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The spin-orbit coupling of the 61Σ+ and 43Π states in KCs: Observation and deperturbation
Journal of Quantitative Spectroscopy & Radiative Transfer 239 (2019) 106650
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The spin-orbit coupling of the 61 + and 43 states in KCs: Observation and deperturbation J. Szczepkowski a,∗, A. Grochola a, P. Kowalczyk b, W. Jastrzebski a, E.A. Pazyuk c, A.V. Stolyarov c, A. Pashov d a
Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, Warsaw 02–668, Poland Institute of Experimental Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, Warszawa 02–093, Poland Department of Chemistry, Lomonosov Moscow State University, Moscow, 119991, Leninskie gory 1/3, Russia d Faculty of Physics, Sofia University, 5 James Bourchier blvd., Sofia 1164, Bulgaria b c
a r t i c l e
i n f o
Article history: Received 2 August 2019 Revised 12 September 2019 Accepted 13 September 2019 Available online 14 September 2019 Keywords: Laser spectroscopy Alkali dimers Electronic states Potential energy curves Spin-orbit coupling Deperturbation
a b s t r a c t Energies of 1655 rovibrational levels of the mutually perturbed 61 + and 43 0+ states in KCs were measured in a series of experiments employing polarization labelling spectroscopy technique. The experimental term values of the 61 + ∼ 43 0+ complex in KCs were used in coupled-channels deperturbation analysis which took explicitly into account the spin-orbit coupling of the 61 + and 43 0+ states. The empirical deperturbation treatment was supported by ab initio electronic structure calculation of adiabatic potentials of the interacting states and of the corresponding spin-orbit coupling matrix element as a function of internuclear distance.
1. Introduction Diatomic alkali metal molecules became particularly interesting for physicists, since a new branch of so called ultracold physics has opened new possibilities of manipulating molecules, and investigating new phenomena in conditions of very low temperatures [1]. A transfer of ultracold molecules from the initial states, in which they are formed, to desired states, particularly to the absolute ground state (v = 0, J = 0 of the lowest electronic state), is usually a necessary step during such experiments. However this process is often connected with a change of the spin state, namely transfer from triplet to singlet electronic states of a chosen molecule. The task is demanding, since optical transitions between states of different symmetries are nominally forbidden. Using levels of mixed singlet-triplet character as intermediate states helps to overcome this problem. Such levels arise due to interaction of electronic states of different multiplicity. Therefore detailed information about energy structure of molecules which are involved in ultracold experiments is very important.
∗
Corresponding authors. E-mail addresses: [email protected] (J. Szczepkowski), [email protected] (A.V. Stolyarov), [email protected]fia.bg (A. Pashov). https://doi.org/10.1016/j.jqsrt.2019.106650 0022-4073/© 2019 Elsevier Ltd. All rights reserved.
© 2019 Elsevier Ltd. All rights reserved.
The spin-orbit coupling (SOC) is a phenomenon responsible for a vast majority of intramolecular perturbations observed in the excited states of alkali metal dimers. Both local and global SO perturbations [2] are especially pronounced in rovibronic spectra of alkali diatomics containing the heaviest Cs atom. The mixed singlet-triplet character of the excited states can be efficiently employed to realize the so-called ”intercombination” (spin-forbidden) transitions in the MCs (M=Li,Na,K,Rb,Cs) molecules. Indeed, in stepwise processes in these molecules, an optical transition from the ground singlet X 1 + or triplet a3 + states, to an intermediate excited state demonstrating the pronounced SOC effect, can be used to reach a manifold of molecular states of multipilicity other than of the initial state. The singlet-triplet complexes, arising in molecules from a local SOC between close-lying states of different multiplicity, are indeed often implemented in schemes for laser cooling and laser assembling of ultracold alkali diatomics in their absolute ground state [3]. One of the molecules for which the SOC is observed and may be useful in terms of manipulating of molecules in ultralow temperatures is KCs [4–6]. Spectroscopic experiments which were aimed at description of its structure have already been performed [7– 12]. One of the investigated states was 61 + , for which only limited information concerning the less perturbed energy levels was provided [13] and interactions with neighbouring states were not
