Jacek Szczepkowski

Anna Grochola

Pawel Kowalczyk

Olivier Dulieu
email: olivier.dulieu@u-psud.fr

Romain Guérout

Piotr S Żuchowski

Wlodzimierz Jastrzebski

Paweł Kowalczyk

Włodzimierz Jastrzebski

Experimental and theoretical study of the B(2) 2 Σ + → X(1) 2 Σ + system in the KSr molecule

Keywords: thermoluminescence spectroscopy, alkaline-earth molecules, electronic states, potential energy curves, ab initio calculations PACS: 31.50.Df, 33.20.Kf, 33.20.Vq, 42.62.Fi

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

In recent years mixed diatomic molecules composed with one alkali-metal atom (Li, Na, K, Rb, Cs) and one alkaline-earth atom (Be, Mg, Ca, Sr, Ba) became particularly interesting for the community of physicists, due to the possibility of performing new experiments in the field of ultracold molecules [START_REF] Pérez-Ríos | External field control of collective spin excitations in an optical lattice of 2 Σ molecules[END_REF][START_REF] Chen | Prospects for transferring 87 Rb 84 Sr dimers to the rovibrational ground state based on calculated molecular structures[END_REF][START_REF] Quéméner | Shielding 2 Σ ultracold dipolar molecular collisions with electric fields[END_REF][START_REF] Barbé | Observation of Feshbach resonances between alkali and closed-shell atoms[END_REF]. Such open-shell diatomics are called "doubly polar molecules" since they have both electric and magnetic permanent dipole moments. At ultralow temperatures they provide unique opportunities for quantum control and new possibilities for quantum simulations [START_REF] Micheli | A toolbox for lattice-spin models with polar molecules[END_REF][START_REF] Baranov | Condensed matter theory of dipolar quantum gases[END_REF][START_REF] Lahaye | The physics of dipolar bosonic quantum gases[END_REF]. Such challenging experiments require guidance based on experimentally known features of the chosen species. However, because of experimental difficulties, only a limited number of studies have been published so far. Low-resolution molecular spectroscopy -complemented by advanced electronic structure calculations -has been achieved on RbSr [START_REF] Lackner | Helium-droplet-assisted preparation of cold RbSr molecules[END_REF][START_REF] Krois | Characterization of RbSr molecules: spectral analysis on helium droplets[END_REF][START_REF] Pototschnig | Ab initio study of the RbSr electronic structure: Potential energy curves, transition dipole moments, and permanent electric dipole moments[END_REF], RbCa [START_REF] Pototschnig | Investigation of the RbCa molecule: Experiment and theory[END_REF], and LiCa [START_REF] Krois | Spectroscopy of cold LiCa molecules formed on helium nanodroplets[END_REF] molecules embedded inside helium nanodroplets as cold as a few kelvins, thus drastically reducing the number of populated levels of the electronic ground state X(1) 2 Σ + (typically the lowest vibrational state, and a few rotational states). The explored spectral range concerned the excited molecular states above the two lowest ones, namely (2) 2 Σ + and (2) 2 Π.

In contrast, high-resolution studies based on Fourier transform spectroscopy and addressing the latter excited electronic states have been recently reported for LiSr [START_REF] Schwanke | Laser and Fourier transform spectroscopy of 7 Li 88 Sr[END_REF] and KCa [START_REF] Gerschmann | Laser and Fourier-transform spectroscopy of KCa[END_REF], and on LiBa in a series of older papers [START_REF] Vergès | Electronic structure of BaLi: the (2) 2 Σ + state[END_REF][START_REF] Stringat | The 2 Σ + , (2) 2 Σ + and (2) 2 Π states of BaLi[END_REF][START_REF] D'incan | Electronic structure of BaLi. II. First observation of the Ba 6,7 Li spectrum: Analysis of the (2) 2 Π → X 2 Σ + system[END_REF]. It is worthwhile also to note the resonant two-photon ionization spectroscopy of LiCa reported in Ref. [START_REF] Russon | Two-photon ionization spectroscopy and all-electron ab initio study of LiCa[END_REF].

Despite the apparent simplicity of alkali-alkaline-earth diatomic molecules, the accurate computation of their electronic structure is cumbersome when heavy atoms are involved, as strong correlations are expected between the valence electrons and those of the closed-shell ionic cores. Numerous exploratory calculations have been performed over the years with almost no possibility to evaluate them against experiment, up to very recently. We can quote papers on barium-alkali-metal-atom compounds [START_REF] Allouche | Electronic structure of BaLi. I. Theoretical study[END_REF][START_REF] Boutassetta | Theoretical study of the low-lying electronic states of the BaNa molecule[END_REF][START_REF] Boutassetta | Theoretical study of the low-lying electronic states of the BaK molecule[END_REF][START_REF] Gou | Theoretical study on the ground state of the polar alkali-metal-barium molecules: Potential energy curve and permanent dipole moment[END_REF], lithium-alkaline-earthatom compounds [START_REF] Kotochigova | Ab initio properties of Li-group-II molecules for ultracold matter studies[END_REF], strontium-alkali-metal-atom compounds [START_REF] Guérout | Ground state of the polar alkali-metalatom-strontium molecules: Potential energy curve and permanent dipole moment[END_REF][START_REF] Żuchowski | Ground-and excited-state properties of the polar and paramagnetic RbSr molecule: A comparative study[END_REF], and other combinations [START_REF] Gopakumar | Ab initio study of ground and excited states of 6 Li 40 Ca and 6 Li 88 Sr molecules[END_REF] up to the recent systematic investigation for sixteen alkali-alkaline-earth compounds of Ref. [START_REF] Pototschnig | Electric dipole moments and chemical bonding of diatomic alkali-alkaline earth molecules[END_REF] Among the alkali-metal-alkaline-earth species, KSr definitely belongs to the most interesting ones. The potassium atom is the only alkali-metal atom with three stable or long-lived isotopes, which together with four isotopes of Sr give 12 possible mass combinations exhibiting both bosonic (e.g. 39 K+ 87 Sr) and fermionic (e.g. 40 K+ 87 Sr) statistics. Such a variety of reduced masses might be used to tune the scattering properties in KSr by proper selection of the isotopologue.

It is well known that the accuracy of quantum chemistry calculations does not generally meet the requirements of ultracold physics. For instance the scattering length for a pair of colliding atoms cannot be predicted without the help of high-resolution experimental data. It is also challenging to approach the spectroscopic accuracy for the ground state and the excited electronic states. In most cases it is essential to combine experimental data and theoretical potential energy curves to derive an "experimental" potential energy curve. On the other hand, experimental techniques like Laser Induced Fluorescence (LIF) or Polarisation Labelling Spectroscopy (PLS), which yield spectroscopic data with rotational resolution, must be initiated with accurate theoretical data, or with data delivered by less demanding experiments, as it was shown in case of KCa molecule [START_REF] Gerschmann | Laser and Fourier-transform spectroscopy of KCa[END_REF]. The thermoluminescence technique used in the present work belongs to this latter category of experimental methods, thus representing an important step to refine potential energy curves.

