1)]^{n}\mbox{
.}$ (1)

In this range we added 291 levels measured in the present experiment to 133
levels listed by Stein et al. Fits with various numbers of parameters have
shown that only 8 Dunham coefficients suffice to reproduce experimental level
energies with an rms error of 0.08 cm-1, consistent with our experimental
accuracy. A considerable reduction of the number of parameters is apparently
connected with exclusion from the fit relatively few levels with
$J^{\prime}\geq 150$ (37 levels) present in the data of Stein et al. The
Dunham coefficients, rounded to the minimum number of significant figures by a
procedure suggested by Le Roy [30], are presented in Table 1. We found that
even addition of the next vibrational level, $v^{\prime}=11$, makes
description by a Dunham series impossible, as then it would require many more
Dunham coefficients, still providing an rms error above 0.1 cm-1.

Table 1: The Dunham coefficients that describe the 1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state of 88Sr2 in the range $1\leq v^{\prime}\leq 10$, $J^{\prime}<150$ wirh rms error 0.08 cm-1. The numbers in parentheses correspond to one standard deviation in units of the last digits. Values of the coefficients are in cm-1.  Coefficient | Value
---|---
$T_{e}$ | 12795.70(4)
$Y_{10}$ | 80.314(23)
$Y_{20}$ | -0.1944(38)
$Y_{30}$ | -0.76(2)$\times 10^{-2}$
$Y_{01}$ | 0.0247559(39)
$Y_{11}$ | -0.8935(12)$\times 10^{-4}$
$Y_{21}$ | -0.226(8)$\times 10^{-5}$
$Y_{02}$ | -0.103(7)$\times 10^{-7}$

As indicated above, the energies of rovibrational levels of the
1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state above $v^{\prime}=10$ display rather
erratic behaviour and starting from $v^{\prime}=15$ levels of more than one
electronic state become visible. All the observed levels in this range are
displayed in Figure 2. Several of them can be arranged into bands apparently
corresponding to separate, albeit strongly perturbed, vibrational levels. It
must be stressed that for a vast majority of the observed rovibrational levels
we are certain about their attribution to given $J^{\prime}$ quantum numbers
and to the main isotopologue 88Sr2 because they were intentionally excited
more than once, via different transitions from at least two initial ground
state levels, what confirmed their assignment. Moreover, in most spectra only
a single ($v^{\prime\prime}$, $J^{\prime\prime}$) level in the ground state
was labelled due to a small linewidth of the probe laser (ca 1 MHz).

Figure 2: Reduced term values $E-0.019\times J(J+1)$ [cm-1] of the observed
rovibrational levels in the 1${}^{1}\Sigma^{+}_{\mathrm{u}}$ $\sim$
2${}^{3}\Sigma^{+}_{\mathrm{u}}$ $\sim$ 1${}^{3}\Pi_{\mathrm{u}}$ $\sim$
2${}^{3}\Pi_{\mathrm{u}}$ complex in Sr2 (black dots) plotted against
$J(J+1)$. The term values calculated from the band parameters of the
1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state listed in Table 2 are represented by
red solid lines and the vibrational numbering is shown for some of them. Green
dashed lines correspond to ’perturber 1’ (Table 3) and blue dash-dot lines to
’perturber 2’ (Table 4).

Despite the observed line shifts amounting in some cases to few cm-1, we tried
to determine approximate band constants of several levels, after discarding
rather arbitrarily the obvious outliers, by fitting level energies to the
formula

$E(v^{\prime},J^{\prime})=T_{v}+B_{v}[J^{\prime}(J^{\prime}+1)]-D_{v}[J^{\prime}(J^{\prime}+1)]^{2}\mbox{
.}$ (2)

The results are listed in Tables 2, 3, 4 and displayed by lines in Figure 2.
We were able to follow systematically vibrational levels of the
1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state up to $v^{\prime}=27$, in spite of
their irregular behaviour (red lines in Figure 2). This left few other bands
which must be attributed to another electronic state (’perturber 1’ – green
lines in Figure 2). Beyond $v^{\prime}=27$ of the
1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state the pattern of rovibrational levels
becomes incomprehensible but above an excitation energy $E\approx 14900$ cm-1
another system of bands emerges (’perturber 2’ – blue lines in Figure 2)
belonging to an electronic state with different characteristics (cf. different
slopes of the blue lines versus the red and green ones, indicating larger
values of vibrational constants $B_{v}$ of the uppermost state).