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sufficiently explored because of lack of suitable theoretical model. In this paper a complex approach to solve this problem is described. It is supported by new calculations and new experimental data concerning the 61 + state, followed by a deperturbation procedure, which takes into account the direct spin-orbit interaction between the 61 + and 43 states. The empirically derived potential energy curve and SOC function describes the experimental data within their uncertainties. The analysis confirms high quality of the ab initio calculations and suggests that the accuracy of the theoretical SOC function is comparable or even better than that of the empirical one. 2. Experimental details We applied the V-type optical-optical double resonance polarization spectroscopy technique. The experimental setup was almost the same as in our previous report [13]. Two lasers were used simultaneously in our experiment. A single mode ring dye laser (Coherent 899-21, pumped by a DPSS Sprout-G laser and actively stabilized to HighFinesse WS-7 wavemeter) delivered the probe beam whereas a high power pulsed dye laser (Lumonics HD500 pumped by LightMachinery IPEX-848 XeCl excimer laser) served as a source of a stronger pump beam. The laser beams were superimposed in a heat-pipe oven placed between two crossed polarizers forming a polarizer-analyzer system for the probe beam. The oven contained KCs vapour typically at a temperature of 400 ◦ C with an addition of about 4 Torr of argon buffer gas. The frequency of the probe beam was fixed on well known (v , J ) ← (v , J ) transitions in the 41 + ← X1 + band system of KCs [14,15], thus labelling the involved (v , J ) levels in the ground state of the molecule. The frequency of the circularly polarized pump beam was scanned over the investigated 61 + ← X1 + transition (see Fig. 1). Whenever it was resonant with a transition originating from the labelled X1 + (v , J ) level, the originally linear polarization of the probe beam became changed. In such cases the probe light was partially passing through the crossed analyzer and detected by a photomultiplier coupled to the boxcar averager (Stanford SR250) and a personal computer. Absolute calibration of the pump laser frequency was achieved by simultaneous recording of optogalvanic spectra of argon and neon and linearity of laser scan was additionally controlled using transmission fringes of a Fabry-Pérot interferometer 0.5 cm long. The precision of the laser wavenumber determination was better than σ = 0.1 cm−1 . 3. Assignment of the spectra The first experimental characterization of the 61 + state [13] led to more than 10 0 0 term energies. They were placed in the region where the SOC to the neighbouring triplet states made the unambiguous assignment possible. The experimental data were described within a single channel (adiabatic) model with deviations between measured and observed energies reaching 10 cm−1 , an accuracy well outside the experimental precision. Moreover, the lowest assigned vibrational level of the 61 + state in [13] was v = 6. Although some transitions to lower vibrational levels were recorded, their analysis was impossible without taking into account interactions with the nearby triplet states (see Fig. 1). Therefore within the present study an additional set of measurements was carried out to supplement the existing data on the 61 + state. The efforts were focused on the energy range 20 0 0 0 − 20700 cm−1 , where the lowest vibrational levels (v < 6) were expected. In some cases two spectral lines were observed instead of a single line, one of them shifted to higher energy than expected and the other to lower energy. In Fig. 2 one can follow two vibrational progressions to the 61 + ∼ 43 complex from lev-
Fig. 1. The ab initio potential energy curves [16] for 1 + (solid black line), 3 + (dotted olive line), 1 (dashed-dotted red line) and 3 (dashed blue line) states in KCs molecule. The energy range investigated in the present work is marked by horizontal red dashed lines. The arrows present schematically transitions induced by the pump and probe lasers.