The present paper originates from a close cooperation between theoretical and experimental groups. We first present in Section 2 the electronic structure calculations performed with two different methods, in order to assess their typical accuracy through the comparison of their results. In Section 3 we describe our experimental approach based on the thermoluminescence technique, which allowed us to extract the first experimental results for the X(1) 2 Σ + and B(2) 2 Σ + electronic states of the KSr molecule. Through the simulations of the recorded spectra (Section 4), we demonstrate in Section 5 that the level of accuracy of the present calculations allowed a better understanding and a proper interpretation of the spectroscopic data. In the future the relatively precise experimental molecular constants should be used to correct the assumptions of the individual theoretical models and to improve the accuracy of the predicted molecular parameters.

Electronic structure calculations

As quoted above, the valence electrons (one for K, two for Sr) and the remaining core electrons of K + and Sr 2+ in the KSr system strongly interact, making the computation of their electronic structure tedious. Therefore the usage of different methods to perform such calculations is mandatory for a documented assessment of their validity. We proceeded here like in our previous collaborative investigations, employing two different methods. Below we recall their main steps, for reader's convenience, while all details and references can be found in Ref. [START_REF] Żuchowski | Ground-and excited-state properties of the polar and paramagnetic RbSr molecule: A comparative study[END_REF].

The first approach, for simplicity hereafter referred to as the FCI+ECP+CPP method, relies on the modelling of the K + and Sr 2+ closed-shell cores by an effective core potential (ECP) completed by a core-polarization potential (CPP) [START_REF] Müller | Ground-state properties of alkali dimers and their cations from ab initio calculations with effective core polarization potentials[END_REF][START_REF] Müller | Treatment of intershell correlation effects in ab initio calculations by use of core polarization potentials. method and application to alkali and alkaline earth atoms[END_REF][START_REF] Foucrault | Nonpertubative method for core-valence correlation in pseudopotential[END_REF], which parameters are reported in Ref. [START_REF] Guérout | Ground state of the polar alkali-metalatom-strontium molecules: Potential energy curve and permanent dipole moment[END_REF]. The Gaussian orbital basis sets spanning the configuration space are also reported in Ref. [START_REF] Guérout | Ground state of the polar alkali-metalatom-strontium molecules: Potential energy curve and permanent dipole moment[END_REF]. Thus the problem is reduced to an effective three-electron system, allowing the resolution of the electronic Schrödinger equation using a full configuration interaction (FCI). This yields the potential energy curves (PECs) of the electronic ground state X 2 Σ + , as well as many excited electronic states of 2,4 Σ + , 2,4 Π, and 2,4 ∆ symmetries, and subsequently the permanent and transition dipole moments (PDM and TDM) involving these states as a function of the internuclear distance R.

The second method employs 19 active electrons whereas the inner electronic shells are treated as effective core potential ECP referred to as ECP28MDF for Sr and ECP10MDF for K, so that the 4s, 4p, and 5s electrons of Sr, and the 3s, 3p and 4s electrons of K are fully correlated. The ECPs used in this calculations were obtained by Lim et al. [START_REF] Lim | All-electron and relativistic pseudopotential studies for the group 1 element polarizabilities from K to element 119[END_REF][START_REF] Lim | Relativistic small-core energyconsistent pseudopotentials for the alkaline-earth elements from Ca to Ra[END_REF] and the basis set tailored for this ECP was uncontracted and augmented by adding one h function to the basis set for each atom (with exponents 3.84 and 0.56 for K and Sr, respectively). The basis set was subsequently augmented by a set of diffusion functions. The midbond basis set was added with exponents 0.9,0.3,0.1 for spd-type functions and 0.6,0.2 for f g functions, and placed on ghost atom in geometric centre of the molecule. The PECs are obtained using counterpoise-corrected supermolecular coupled-cluster method based on spin-restricted reference [START_REF] Knowles | Coupled cluster theory for high spin, open shell reference wave functions[END_REF] with single and double excitations and non-iterative correction for triply-excited configurations [RCCSD(T)]. For the ground-state closed-shell system, this method is considered as the golden standard of the quantum chemistry calculations of interaction potential. However, for the ground state open-shell systems RCCSD(T) were studied far less in terms of uncertainty of calculations of PECs compared to closed-shell systems. For the first excited states, the calculations of potential energy curves which employ such approach are explored very little (note for example the work of Klos et al. [START_REF] Kłos | A new potential energy surface for OH(A 2 Σ + )-Ar: The van der Waals complex and scattering dynamics[END_REF]. In this respect it is very appealing to confront RCCSD(T) method applied to the ground state and the first excited 2 Σ + state, and compare it with alternative methods. Such a comparison will be profitable for future studies of similar systems relevant in ultracold molecular physics. While the former state is easy to converge and maintains the singlereference character for broad range of distances, the supermolecular RCCSD(T) calculation for the second 2 Σ + state was a bit more demanding. To this end, the orbital rotation of 4s and 4p potassium orbitals were applied on top of the ground state reference state, and subsequently converged. The coupled-cluster equations could be easily converged down to about 2.5. It is worth noting that the value of T1 diagnostics of Lee [START_REF] Lee | Comparison of the T1 and D1 diagnostics for electronic structure theory: a new definition for the open-shell D1 diagnostic[END_REF] used commonly for assessment of multireference character of CCSD calculation is at most 0.03 in the energy well, slightly more than the value indicating single-reference character. The molpro 2012 package was used for this part of ab initio calculations [START_REF] Werner | Molpro 2012-1, a package of ab initio programs[END_REF].

In both methods, we model the long-range part of the potential curves using the standard -C 6 /R 6 -C 8 /R 8 expansion smoothly matched to the curves obtained above (at 30 a.u. for the FCI+ECP+CPP method, and at 20 a.u. for the RCCSD(T) method) with switching function taken from Ref. [START_REF] Janssen | Ab initio potential energy surfaces for NH ( 3 Σ -)-NH ( 3 Σ -) with analytical long range[END_REF]. The van der Waals coefficients were taken from the paper of Jiang et al. [START_REF] Jiang | Long-range interactions between alkali and alkaline-earth atoms[END_REF].

The results for the PECs of the X(1) 2 Σ + ground state and the lowest A 2 Π Table 1: Spectroscopic constants of the KSr system obtained from theoretical calculations. The vibrational constants correspond to 39 K 88 Sr isotopic combination. See text for the details.

state and B 2 Σ + states relevant for the present study are displayed in Fig. 1 and the related transition dipole moment (TDM) in Fig. 2, together with the results reported previously in Ref. [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF] (by the group henceforth referred to as the Graz group) based on the multireference configuration interaction (MRCI) approach.

T e [cm -1 ] D e [cm -1 ] ω e [cm -1 ] B e [cm -1 ] R e [Å] RCCSD(T) (present paper) X(1)
The agreement between the RCCSD(T) approach and the FCI+ECP+CPP method for the ground state is remarkable, and confirms the trend already reported in Ref. [START_REF] Żuchowski | Ground-and excited-state properties of the polar and paramagnetic RbSr molecule: A comparative study[END_REF] for RbSr. In contrast, the MRCI approach of Ref. [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF] predicts a significantly deeper potential well with a rather strong harmonic constant (see Tab. 1). As the RCCSD(T) approach is currently considered as the most reliable method for such ground state calculations, an as its results are confirmed by a completely different semi-empirical approach, we recommend these results for further use. The situation is less clear for the lowest excited electronic states: all methods provide consistent results, but again the well depth reported in Ref. [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF] is significantly deeper than in the two methods.