Table 2: Approximate band constants (all values in cm-1) of vibrational levels $v^{\prime}=11-27$ of the 1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state (with no deperturbation attempted). $v^{\prime}$ | $T_{v}$ | $B_{v}$ |  $D_{v}$ $\times 10^{8}$ | rms | range of $J^{\prime}$
---|---|---|---|---|---
11 | 13681.66 | 0.023464 | 1.31 | 0.07 | $65-149$
12 | 13753.65 | 0.023283 | 1.46 | 0.04 | $23-149$
13 | 13823.81 | 0.023084 | 1.80 | 0.04 | $25-145$
14 | 13891.51 | 0.022832 | 2.32 | 0.05 | $53-149$
15 | 13956.14 | 0.022432 | 2.42 | 0.08 | $45-149$
16 | 14017.37 | 0.021914 | 1.78 | 0.06 | $71-149$
17 | 14074.47 | 0.021593 | 2.38 | 0.06 | $35-149$
18 | 14129.15 | 0.021134 | 1.64 | 0.04 | $57-149$
19 | 14180.14 | 0.020985 | 1.86 | 0.08 | $75-147$
20 | 14228.77 | 0.021433 | 3.78 | 0.08 | $61-147$
21 | 14280.05 | 0.020059 | — | 0.03 | $75-95$
22 | 14325.14 | 0.020337 | 1.98 | 0.08 | $75-139$
23 | — | — | — | — | —
24 | 14414.29 | 0.019939 | 2.08 | 0.07 | $77-139$
25 | 14458.99 | 0.019940 | 2.97 | 0.08 | $65-145$
26 | 14508.10 | 0.019481 | 0.14 | 0.12 | $81-139$
27 | 14543.83 | 0.019075 | 1.36 | 0.04 | $65-137$
Table 3: Approximate band parameters (all values in cm-1) of vibrational levels of the electronic state referred to as ’perturber 1’. The absolute vibrational numbering is unknown. $v^{\prime}$ | $T_{v}$ | $B_{v}$ |  $D_{v}$ $\times 10^{8}$ | rms | range of $J^{\prime}$
---|---|---|---|---|---
n | 13980.74 | 0.020312 | 0.82 | 0.05 | $75-149$
n+1 | 14039.99 | 0.020472 | 1.33 | 0.07 | $75-149$
n+2 | 14099.64 | 0.021116 | 5.04 | 0.06 | $75-87$
n+3 | 14165.24 | 0.020682 | 1.45 | 0.04 | $75-149$
n+4 | 14232.05 | 0.020015 | 0.21 | 0.04 | $75-135$
n+5 | 14296.62 | 0.020852 | 2.37 | 0.12 | $63-135$
n+6 | 14364.31 | 0.020055 | 0.85 | 0.08 | $61-147$
n+7 | 14433.28 | 0.020145 | 0.19 | 0.12 | $77-137$
n+8 | 14494.29 | 0.019914 | 2.51 | 0.06 | $63-131$
Table 4: Approximate band parameters (all values in cm-1) of vibrational levels of the electronic state referred to as ’perturber 2’. The absolute vibrational numbering is unknown. $v^{\prime}$ | $T_{v}$ | $B_{v}$ |  $D_{v}$ $\times 10^{8}$ | rms | range of $J^{\prime}$
---|---|---|---|---|---
p | 14944.49 | 0.021006 | 0.85 | 0.08 | $63-137$
p+1 | 15008.01 | 0.021072 | 1.02 | 0.20 | $63-147$
p+2 | 15071.62 | 0.021202 | 1.80 | 0.14 | $93-137$
p+3 | 15136.38 | 0.021074 | 1.53 | 0.08 | $63-111$
p+4 | 15200.54 | 0.020924 | 1.08 | 0.13 | $63-109$

It must be noted that Leung et al. [6] claimed to measure energies of 37
lowest vibrational levels of the 1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state but
the measurements, as usually in the ultracold physics experiments, were
limited to only one rotational level $J^{\prime}=1$ for each vibrational
quantum number. We find that energies of these levels agree with values
calculated from parameters of Table 2 within $\pm 0.2$ cm-1 for
$v^{\prime}=11-18$, with discrepancies growing irregularly up to $\pm 2$ cm-1
for strongly perturbed levels $v^{\prime}=19-27$. This agreement can be
considered good taking into account long extrapolation from the range of
rotational quantum numbers explored in our experiment to $J^{\prime}=1$ as
well as somewhat arbitrary choice of ’unperturbed’ levels used in our fits.