els labelled in the ground state: (v = 4, J = 110) and (v = 5, J = 54). An irregular vibrational spacing and appearance of extra spectral lines are visible. At the same time an intricate pattern of intensities of the lines was observed: P and R lines often had different intensity, for example in Fig. 2 for transitions from (v = 5, J = 54) the R lines are completely missing. Those symptoms are characteristic for a situation where rovibrational levels of the observed electronic state interact locally with levels of other electronic states. As it was reported in [13] and observed during present measurements, the largest shifts of positions of the rovibrational levels were found in regions where the energy levels of the 61 + state are close to levels of the 43 state, which enhances the effect of interaction between these two states. During the final analysis an iterative deperturbation approach was employed, a detailed procedure of which is described in the following section. At each consecutive step of deperturbation the two channels model was used to predict positions of spectral lines and then the spectra were reanalyzed to identify new lines or, if necessary, to reassign these for which the previous assignment was incorrect. The procedure was repeated until all lines were properly assigned and the final model was built. Eventually more than 600 new term energies were added to the data field forming a total of 1655 term energies (see Fig. 3(a)) based on 2551 spectral lines only of the main isotopologue 39 K133 Cs. In Fig. 3(b) one can see a comparison between the experimental energy levels and their positions cal-
J. Szczepkowski, A. Grochola and P. Kowalczyk et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 239 (2019) 106650
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Fig. 2. A portion of the experimental spectrum of KCs corresponding to transitions from two levels labelled in the ground state, (v = 4, J = 110) and (v = 5, J = 54), to the 61 + ∼ 43 complex. The lines above the spectrum indicate transitions to consecutive vibrational levels belonging to the P or R branch.
Fig. 3. (a) Reduced energies Ered = E − 0.015 × J (J + 1 ) [cm−1 ] of rovibrational levels taken into account in the deperturbation procedure. Red dots indicate levels observed in the present experiment, while black stars correspond to the levels assigned before [13]. (b) Comparison between the experimental energy levels and their positions calculated with potential curves obtained in the single channel model. Observed deviations exceed significantly the estimated experimental uncertainty of 0.1 cm−1 .
culated with the potential curves obtained in the single channel model. In the wavenumber range up to 21200 cm−1 the observed deviations show a structure which suggests a complete mixing of the original Hund’s case (a) states, since two separate groups of levels can be seen which can be attributed to two Hund’s case (c) electronic states. Beyond 21200 cm−1 the mixing becomes more local and typical patterns of avoided crossings appear.
H = T + U (R ) + H ROT , where
T =− H ROT =
4. The coupled-channels deperturbation analysis The 61 + state is spin-orbit coupled directly only to the = component of the triplet 43 state. The spin-rotational interaction between the different components of the 43 state and its interactions with the two neighbouring 53 + and 63 + states were ignored when the initial Hamiltonian of the problem was set up. The diagonal elements H (corresponding to a single-channel model of the decoupled, i.e. isolated states) can be written as [2]: 0+
(1)
h ¯ 2 d2 ; 2μ dR2
h ¯2 (J (J + 1 ) − 2 + S(S + 1 ) − 2 − 2 ), 2 μR 2
(2)
and U (R ) is the potential energy of a given state. Since only the = 0+ component of the 3 state is considered, the relevant spin-orbit splitting function A(R) (arising from the diagonal part of the SO operator) is implicitly involved in the potential curve for the 43 0+ component: U3 + (R ) = U3 (R ) − A(R ). The 0
1
off-diagonal part of the spin-orbit term is represented by a single radial function:
HSO = ξ (R ).
(3)
4
J. Szczepkowski, A. Grochola and P. Kowalczyk et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 239 (2019) 106650 Table 1 Scheme of the Hamiltonian matrix for the present problem.
61 + 4 3 0
61 +
4 3 0
T + H ROT + U (R ) + Hq,
ξ (R) T + H ROT + U (R ) + Hq,
ξ (R)
indirectly through their SOC with the 43 0+ state. We tried to decrease the systematic deviations by extending the model to three coupled channels, including the = 1 component of the 53 + state, but this did not seem to cause even qualitatively the needed J dependent shifts of the energy levels. Moreover, by adding a new electronic state we observed that the model became too overdetermined. Therefore, given the present data set, we decided to extend the model with J-dependent second order corrections reflecting rotational interactions with distant electronic states of the phenomenological form [18]:
Hq, =
Fig. 4. Comparison between the theoretical and fitted potentials (left axis). Both theoretical curves have been shifted up by 65 cm−1 in order to match closely the fitted PECs. The theoretical diagonal SO radial function is also shown (right axis).