The TDM is well known as a quantity very sensitive to the details of the electronic wave functions. The agreement between the FCI+ECP+CPP method and the one of the Graz group is remarkable (see Fig. 2) especially in the 4Å-6Å range, where the oscillation for the B state originates from a broad avoided crossing-as large as about 1500 cm -1 between the B(2) 2 Σ + and D(3) 2 Σ + excited states (see Fig. 3). Such an agreement clearly expresses that both methods, while very different, describe the electronic wave functions in the same way, thus representing an indication of their validity. However it is striking that the asymptotic values of the B-X and A-X TDM reach the atomic K(4s 2 S)-K(4p 2 P ) TDM in the FCI+ECP+CPP approach, while it is overestimated by 8 % in the results of the Graz group.

The data for the PECs and TDMs computed in the present work are reported in the Supplementary material [40]. Note that according to our calculations with the FCI+ECP+CPP method, the PECs correlated to the next dissociation limit, K(4s 2 S)+Sr(5s5p 3 P ), have all their minima well above the one of the PECs correlated to K(4p 2 P )+Sr(5s 2 1 S) studied here (see Fig. 3). Beside the

B(2) 2 X(1) 2 E[cm -1 ] R [Å] K(4p)+Sr(5s 2 ) K(4s)+Sr(5s 2 ) (1) 2
Figure 1: Comparison of three sets of theoretical potential energy curves for the ground and two lowest exited states of KSr. Results are displayed for the FCI+ECP+CPP approach (red solid lines), for the RCCSD(T) approach (black dashed lines), and for the MRCI potentials of the Graz group [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF] (green dotted lines). The original energy scales used in the calculations, locating zero of energy at the dissociation limit of the ground state, are preserved.

Σ + -A(1) 2 Π, X(1) 2 Σ + -B(2) 2 Σ + and X(1) 2 Σ + -D(3) 2 Σ +
transitions in the KSr molecule. Results are displayed for the FCI+ECP+CPP approach (red solid lines), and for the MRCI results (green doted lines) of the Graz group [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF]. D(3) 2 Σ + PEC quoted above, the other PECs (1) 4 Σ + , (2) 2 Π and (1) 4 Π do not cross the one of the B and A state, and thus are not expected to play a role in the present results. Moreover, the minimum of the (1) 4 Π PEC is predicted around 10500 cm -1 , thus largely separated from the B(2) 2 Σ + PEC, and should not significantly contribute to the spin-orbit energy in the B state (see Fig. 3).

The thermoluminescence experiment

At temperatures high enough a thermal population of electronically excited molecular states is possible. Since working temperatures for potassium and strontium are significantly different, a special dual-temperature stainless steel heat-pipe oven was constructed to produce a vapor of KSr molecules. A detailed description of this device can be found in Ref. [START_REF] Bednarska | A three-section heat-pipe oven for heteronuclear alkali molecules[END_REF]. In general, the heat-pipe was spatially divided into three sections. The central part of the pipe was filled with 10 g of strontium and heated to the temperature T Sr = 1100 K, while the outer parts were filled with 10 g of potassium and maintained at the temperature T K = 820 K. Only potassium parts of the oven were covered inside with a stainless steel mesh to provide a proper metal circulation in the heat-pipe. Helium at pressure of 30 Torr was used as a buffer gas to assure stability of the heat-pipe operation and to prevent metal deposition on the oven windows. The atomic vapors were mixing and reacting in the central part of the heat-pipe, producing KSr molecules. Thermally excited fluorescence spectra The potential energy curves calculated with FCI+ECP+CPP method for all states correlating to the K(4p 2 P )+Sr(5s 2 1 S) and K(4s 2 S)+Sr(5s5p 3 P ) asymptotes. The solid black lines denote 2 Σ + states, the dashed red lines denote 2 Π, the dash-dot blue line refers to 4 Π and the dotted green line denotes 4 Σ + state. The zero energy is the same as in Fig. 1 were recorded using Bruker Vertex V80 Fourier Transform Spectrometer with a spectral resolution 0.16 cm -1 , limited by an aperture size. At the applied temperatures, the KSr second excited state, namely the B(2) 2 Σ + state, (the first excited state being the A(1) 2 Π state) was populated thermally, and fluorescence to the ground state was recorded. The population of the B(2) 2 Σ + state rovibrational levels was determined by the temperature of the KSr molecules, particularly of the central part of the heat-pipe, according to the Boltzmann distribution. The band heads corresponding to transitions between different vibrational levels of the B(2) 2 Σ + and X(1) 2 Σ + electronic states were clearly seen (see Fig. 4d). At the resolution of the present experiment, each unresolved band head consists of a group of about 30 rotational lines.

Simulations of the recorded spectra

Simulations of molecular spectra were performed using theoretical potentials in order to compare them with the experimental results. Such comparison was necessary to enable the correct assignment of the band heads and further analysis of the experimental data. Calculations were based on three sets of theoretical potential energy curves, generated independently with FCI+ECP+CPP and RCCSD(T) methods (see Section 2) as well as originating from the Graz group [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF].

In the first step of the simulation energies of rovibrational levels of the B(2) 2 Σ + and X(1) 2 Σ + states were calculated by solving the radial Schrödinger equation with theoretical potentials. All bound levels in the X(1) 2 Σ + state were taken into account, spanning the range v ′′ =0-55, N ′′ =0-217 (the upper value corresponding to the highest bound rotational level for v ′′ =0), where N denotes the quantum number of the total angular momentum apart from spin, being the natural rotational quantum number for 2 Σ + levels. The highest vibrational level to be considered in the B(2) 2 Σ + state was determined by calculated Franck-Condon factors (FCF). We included in the simulations all B(2) 2 Σ + vibrational levels with non-zero FCF values for transitions to the ground state levels (i.e. v ′ =0-80). Note that we checked that the contribution of the A(1) 2 Π state to the observed spectra is found typically 2000 times smaller than the one of the B state, due to the combined effect of smaller TDM and unfavorable FCFs.

Several assumptions were made to simulate the spectra. The fine structure splitting of molecular levels (i.e. splitting between F 1 and F 2 levels due to the spin-rotation coupling) was neglected, since the distance between fine structure components for low rotational quantum numbers N ′ , for which the band heads are formed, is smaller than the spectral resolution. The intensities of the transitions B(v ′′ , N ′′ ) → X(v ′′ , N ′′ ) were calculated assuming thermal equilibrium, according to the formula

I trans (v ′ , N ′ → v ′′ , N ′′ ) ∼ HL(N ′ → N ′′ )•P (v ′ , N ′ )• ψ ′ v ′ ,N ′ |µ|ψ ′′ v ′′ ,N ′′ 2 , (1)
where

ψ ′ v ′ ,N ′ |µ|ψ ′′ v ′′ ,N ′′
2 corresponds to FCF including theoretical transition dipole moment µ (taken either from the present calculation or from the Graz group [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF], see Fig. 2), HL(N ′ → N ′′ ) denotes normalized Hönl-London factors for P and R branches given by the formula [START_REF] Ochkin | Spectroscopy of Low Temperature Plasma[END_REF] HL

P (N ′ ) = 1 2(2N ′ + 1) N ′ - 1 4N ′ , (2)
HL R (N ′ ) = 1 2(2N ′ + 1) (N ′ + 1/2)(N ′ + 3/2) (N ′ + 1) , (3)
and P (v ′ , N ′ ) describes population of energy levels in the B(2) 2 Σ + state at thermal equilibrium

P (v ′ , N ′ ) = 2(2N ′ + 1) exp - B B e N ′ (N ′ + 1) kT exp - T B e + ω B e (v ′ + 1/2)
kT .