##  IV Discussion

Because of large discrepancies in potential energy curves derived by different
authors and methods, the discussion in this section can be only of qualitative
nature. We believe that the oldest calculations by Boutassetta et al. [10, 11]
(Figure 1) provide a plausible explanation of the experimental findings.
According to Figure 1, in a simplistic picture, the
1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state should be free of perturbation by any
other electronic state up to $E\approx 13650$ cm-1 where the origin of the
2${}^{3}\Sigma^{+}_{\mathrm{u}}$ state is located. However, a direct
perturbation 1${}^{1}\Sigma^{+}_{\mathrm{u}}$ $\sim$
2${}^{3}\Sigma^{+}_{\mathrm{u}}$ is not possible [31] and it has to be
mediated via mixing of the 2${}^{3}\Sigma^{+}_{\mathrm{u}}$ and
1${}^{3}\Pi_{\mathrm{u}}$ states. Our experimental observations suggest that
this is the 1${}^{3}\Pi_{\mathrm{u}}$ state which becomes visible in the
spectra as ’perturber 1’ (note smaller $B_{v}$ values corresponding to levels
marked in green in Figure 2, which indicate larger $R_{v}$ values than in the
1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state). With increasing excitation energy,
the inner turning points of the 1${}^{3}\Pi_{\mathrm{u}}$ and
2${}^{3}\Sigma^{+}_{\mathrm{u}}$ potentials gradually draw closer and mutual
perturbation of both triplet states makes the pattern of vibrational levels
even more complicated (cf. the region of reduced energies $14500-15000$ cm-1
in Figure 2). Finally, at approximately 14900 cm-1 another state emerges,
namely 2${}^{3}\Pi_{\mathrm{u}}$, which can be visible in the spectra due to a
direct 1${}^{1}\Sigma^{+}_{\mathrm{u}}$ $\sim$ 2${}^{3}\Pi_{\mathrm{u}}$
perturbation (’perturber 2’).

An interesting, alternative explanation of the experimental observations may
be provided by potential curves of Sr2 calculated in Hund’s case (c) by
Skomorowski et al. [14] (Figure 3). Note that in absence of singlet–triplet
symmetry of electronic states all transitions between them should be visible,
as observed in our experiment. In Hund’s case (c) the
1${}^{1}\Sigma^{+}_{\mathrm{u}}$ state corresponds to (1)$0^{+}_{u}$ and then
the sequence of states coming into play when excitation energy increases is
similar as in case (a), here ’perturber 1’ being the (2)$0^{+}_{u}$ state and
’perturber 2’ – the (3)$1_{u}$ state. Skomorowski et al. claim very good
agreement between the low part of their (1)$0^{+}_{u}$ state potential and the
experimental one constructed by Stein et al. [9] but again we find that
theoretical results describe our findings only qualitatively, particularly
that no numerical data accompany Ref. [14].

Figure 3: Potential energy curves of Sr2 relevant to the present work,
calculated in Hund’s case (c) by Skomorowski et al. [14]. (As Ref. [14] does
not provide numerical results referring to the potential curves, this Figure
is an illustrative redrawing of their Fig.3.)

The above analysis shows clearly that a rigorous deperturbation of the coupled
1${}^{1}\Sigma^{+}_{\mathrm{u}}$ $\sim$ 2${}^{3}\Sigma^{+}_{\mathrm{u}}$
$\sim$ 1${}^{3}\Pi_{\mathrm{u}}$ $\sim$ 2${}^{3}\Pi_{\mathrm{u}}$ complex of
states is necessary to explain the experimental results quantitatively. This
is a serious numerical challenge, but first of all it would require reliable
theoretical potentials as starting points. In the lack of them, in addition to
qualitative analyses we make available the full list of the experimentally
determined energies of rovibrational levels as the supplementary material
accompanying this paper.

##  V Acknowledgements

Partial financial support from the National Science Centre Poland (Grant No.
2021/43/B/ST4/03326) is gratefully acknowledged.

##  VI Appendix A. Supplementary data

Supplementary data associated with this article can be found online at the
address http://dimer.ifpan.edu.pl.