The eigenvalues of the model Hamiltonian were calculated by the collocation (Fourier Grid Hamiltonian) method [17,18]. The grid ˚ With a mapping function, 150 grid points spans from 2.6 A˚ to 10 A. were sufficient to reach the accuracy better than 0.001 cm−1 . The initial fits of the experimental data with U (R ), U (R ) and ξ (R) being adjusted showed that the present model is adequate, leading to residuals which dimensionless standard deviation
σ¯ =
exp N 1 (Ei − Eicalc )2 N σi2 i=1
is about 1.5. The value slightly larger than 1 is mainly due to systematic deviations for larger J’s, reaching 0.5 cm−1 for J = 136. The only J-dependent operators which are missing in the model are the L-uncoupling operator which couples the 43 0 state with the 3 1+ states and the S-uncoupling operator which mixes the components of the 43 state since the spin-orbit splitting function A(R) is relatively small (as the present theoretical calculations show, see Fig. 4). These interactions influence the 61 + levels only
h2 ¯ 2 J ( J + 1 )q ( R ), 2 μR 2
(4)
and this corrections were added to the diagonal part of the Hamiltonian (1), thus introducing two more fitting functions to the model, one for the 1 + state, q (R), and one for the 3 state, q (R ). A simplified scheme of the Hamiltonian matrix is given in Table 1. With the second order corrections it was possible to reduce the dimensionless standard deviation to below 1. The effect of the second order corrections can be estimated as q = E/(J (J + 1 )), where E is the energy shift which they cause. This quantity is v and J dependent, but on average it varies between 5 × 10−5 and 2 × 10−4 cm−1 both for the 61 + and the 43 states. It is difficult to point which interactions are accounted for through these second order functions. They should be treated rather as effective phenomenological corrections, which include several intramolecular interactions in an implicit form. A total of 1635 energy levels of the coupled singlet-triplet system are reproduced by this model with an rms of 0.089 cm−1 , which is within the expected experimental uncertainty (see Fig. 5). About 20 experimental levels, which show systematic deviations above 0.5 cm−1 were excluded from the fit, but for such highly condensed structure of electronic states in a heavy molecule like KCs it is not unusual that other local perturbation may cause deviations from the present model. It may be noted that 80% of the residuals lie within ± 0.1 cm−1 , i.e. within one standard deviation. This is more than expected for a normal distribution ( ≈ 68%)
Fig. 5. Residuals from two fits of the model applying the same Hamiltonian from Table 1 but two different sets of model functions. PECs in case (b) have less parameters than in case (a) (see the text for details). The shapes of the fitted SOC are shown in Fig. 6.
J. Szczepkowski, A. Grochola and P. Kowalczyk et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 239 (2019) 106650
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ECP-CPP-FCI approach is based on the full configuration interaction (FCI) treatment of two-valence-electron problem defined by the large-core two-component relativistic pseudopotentials (ECPs) and the core-valence correlation treatment through semi-empirical core-polarization potentials (CPPs) for both K and Cs atoms. The SO coupling matrix element between the 61 + and 43 states, ξ ab (R), and equidistant SO splitting of the triplet 43 state, Aab (R), were estimated as first-order SO interactions of scalar-relativistic states obtained during the ECP-CPP-FCI calculation. To take into account implicitly a systematic R-dependent error in the calculated energy [22] the potential energy curves (PECs) for the excited states were constructed by adding the vertical excitation energies UiXF CI = UiF CI − UXF CI calculated as functions of the internuclear distance to the highly accurate empirical ground X 1 + state potential emp UX from Ref. [7]: F CI Uiab (R ) = UiX (R ) + UXemp (R )
Fig. 6. Comparison between √ the theoretical SOC function ξ (R) (red line with circles, multiplied by a factor − 2) with two fitted curves: black, dashed line - from the first fit, blue, solid line - from the second fit. The theoretical curve was used as the initial guess for the second fit.