(4) The values of spectroscopic parameters T e , ω e and B e of the excited state were taken from the theoretical predictions. The line profiles were assumed to have a Gaussian shape with FWHM = 0.16 cm -1 , that results from the Fourier Transform Spectrometer working parameters. The simulated spectra were used in the process of identification of the experimental results.

Data Analysis

A specific method of experimental spectra analysis was used. At the beginning three theoretical spectra of the B→X transition were simulated according to the procedure described in the previous section, relying on three sets of potential curves, either from Ref. [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF] or from the present calculations. These simulated spectra were compared with the experimental spectrum, as shown in Fig. 4. It can be noticed that the general shape of the spectra is similar and the main difference comes from different theoretical values of term energy of the B state (T B e ) in each calculation(see Tab.1), resulting in different positions of the spectra on the energy (wavenumber) scale. The relative positions of the individual band heads and their intensities are also somewhat different for the three theoretical calculations of interest. Nevertheless, by careful comparison, we were able to assign several band heads in the experimental spectrum, as shown in Fig. 4(d). The wavenumbers of the assigned band heads can be arranged neatly into a Deslandres table (Table 2), which supports our assignment.

At the next stage of the analysis we ascertained which rotational transitions were responsible for formation of each band head, taking into account the experimental width of the band heads, typically 0.3 cm -1 (see Fig. 5). All the observed vibrational bands were red shaded which means that the band heads were formed in the P branches, the R branches providing merely a background in the recorded spectra. (This observation implies that the equilibrium distance R ′ e of the B state is smaller than R ′′ e of the ground state). A range of the B state rotational levels N ′ involved in formation of each band head was identified on the basis of theoretical calculations and only transitions starting from these levels were taken into account in further steps of the analysis. Note that a different range of N ′ values was attributed not only to each band head but also to each theoretical model. In this way a group of P lines responsible for band head formation was specified for each band and we assigned to these lines the same wavenumber, namely that of the corresponding band head. As a result we obtained a set of about 350 P lines, split into 13 groups, with the same wavenumber within each group. The wavenumbers of these lines were fitted to differences of term values of the upper and lower levels,

ν = T B (v ′ , N ′ ) -T X (v ′′ , N ′′ ) , (5)
which were expressed by a standard Dunham expansion

T B = T B e +Y B 10 (v ′ +0.5)+Y B 01 (N ′ (N ′ +1))+Y B 20 (v ′ +0.5) 2 +Y B 11 (v ′ +0.5)(N ′ (N ′ +1)), (6)
T X = Y X 10 (v ′′ +0.5)+Y X 01 (N ′′ (N ′′ +1))+Y X 20 (v ′′ +0.5) 2 +Y X 11 (v ′′ +0.5)(N ′′ (N ′′ +1)), (7)
with the Dunham coefficients Y ij having their usual meaning. The value of the rotational constant of the ground state Y X 01 ≡ B X e had to be taken from theoretical predictions because of the unresolved rotational structure of the experimental thermoluminescence spectra. Three sets of Dunham coefficients were generated for both the B(2) 2 Σ + and the X(1) 2 Σ + states assuming three different values of rotational constant Y X 01 resulting either from the present calculations or these from Ref. [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF] (Table 3). Comparison of the three sets of coefficients for each state shows that their values (except of the enforced values of rotational constants Y 01 ) agree within their uncertainties despite significant differences between theoretical models. Therefore one can expect that the obtained Dunham coefficients may be used to predict fairly accurately values of energies of vibrational levels in the X(1) 2 Σ + and B(2) 2 Σ + states also beyond the range of this experiment. Finally we note that the Y X 10 ≡ ω X e values in Table 3 lie in between the one of the present calculations, and the one from Ref. [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF]. In contrast, the obtained value for Y B 10 ≡ ω B e is larger than the theoretical values from RCCSD(T) and the Graz results, while it is in remarkable agreement with the value from the FCI+ECP+CPP method. The T e value derived from the Dunham analysis is also in good agreement with the results of the FCI+ECP+CPP calculation and that of Ref. [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF], and smaller than the one obtained by the RCCSD(T) approach. It is probable that this large deviation of T e in RCCSD(T) results from the strong multireference character of the B state -which exhibits a marked avoided crossing with the next electronic state, a quoted above -that is not accounted for in the coupled-cluster method.

It should be noted that the uncertainties of the Y ij values result mainly from the problem of determination of accurate positions and widths of the band heads, as many lines corresponding to transitions to different rovibrational levels in the B(2) 2 Σ + and X(1) 2 Σ + states overlap in the spectra and observation of only top parts of the band heads is possible. A Monte-Carlo method was used to find the error associated with this problem. With the half-width of each band head assumed somewhat arbitrarily to be 0.3 cm -1 , the positions of the individual band heads were randomly changed within the range of ±0.3 cm -1 (see Fig. 5) and also their widths were changed within ± 0.16 cm -1 (that influenced the number of rovibrational lines taken into consideration). A set of Dunham coefficients was determined for each random combination of the positions and widths of the band heads. The procedure was repeated until the average values of all coefficients became equal to the values determined in a straight fit described in the previous paragraph. The final errors (listed in Table 3) were defined as tripled values of the standard deviations obtained for each Dunham coefficient. Note that the Y 11 coefficients for both states are statistically meaningless but their inclusion has improved a quality of the fit. Finally, three sets of rovibrational levels were generated for each of the B(2) 2 Σ + and X(1) 2 Σ + states basing on the experimental Dunham coefficients of Table 3, up to 350 levels per each set, in the range of quantum numbers v ′′ = 0; 4 , N ′′ = 0; 32 and v ′ = 0; 7 , N ′ = 0; 33 . With these input data potential energy curves were built for each state using the point-wise Inverted Perturbation Approach (IPA) method [START_REF] Pashov | Construction of potential curves for diatomic molecular states by the IPA method[END_REF]. As a starting potentials the corresponding theoretical potential energy curves were used. The resulting IPA potentials are determined reliably by the experimental data in the energy range between 0 and about 250 cm -1 for the X state and approximately between 8900 and 9550 cm -1 for the B state. In Fig. 6 the IPA potentials are compared with the bottom parts of the theoretical potential energy curves. For the theoretical potentials we assumed that the minima of the ground state curves correspond to zero energy (as in case of the experimental ones) but the calculated values of T B e have been retained (see Table 1). The shapes of experimental IPA curves confirm the conclusions drawn from the obtained values of the Dunham coefficients. The shape of all three experimental potentials for both states is almost identical and does not depend on the starting theoretical potential. The main difference lies in the positions of their minima and relates to the theoretical R e value. For the ground state positions of the potential minima differ by less than 0.1 Å. In the case of the B(2) 2 Σ + state the differences are even smaller. All IPA potentials are presented in the supplementary materials [40].