## References

  * [1] W. Skomorowski, R. Moszynski, Ch. P. Koch, Phys. Rev. A 85, 043414 (2012).
  * [2] S. Stellmer, B. Pasquiou, R. Grimm, F. Schreck, Phys. Rev. Lett. 109, 115302 (2012).
  * [3] G. Reinaudi, C. B. Osborn, M. McDonald, S. Kotochigova, T. Zelevinsky, Phys. Rev. Lett. 109, 115303 (2012).
  * [4] A. Ciamei, A. Bayerle, C.-C. Chen, B. Pasquiou, F. Schreck, Phys. Rev. A 96, 013406 (2017).
  * [5] S. S. Kondov, C.-H. Lee, K. H. Leung, C. Liedl, I. Majewska, R. Moszynski and T. Zelevinsky, Nature Phys. 15, 1118–1122 (2019).
  * [6] K. H. Leung, E. Tiberi, B. Iritani, I. Majewska, R. Moszynski and T. Zelevinsky, New J. Phys. 23, 115002 (2021)
  * [7] A. Stein, H. Knöckel, and E. Tiemann, Phys. Rev. A 78, 042508 (2008).
  * [8] A. Stein, H. Knöckel, and E. Tiemann, Eur. Phys. J. D 57, 171–177 (2010).
  * [9] A. Stein, H. Knöckel, and E. Tiemann, Eur. Phys. J. D 64, 227–238 (2011).
  * [10] N. Boutassetta, A. R. Allouche, and M. Aubert-Frécon, Phys. Rev. A 53, 3845–3852 (1996).
  * [11] A. R. Allouche, private communication.
  * [12] E. Czuchaj, M. Krośnicki, H. Stoll, Chemical Physics Letters 371, 401–409 (2003).
  * [13] S. Kotochigova, J. Chem. Phys. 128, 024303 (2008).
  * [14] W. Skomorowski, F. Pawłowski, Ch. P. Koch, R. Moszynski, J. Chem. Phys. 136, 194306 (2012).
  * [15] J. Szczepkowski, M. Gronowski, A. Grochola, W. Jastrzebski, M. Tomza, P. Kowalczyk, J. Phys. Chem. A (2023), https://doi.org/10.1021/acs.jpca.3c02056 .
  * [16] R. Teets, R. Feinberg, T. W. Hänsch, and A. L. Schawlow, Phys. Rev. Lett 37, 683–686 (1976).
  * [17] A. Grochola, W. Jastrzebski, P. Kortyka, P. Kowalczyk, J. Mol. Spectrosc. 221, 279–284 (2003).
  * [18] R. E. Teets, N. W. Carlson, A. L. Schawlow, J. Mol. Spectrosc. 78, 415–421 (1979).
  * [19] G. Ehret, G. Gerber, Opt. Commun. 51, 145–150 (1984).
  * [20] P. Niay, P. Bernage, H. Bocquet, J. Mol. Spectrosc. 128, 502–508 (1988).
  * [21] W. Jastrzȩbski, P. Kowalczyk, Chem. Phys. Lett. 206, 69–72 (1993).
  * [22] A. Pashov, I. Jackowska, W. Jastrzȩbski, P. Kowalczyk, Phys.Rev. A58, 1048–1054 (1998).
  * [23] N. H. Bang, A. Grochola, W. Jastrzebski, P. Kowalczyk, Chem. Phys. Lett. 440, 199–202 (2007).
  * [24] N. H. Bang, A. Grochola, W. Jastrzebski, P. Kowalczyk, J. Chem. Phys. 130, 124307 (2009).
  * [25] W. Jastrzebski, J. Szczepkowski, P. Kowalczyk, A. R. Allouche, P. Crozet, A. J. Ross, J. Chem. Phys. 143, 044308 (2015).
  * [26] A. Pashov, P. Kowalczyk, W. Jastrzebski, Phys. Rev. A 100, 012507 (2019).
  * [27] R. Ferber, W. Jastrzȩbski, P. Kowalczyk, J. Quant. Spectrosc. Rad. Transfer 58, 53–60 (1997).
  * [28] A. Ciamei, J. Szczepkowski, A. Bayerle, V. Barbé, L. Reichsöllner, S. M. Tzanova, C.-C. Chen, B. Pasquiou, A. Grochola, P. Kowalczyk, W. Jastrzebski, F. Schreck, Phys. Chem. Chem. Phys. 20, 26221–26240 (2018).
  * [29] A. A. Nayeb-Hashemi, J. B. Clark, Bull. Alloy Phase Diagr. 7, 149–156 (1986).
  * [30] R. Le Roy, J. Mol. Spectrosc. 191, 223–231 (1998).
  * [31] I. Kovacs, _Rotational structure in the spectra of diatomic molecules_ , Elsevier, New York 1969.