which points out that the experimental error has been assumed rather conservatively by us. All model functions are defined in a pointwise form and natural cubic spline functions were used for interpolation. This representation is very flexible and allows functions with various shapes to be modelled with relatively few parameters. However, the flexibility of the pointwise functions is also their disadvantage, as it may lead to unphysical oscillations. In the present version of the fitting code the minimization routine was based on the Singular Value Decomposition (SVD) technique [19], which orders the parameters (or their linear combinations) according to their influence on the positions of the calculated energy levels. During the fit, one can select and adjust only parameters which contribute significantly to the reduction of the standard deviation. The rest of the parameters are kept fixed to their initial values instead of giving them some arbitrary value. Such approach was successfully applied in single channel fits [20] and it was very useful in case of coupled channels [18] as well. However the increased number of fitted functions by the coupled channels treatment and strong correlation between some of the interactions make the shapes of the fitted functions somewhat ambiguous. While the potential energy curves may be kept smooth and close to our expectations (based on ab initio calculations), the shape of the other radial functions is usually unknown. Nearly identical sets of energy levels (within their uncertainty) may be calculated from Hamiltonians with very different off-diagonal spin-orbit functions. As an example in Fig. 5 two sets of residuals from the fits are shown. Within the experimental uncertainty (indicated with horizontal red lines) they look reasonably similar. However the SOC function (see Fig. 6) as well as the second order functions (Eq. 4) in both cases are different. The correlations between the model functions and the non-linear nature of the fit are the reason that the agreement between the experimental and the calculated energies cannot be the only measure for the goodness of the fit. In order to probe the physical reliability of the derived empirical PECs, and especially the radial SOC function ξ (R), we performed ab initio electronic structure calculations of the adiabatic potentials for the interacting 61 + and 43 states and the corresponding SO matrix elements in the framework of the scalar-relativistic ECP-CPP-FCI method described in detail in Ref. [21]. Briefly, the
(5)
It should be noted that only the energy of the = 1 component of the triplet 43 state can be directly obtained under the scalar-relativistic approximation. The ab initio potential required for the = 0+ component, which actually interacts via the SO coupling with the 61 + state, was determined from the difference: U3ab (R ) = U3ab − Aab (R ). 0+
1
The comparison between shapes of the theoretical and fitted potentials shows a good agreement (Fig. 4) in the region defined by the experimental data (marked with red horizontal lines). It is not uncommon that the theoretical calculations may suffer from systematic R-independent error, so both theoretical PECs were shifted by 65 cm−1 in order to achieve the observed agreement for the 61 + state. Outside this region the differences are not significant. At small internuclear distances the shapes of the fitted curves deviate from the theoretical ones because at some intermediate steps the repulsive branch has been smoothed manually. It is not unlikely that in this region the same fit quality may be achieved by potentials close to the shifted theoretical ones. The theoretical diagonal spin-orbit function Aab (R) for the 43 state is also shown in Fig. 4. In Fig. 6 we compare the theoretical spin-orbit coupling function with two fitted functions. √ To do this the theoretical function is multiplied by a factor√− 2 because the theoretical spin-orbit coupling is defined as − 2ξ ab (R ). The first experimental function (black dashes) was obtained when the fit started from the IPA potential for the 61 + state and the ξ (R) function taken as a constant. The second function (solid blue) resulted when starting from the theoretical potentials and the theoretical SOC function (red). The fit quality is nearly the same (0.089 cm−1 and 0.087 cm−1 ). All three functions have similar values in the region of Rc ≈ 6.11 A˚ where the potentials of the two states cross, but one can see that at smaller and larger internuclear distances the difference between the experimental curves ξ (R) is comparable with the differences between each of them and the theoretical one. This observation demonstrates strong correlation between the fitted functions, which leads to some ambiguity in their shapes. With the present set of experimental data, mainly of 61 + character, it is impossible to fix unambiguously the shape of the off-diagonal spin-orbit coupling function in order to compare it with the theoretical calculations. On the other hand we did not manage to fit the experimental data with the ξ (R) function fixed to the theoretical prediction. This may indicate that the SO calculations need a second order correction reflecting the SO coupling with remote states, but also that within the simplified model Hamiltonian the fitted spinorbit function includes effectively additional interactions, omitted in the model. As a final result, we recommend the second fitted spin-orbit function (solid blue line in Fig. 6), because it is closer to the theoretical expectation. Additional reason to chose the second
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J. Szczepkowski, A. Grochola and P. Kowalczyk et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 239 (2019) 106650
set of model functions is that the potential energy curves are described in this case by less parameters (20 versus 30 for the 61 + state and 15 versus 22 for the 43 state). It should be also noted that the empirical SOC function can be arbitrary outside the range of internuclear distances covered by the present experimental data set (which approximately coincides with the classically allowed region of vibrational motion within the PECs). The resulting model functions can be found in the electronic supplementary material [23].