To confirm the validity of the procedure described above, the molecular spectra were calculated again, basing on the final experimental potential curves. The exemplary result, obtained for the curves assuming the ground state R e value taken from the FCI+ECP+CPP calculations, is shown in Fig. 7. For all identified band heads there is a good agreement between their experimental and calculated positions. Moreover, the calculated spectrum allows to identify an additional band head in the observed spectrum, indicated by a question mark in Fig. 4(d). This proves that a good estimation of the potential energy curves of the investigated states has been obtained.

Conclusions

With the guidance of three sets of theoretical potential energy curves and transition dipole moments -two sets being presented in this work, and one from a previous work Ref. [START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF] -we derived for the first time the fundamental spectroscopic constants of the KSr ground state, and of its second excited electronic state. We performed an experiment involving thermoluminescence spectroscopy, which provided molecular spectra in the X-B system with resolution 0.16 cm -1 given by the Fourier Transform Spectrometer.

The three sets of theoretical potentials curves were used first to assign the intricate experimental spectra. Then they were employed as starting curves of the IPA routine. In each case the curves have been modified by the routine to fit the experimental data. As it turned out, the resulting potentials are of almost identical shape, irrespective of the starting curve. Only relative positions of the equilibrium internuclear distances R e differ noticeably and depend on the theoretical model, being the main limitation of our study. Nevertheless the procedure provides an interesting alternative to complicated and time-consuming experimental techniques if quick evaluation and coarse optimization of theoretical potential curves is required. Among the three theoretical approaches examined in the present work, the FCI+ECP+CPPP one seems to provide the best possible agreement with the present measurements. Further studies should be performed to confirm to which extent those three approaches could indeed be discriminated with respect to their accuracy. We note however that their overall agreement with the derived experimental data is satisfactory, confirming that they could serve well for modeling possible schemes for ultracold molecule formation starting from ultracold atom pairs.

Figure 2 :

 2 Figure 2: The transition dipole moments for the X(1) 2Σ + -A(1) 2 Π, X(1) 2 Σ + -B(2) 2 Σ + and X(1) 2 Σ + -D(3) 2 Σ +transitions in the KSr molecule. Results are displayed for the FCI+ECP+CPP approach (red solid lines), and for the MRCI results (green doted lines) of the Graz group[START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF].

Figure 3 :

 3 Figure 3:The potential energy curves calculated with FCI+ECP+CPP method for all states correlating to the K(4p 2 P )+Sr(5s 2 1 S) and K(4s 2 S)+Sr(5s5p 3 P ) asymptotes. The solid black lines denote 2 Σ + states, the dashed red lines denote 2 Π, the dash-dot blue line refers to4 Π and the dotted green line denotes 4 Σ + state. The zero energy is the same as in Fig.1

Figure 4 :

 4 Figure 4: Comparison of the B(2) 2 Σ + ← X(1) 2 Σ + spectra in the KSr molecule simulated using theoretical potential curves from the present paper (obtained by RCCSD(T) method (a) and by FCI+ECP+CPP approach (b)), and from Ref. [39] (c) with the experimental spectrum from this work (d). The assignment of band heads is based on theoretical calculations (vertical dashed line).

Figure 5 :

 5 Figure 5: An example of band head formation in the 0-0 band of the B-X system of KSr. Vertical dashed lines indicate the range of rotational levels involved in the band head formation and assumed to correspond to the same wavenumber in further analysis.

( a )

 a Values taken from the theory and fixed during the fit.(b) Strongly correlated with Y 01 value of the X(1) 2 Σ + state.

Figure 6 :

 6 Figure6: Comparison of the bottom parts of the theoretical curves according to the calculations with the methods FCI+ECP+CPP (dashed lines, green in colour) and RCCSD(T) (solid lines, black in colour), presented in this paper, and from the Graz group[START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF] (dotted lines, blue in colour) with the experimental IPA curves of the B(2) 2 Σ + (upper panel) and X(1) 2 Σ + (lower panel) electronic states. The three IPA potentials for each state employ the ground state Re values predicted by the present calculations (FCI+ECP+CPP -open circles and RCCSD(T) -rectangles) and that reported in Ref.[START_REF] Pototschnig | Vibronic transitions in the alkali-metal (Li, Na, K, Rb) -alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals[END_REF] (triangles).

1 ]Figure 7 :

 17 Figure 7: Comparison of the original experimental spectrum (upper curve, black in colour) with the spectrum simulated using the experimental IPA potential employing ground state Re value from the present FCI+ECP+CPP calculations (lower curve, green in colour).

Table 2 :

 2 Deslandres table constructed for the observed band heads in the experimental spectrum of KSr.

		v ′′ = 0	1	2	3	4
	v ′ = 0 8943.67 53.54 8890.13
		77.21	77.38
	1	9020.88 53.37 8967.51 52.31 8915.2
			76.08
	2		9043.59
			77.33
	3		9120.92 52.18 9068.74
			75.51	76.27
	4		9196.43 51.42 9145.01
				75.08
	5			9220.81 51.18 9169.63
					75.48
	6				9245.11
	7					9270.29

Table 3 :

 3 The experimental Dunham coefficients for the B(2) 2 Σ + and X(1) 2 Σ + states of the KSr molecule. The three different sets of coefficients for each state rely on different values of the ground-state equilibrium distance Re taken from three theoretical calculations. All values are in cm -1 .

	Y i, j	MRCI [39]	FCI+ECP+CPP	RCCSD(T)
			X(1) 2 Σ +
	Y 10	54.83 ± 1	54.58 ± 1	54.59 ± 1
	Y 20 Y 01 × 10 2 Y 11 × 10 4	-0.59 ± 0.19 3.23557 a -2.2 ± 4	-0.56 ± 0.18 3.22434 a -2.7 ± 5	-0.56 ± 0.18 3.12098 a -2.9 ± 5
			B(2) 2 Σ +
	T B e	8933.24 ± 0.9	8932.49 ± 0.9	8932.53 ± 0.9
	Y 10	77.61 ± 0.54	77.55 ± 0.54	77.56 ± 0.54
	-0.17 ± 0.04 Y 01 × 10 2 3.311 ± 0.013 b Y 20 Y 11 × 10 5 -4.8 ± 15	-0.16 ± 0.04 3.397 ± 0.014 b -5.4 ± 17	-0.16 ± 0.04 3.266 ± 0.013 b -8.2 ± 25

Acknowledgements J.Sz acknowledges partial support from the Miniatura I programme founded by the National Science Centre of Poland [grant number 2017/01/X/ST2/00057]. P.S.Z is grateful for the Grant OPUS 13 founded by the National Science Centre of Poland [grant number UMO-2017/25/B/ST4/01486]. Authors acknowledge


Maciej B Kosicki

Piotr S Żuchowski

Maykel L González-Martínez

Olivier Dulieu
email: olivier.dulieu@u-psud.fr

Product-state distribution after isotopic substitution in ultracold atom-molecule collisions

We show that products of the isotopic substitution reactions in experimentally accessible molecules such as NaK, RbCs and SrF are cold according to their translational energy below hundreds of mK. For these chemical reactions molecular products may occupy only the lowest rotational states. We also discuss the possibility of controlling the chemical reactions by the electric field in ultracold mixtures of molecules and atoms with low kinetic energy release, where one of the constituent atoms of colliding molecule is replaced by its isotope. This letter open new avenues in investigating the branching ratios of chemical reactions in ultracold conditions.

The past decade has witnessed rapid progress in the investigation of chemistry at ultracold temperatures. Ospelkaus et al. [1] demonstrated field-free chemical reactions between KRb molecules and constituent atoms. By controlling the internal state of KRb molecule, it was possible to enhance the reaction rate between two KRb molecules by allowing s-wave collisions. They have also described collisions of KRb with ultracold K or Rb atoms finding that a formation of K+Rb 2 products is energetically forbidden. Later on, the same experimental group [2,3] has also shown that an external electric field may be applied to orient KRb molecules and to influence their reaction rate. Recently, a mechanism to monitor the chemical reactivity by the selection of reactant vibrational state has been utilized in the reaction between NaRb molecules [4]. Other experimental attempts involve Feshbach molecules and ultracold atoms in the presence of an external magnetic field [5,6].

Measurements of the reactant loss is a straightforward way to probe reactivity at ultracold temperatures. However, the detection of products would provide much more details about the underlying dynamics. A basic difficulty in detecting product states of such reactions is that the kinetic energy release is often larger by several orders of magnitude than the depths of most traps used in experiments (see e.g. [7,8]). Despite several successful realizations of ultracold molecular gases of ground-state heteronuclear alkali-metal diatomic species [9,10,11,12,13,14,15,16,17] and promising prospects [18,19], no detailed study of product-state distribution has yet been reported so far. However two important steps in this direction were recently achieved: the detection of weakly-bound 87 Rb 2 molecules by three-Rb-atom recombination [20,21], and the detection of polyatomic ions after the collision of two ultracold KRb molecules [22].

It would be desirable to explore cases in which the products can be accumulated in the trap after the chemical reaction. Inspired by a recent proposal for chemical reaction with very small energy release, involving isotope exchange in ultracold molecular collisions [23], we investigate in the present work the isotopic substitution reactions between a ground-state alkali-metal atom a A with atomic mass a and a ground-state heteronuclear diatomic molecule containing a different isotopic species b A (with a > b), and another atom B

a A + b AB -→ b A + a AB. (1)
If the molecule is initially in its lowest rovibrational level v = 0, n = 0, (neglecting the hyperfine structure for now), the energy release ∆E a-b for the reaction (1) can be expressed in terms of the change of the zero-point energy

∆E a-b = ω a -ω b 2 , (2)
where ω a (ω b ) is the harmonic constant of the a AB ( b AB) molecule. Values of ∆E a-b for a selected series of species of experimental interest are found small enough to be considered in future experiments based for instance on electrostatic or microwave traps (Table 1).

Our previous studies show that atom-exchange reactions involving alkali-metal dimers and atoms, as well as strontium monofluoride and strontium atoms, are barrierless processes [7,8]. In all these cases the potential energy surfaces (PES) are very deep and support tens of vibrational levels and hundreds of rotational levels. Full quantum scattering calculations for atom-molecule reactive collisions, even for the field-free case, are cumbersome for strongly interacting species as alkali-metal atoms [24,25,26,27,28], due to the huge number of involved channels. Interestingly enough, the isotope-exchange of lithium in field-free atom-diatom collisions has been investigated in [26], and the product-state distribution for this system has been analyzed.

Recently, it was demonstrated [START_REF] Croft | Universality and chaoticity in ultracold k+krb chemical reactions[END_REF][START_REF] Croft | [END_REF] that the distribution of product rotational states after the atom-exchange reaction between ultracold K atom and KRb molecule, calculated with rigorous quantum scattering calculations exhibits the Poissonian distribution, while the of positions and widths of scattering resonances display signatures of quantum chaos: Wigner-Dyson distribution of resonances spacings in energy domain, and Porter-Thomas distribution of widths of resonances. A similar conclusion has been reached for the D + + H 2 → H + + DH reaction [31,32]. Furthermore, it was noticed in [33] that the collisions of alkali-metal dimers with atoms exhibit strongly resonant character: near the collision threshold, the density of bound states supported by closed channels (related to excited hyperfine, rotational and vibrational states of products) is very high. Because of that, long-lived collision complexes are predicted to be formed, and full randomization of the reaction energy (via internal modes) should occur before the complex is destroyed. This essentially makes the product state distribution of the complex statistical, i.e. without featuring correlation between products and reactants. We then use the statistical approach adopted in [34] for ultracold molecular collisions to investigate such isotope-substitution reactions when all particles are in their absolute ground state in free space. We will show how the electric field can be used to control their reactivities by blocking the reaction channels and by tuning product state distributions. Although we focus here on the electric field control, one could imagine a similar scenario with the magnetic field for paramagnetic molecules or microwave radiation fields.

For the chemical reactions addressed in Table 1, the kinetic energy release is small. Values for ∆E a-b values have been obtained using experimental data [35,36,37,38,39,40,41] and mass-scaled harmonic frequencies of the ground state. Non-Born-Oppenheimer (NBO) effects, leading to an estimated shift of about dozens of MHz between the two isotopologues a AB and b AB for alkali-metal dimers [42], are neglected, so that they have both the same dissociation limit before including hyperfine structure. The same origin is considered for the hyperfine manifold of the colliding atomic partner. The values of ∆E a-b are obtained after setting the origin of energies at this dissociation limit, which is the center of gravity of the hyperfine manifold of the constituent atoms. The hyperfine structure of the (v = 0, n = 0) ground-state level is also neglected for these evaluations. The 88 Sr+ 86 Sr 19 F reaction must be considered with care, as ∆E a-b is only about 300 MHz larger than the n = 0 → n = 1 rotational excitation in 88 Sr 19 F. Using the same methodology as in [42], based on ADF quantum-chemistry code [43] with the density functional method (PBE96) [44], we found that the NBO shift between the ground-state (v = 0, n = 0) level energies of 88 Sr 19 F and 86 Sr 19 F, is smaller than 10 MHz, and thus, can be safely neglected.

Table 1. The calculated reaction energy ∆E a-b (references to used data in square brackets), and the rotational constant B 0 , the hyperfine coupling constant A and spinorbit coupling constant γ SR of products are shown. Symbols n max and max denote the maximum rotational state of molecule for a collision energy of 1 µK and the maximum end-over-end angular momentum which can be populated by ∆E a-b , respectively. See supplementary material for complete compilation for all possible alkali-metal dimers reactions.

Products ∆E a-b [mK] B 0 /k b T [mK] A/k b T [mK] γ SR [mK] n max max 39 K + 23 Na 40 K
415 [36] 136 [45] 11 [ The ∆E a-b values displayed in Table 1 range from 123 mk to 811 mK, corresponding to the 87 Rb+ 7 Li 85 Rb and 41 K+ 23 Na 39 K collisions, respectively. These energy releases are larger than the typical depth of optical dipole traps, but it should be possible to retain the products in electrostatic traps or in microwave cavities, or even in magnetic traps for the case of monofluoride reactants. The maximal value n max of the molecular rotational quantum number n allowed by the energy release is deduced within the rigid-rotor approximation for the rotational energies E(n) = n(n + 1)B 0 , where B 0 is the rotational constant of the (v = 0, n = 0) ground-state level. We neglected collision energy of 1 µK in the n max estimation. Interestingly, very few rotational levels can be populated in the products, and only n = 0 in the cases of 87 Rb+ 85 Rb 23 Na, and 87 Rb+ 85 Rb 7 Li collisions. In contrast, very large number of hyperfine levels can be populated in the alkali-metal. Specifically, there are 144 hyperfine levels for both n = 0 and n = 1 rotational states of the Na 40 K and even more for the 87 RbCs due to the presence of n = 2 rotational state (both in X 1 Σ + ground-state). For the alkali-metal-atom+ alkali-metal-dimer system, these levels are additionally split by the hyperfine interactions of the atomic collision partner. The rotational splitting in the SrF molecule in the X 1 Σ -ground-state needs additional comments. Firstly, due to non-zero electronic spin, each rotational level is split by the spin-rotation interaction into states described by j = n + s. Secondly, these states are split by the fluorine hyperfine interaction. Overall, there are 16 hyperfine levels in the 88 SrF which can be populated for n = 0 and n = 1 rotational states. In field free case we use f = j + i quantum number to label them, accordingly.

Cold polar molecules considered in this paper exhibit a permanent dipole moment (PDM) in their own frame (identical for all isotopologues to a very good approximation) large enough (see, e.g. [51]) to strongly interact with a static electric field. Noticeable Stark shifts comparable to ∆E a-b at zero field can be induced by applying an experimentally achievable electric field. Figures 1(a), 2(a), and 3(a) show the variation of the level energies of the reactant and product complexes with the electric field for three typical cases, 40 K+ 39 K 23 Na, 87 Rb+ 85 Rb 133 Cs, and 88 Sr+ 86 Sr 19 F collisions, respectively. The energy levels of reactants and products are parallel, with different splittings within each n manifold due to the change of hyperfine states. In case of molecules with lighter atoms, such as NaK or SrF molecules, the hyperfine splittings of the reactant atoms are much smaller than rotational energy of the molecules. On the other hand, the Rb hyperfine splitting is much larger than the RbCs rotational constant, so the pattern of thresholds of reactants and products is quite complex. Note that the hyperfine states of molecules are superimposed at the resolution of the plot.

As can be seen in the panels (a) of the figures, it is possible to induce crossings between the energies of reactants and products by tuning the electric field, thus closing or opening reaction thresholds for the related collisions. For the 39 K+ 40 K 23 Na (Fig. 1 (a)), close to 4 kV/cm, the ground state of the reactant 23 Na 39 K approaches m n = 0 states of 23 Na 40 K molecule (which for zero field correlate with n = 1), and becomes a closed channel while the electric field is increased further. The same situation can be seen close to 7.5 kV/cm: the channels with |m n | = 1 close as the electric field is swept across the state crossing. On the other hand, there are multiple crossings of the ground state of reactants with rotationally excited states of product molecules up to n = 2 in case of the RbCs system (Fig. 2 (a)). A magnitude of electric field above 15 kV/cm closes channels corresponding to n = 2 states of product 87 Rb 133 Cs. The reaction starting with 88 Sr+ 86 SrF (see Fig. 3 (a)) is very peculiar, since the reactant ground-state energy is very close to the one of rotationally-excited product states, and an electric field of about 2.5 kV/cm is sufficient to close all channels except ground rotational n = 0 state. The molecules which are produced in n = 1 state are very slow with total translational energy on the order of 20 mK or less.

The statistical approach for collisions was initially formulated in the case of nuclear scattering by Feshbach [52,53,54], and adapted for molecular collisions by Bernstein, Light, and Miller [55,56,57]. It has been since then extended and combined with more rigorous close-coupling schemes to study the reactive scattering in many-partial wave regime for a number of important chemical reactions [58,59,60]. Recently, the statistical approach was used for ultracold collision studies between Li and LiYb, and compared to quantum reactive scattering [61]. Here, the statistical approach of [34] is extended to the case of ultracold atom/molecule collisions in a presence of an electric field, and is used to analyze the product state distributions in reactions with exothermicity on the order of tens to hundreds of kelvins.

An essential assumption of the statistical method is that collisions proceed via the long-lived intermediate complex, so that a full randomization of quantum state populations in the interaction region occurs. Then the reactants and products are uncorrelated, and the transition probability for reactant state α to product state β is expressed as

P M α→β = p M α p M β / γ p M γ , (3)
where γ denotes all possible values β. The quantities p M α and p M β are the probabilities for capture in the entrance channel, and isotopologue formation in the outgoing channel. All these probabilities depend on the collision energy E and on the electric field magnitude ε. Note that in the external field the total angular momentum J of the collisional complex is not conserved but its projection M onto the space-fixed axis (along the direction of the electric field) is.

The statistical method should be particularly useful for collisions of heavy atoms with alkali-metal dimers and systems with monofluorides of atoms with an external s 2 shell, like alkaline-earth atoms. The method relies on complex formation due to a presence of resonances and for systems with large reduced mass, small rotational constants, and a rich hyperfine structure. The initial estimations of the density of states near the collision threshold were performed by Mayle et al. [62] and recently revisited by Christianen et al. [63] for the systems with neglected hyperfine structure. In the latter paper, the density of states which do not conserve the total angular momentum but conserve its projection (which is the case for electric field) for K 2 +Rb collisions was calculated to be on the order of 10 4 states per inverse Kelvin which is two orders of magnitude more than in case when the total angular momentum is conserved (in fieldfree case). This number is possibly smaller for NaK+K system but larger for RbCs+Rb. The density of states in all systems concerned here is, in fact, more significant due to complex hyperfine structure. The strongest effect which couples different hyperfine states is nuclear quadrupole interaction with the electric field gradients at nucleus which changes with geometry, but also interaction-induced modification of hyperfine coupling of alkali-metal atoms [64] or SrF molecule. The detailed analysis of resonances in these complexes in an electric field with hyperfine couplings included is very challenging. However, for the sake of the present paper, one can safely assume that statistical regime is achieved in all systems studied in the present paper.

The reaction cross-section σ M α→β for a collision at energy E (taken from an arbitrary origin) and an electric field magnitude ε can be expressed as

σ M α→β (E, ε) = πh 2 2µ(E -E α (ε)) M p M α (E, ε)p M β (E, ε) γ p M γ (E, ε) (4)
where µ is the reduced mass of the reactants, E -E α (ε) gives the effective collisional energy above the reactant energy E α at a given ε. In the studied cases, the initial channel corresponds to a single hyperfine state of the reactant molecule and of the atom partner, colliding in the s-wave regime (i.e., the quantum number associated to the atom-molecule mutual rotation is equal to 0, and it is assumed to be decoupled from the other angular momenta). Following [34], we obtained the energy levels of molecules concerned by diagonalizing the Hamiltonian of the system including the hyperfine structure and the Stark operator (see e.g. [65,66]). We used accurate atomic and molecular constants to obtain the energy levels of the systems [46,49,50,67,37,45,11,38,68,36,39,40,48,69,70,65,66,41,35,47].

Although the spacing between hyperfine energy levels of 1 Σ + molecules is several orders of magnitude smaller than the one between rotational levels, we included them to properly account for state counting. To calculate complex formation probability P M α,β , we used the semi-classical approach of tunneling through the reaction barrier (which in this case reduces to the centrifugal barrier): unit probability for E -E γ above the centrifugal barrier (similarly as in Langevin model), and a non-zero probability of the tunneling effect expressed in the Wentzel-Kramers-Brillouin model (WKB) otherwise, which can be written as [START_REF] Balint-Kurti | Theory of Molecular Collisions[END_REF]34]:

p M γ (E, ε) = exp - 2 h Rmax R min 2µ γ γ ( γ + 1)h 2 2µ γ R 2 - C 6 R 6 -(E -E γ (ε)) dR , (5)
where the R min and R max are classical turning points characterizing a position of the centrifugal barrier and C 6 is the long-range dispersion coefficient. We took van der Waals C 6 coefficients from Ref. [69] to represent long-range potentials. Figures 1b,2b, and 3b show the probabilities of forming the atom+diatom products with the specific translational energy for the reactants with an initial collisional energy of 1 µK, while the panel (c) therein shows the formation probability of the molecules in a given |m n | quantum state. Each curve in the panels (c) is a sum of all contributions to the probability presented in the panels (b). Interestingly, for zero electric field, the largest propensity of forming the product molecule corresponds to the state with the largest energy difference between ground state n = 0 and product state. That is due to the distribution of products among all accessible quantum numbers in an outgoing channel is the largest. This result can be rationalized using the phasespace theory (PST) of Pechukas and Light [56], which is essentially a particular case of Eq. 3 in which step-function probabilities (equal to 0 for barrier reflection, and equal to 1, for transmission over the barrier) are used. If we neglect the hyperfine states, one can easily show that for final rotational state n of products, one should populate (2n + 1)( max (n) + 1)/( n≤nmax (2n + 1)( max (n) + 1)), where max (n) is the maximum end-over-end angular momentum for which the centrifugal barrier fits below ∆E a-b -B 0 n(n+1). In Table 1, we specified the largest possible max value corresponding to n = 0 in the product state. These values are on the order of 20, so clearly, in the outgoing channel, we are no longer in the quantum regime. Approximate estimation of branching ratios in the zero-field with the PST theory agree well with calculations using the theory developed in Ref. [34] for K+NaK and Sr+SrF reactions, for which the energy level splitting due to hyperfine coupling is much smaller than the rotational spacing.

An inspection of Figure 1b reveals that the zero-field probabilities for the K+NaK system are 63% for the channels corresponding to n = 1 state with a translational energy order of 100 mK and 37% for n = 0 state with energy about 400 mK. Interestingly, product distributions for the K+NaK system are less complicated than in case of the Rb+RbCs reaction(see Figure 2b). For the latter one, probabilities are 22% for 60 mK, and 34% for 110 mK in a mixture of products with n = 1, and n = 2 states, and 26% for separated n = 1 state with 210 mK, and 9% for n = 0 states with energy about 100 mK, and 250 mK. In case of the Sr+SrF reaction, probabilities are 60% for n = 0 state with an energy order of 750 mK, and 40% for energies between about 10 mK, and 20 mK which correspond to n = 1, and n = 0 states, respectively.

Reaction products and their probability distributions can be controlled by tuning of an external electric field, since the energy levels have different effective dipole moments (defined as energy derivative with respect to electric field). The Stark shifts of the rotational energy of reactants close reactive channels if product states are above reactant states in the energy. As the energy gap between reactant and product channel changes the number of possible L max states, which can cross the centrifugal barrier decreases. Hence the number of product states also decreases with the energy gap, as the

L max ≈ ( √ 108∆E/E vdw ) 1 3
. For this reason it is rather inefficient to tune the difference between products and reactants arbitrarily close with the electric field as a very small quantity of very cold products will be produced in such reaction. For the 40 K (F=9/2) + 23 Na 39 K (v=0, n=0) reaction, the intensity of the electric field of 8.16 kV/cm is sufficient to produce molecules in the pure rotational ground state. For a slightly detuned value of the electric field, e.g., for the intensity of about 7.3 kV/cm, one can produce the molecules in the |m n | = 1 state with translational energy of products equaled 16 mK with a probability of 10%. Translational energy of products may even be smaller than 100 µK for about 8.1 kV/cm but with a population of about 3%. Interestingly, the probability is 23% for molecules in n = 1 state with translational energy about 10 µK in the Rb+RbCs system. In this reaction, the Stark shifts close reactive channels corresponded to n = 1 state for intensities above 36.3 kV/cm. The probability for production of samples of 88 SrF molecules with the translational energy smaller than 100 µK is 4% for an electric field 2.6 kV/cm which is a slightly detuned value of the field at which all reactive channels close, except ones for the ground rotational state.

In summary, we have shown that products of the isotopic substitution reactions are cold according to their translational energy below hundreds of mK, and molecular products may occupy the lowest rotational states. We have also discussed the possibility of controlling the chemical reactions by the electric field in ultracold mixtures of molecules and atoms with low kinetic energy release, where one of the constituent atoms of colliding molecule is replaced by its isotope. Such experiments could be implemented with present experimental techniques, using a modest electric field (up to 20 kV/cm). With such an experimental setup it is possible to trap a large amount of the product molecules in the trap, if the external microwave cavity trap is used, or -in case of 2 Σ - molecules -in a magnetic trap. This can open new avenues in investigating the branching ratios of chemical reactions in ultracold conditions. Given that full quantum dynamics calculations in external fields are very challenging and at present approximations need to be introduced, such experiments can shed new light on untangling complicated dynamics [39] 100 [11] 49 [49] 0 20 85 Rb + 39 K 87 Rb 198 [37] 55 [47] 49 [49] 1 22 85 Rb + 40 K 87 Rb 199 [37] 55 [47] 49 [49] 1 22 85 Rb + 41 K 87 Rb 201 [37] 55 [47] 49 [49] 1 22 85 Rb + 87 Rb 133 Cs 253 [40] 23 [50] 49 [49] 1 27

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. The 40 K + 23 Na 39 K -→ 39 K + 23 Na 40 K reaction. (a) Splitting of the molecular rotational levels in a absence and presence of external electric field. The diagram shows field-free rotational levels which are labeled by n 39 (indicating 23 Na 39 K) and n 40 (indicating 23 Na 40 K) quantum numbers as well as by f = i+s quantum number describing the hyperfine interactions of the atomic partner( 40 K for reactants and 39 K for products). The plot shows a splitting of these levels due to the interaction with external electric field. Note that, the hyperfine states of the system are superimposed the resolution of the diagram and plot. The ∆E 40-39 is the reaction exothermicity. The red thick line corresponds to initial state of the reactants. The dashed vertical lines in the plot indicate crossings between reactant and product states due to the Stark effect. (b) Product distribution of the translational energy. (c) Distribution of the projection of the rotational quantum number, m n , corresponding to 23 Na 40 K molecule. The results presented in panels (b) and (c) are obtained with an initial collisional energy of 1 µK.

Table 2 .

 2 The calculated reaction energy ∆E a-b (references to used data in square brackets), and the rotational constant B 0 , the hyperfine coupling constant A of products are shown. Symbols n max and max denote the maximum rotational state of molecule for a collision energy of 1 µK and the maximum end-over-end angular momentum which can be populated by ∆E a-b , respectively.Products∆E a-b [mK] B 0 /k b T [mK] A/k b T [mK] n max Rb + 23 Na 87 Rb 188

	max

Supplementary material

As mentioned in Table 1, we present complete list of the kinetic energy release, ∆E a-b , for all possible isotopic substitution reactions in the alkali-metal dimers. These values are obtained using the Eq. 2.