5. Conclusion We present an extension of the previous study of the 61 + state, coupled mainly to the = 0+ component of the 43 state [13]. A coupled channels treatment of the problem enabled the assignment of new experimental data and thus extended the number of experimental term energies by about 50%, up to 1655. Very often the recorded spectral lines were doubled due to strong perturbations. About 240 levels have leading or significant triplet contribution. A comparison between the single channel (Fig. 3(b)) and the coupled channels residuals (Fig. 5) convinces about the quality of the presented model. The experimental data come from a region with high density of electronic states, but nevertheless it turned out that almost all experimental observations can be reproduced within their uncertainty by a relatively simple model of two coupled channels, which includes two potential curves, one SOC function and two secondorder corrections. The structure of the experimental data and/or the nature of the problem did not allow us to fix unambiguously the shape of the SOC function ξ (R). We managed to fit the experimental data with two different functions (see Fig. 6), which differ also from the ab initio function. In spite of the apparent arbitrariness of this function it was not possible to fit the experimental data by fixing the SOC function to the theoretical one. For lower electronic states, SOC functions from similar electronic structure calculations turned out to be accurate enough to model the experimental data [9]. As already mentioned, the lower accuracy of the calculations for such high lying electronic states might be the reason. Another plausible explanation may be that the fitted SOC function is in fact an effective function (like every model function) which tries to account for all second order interactions neglected in the present two channels Hamiltonian. Since the value of the ξ (R) function is relatively small and the other electronic states are close, the fitted curve may differ significantly from the calculated one. A test for consistency of the presented model, along with its predictive power, would be a comparison of the experimental and calculated line intensities. It must be noted that the intensities of polarization labelling spectra should be handled with care, because they depend on many factors (variable intensity and absorption of the pulsed pump laser beam, intensity of the probe laser beam, frequency dependence of the detector response and so on) but at least minima in the intensity distribution of the experimental vibrational progressions should be reproduced by the model. This was done in Ref. [13] within a single channel model and therefore we tried to repeat this test using non-adiabatic vibrational wavefunctions calculated from the coupled channels model. Unfortunately, it was not possible to reproduce the minima of the experimental intensities with either of the fitted models. In fact their intensity predictions agree with each other, but not with the experiment. We would like to underline this discrepancy since so far, especially in single channels fits, the model parameters were adjusted only to match the experimental energies. Usually line intensities were calculated post factum and very rarely were they used as input data for the fit.
At present we believe that the ambiguity of the fitted model functions (and also the associated discrepancy of line intensities) is due to the composition of the experimental data (only one component of the 43 state observed) and also to the nature of the non-linear problem which involves correlated model functions. It is worth mentioning that contrary to the single channel fit, where one model function is searched (the PEC), not only more model functions are fitted by coupled channels problems, but at the same time less quantum numbers can be assigned to the experimental data (usually only J and parity, whereas most quantum numbers associated with the electronic state and vibrational level are missing). Quite often matching experimental levels with the calculated ones based just on J value, parity and the proximity of the calculated energy level may lead to misassignments, especially at the initial stages of fitting when the calculated energies are still far from the experimental ones. Therefore in the future it would be necessary to find a fitting strategy which includes additional experimental information, for example line intensities, or which approximates the shapes of the model functions by results of more accurate ab initio calculations. Whenever possible (and reasonable), second order corrections should be introduced explicitly in the model. Otherwise the fit would try to compensate their effects by other model functions and this may lead to effective functional forms which have unexpected behaviour and/or cannot be compared directly with results of ab initio calculations. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Acknowledgments AP acknowledges partial support by the Grants DN18/12/11.12.2017 and DTNS/Russia/01/10 of the Bulgarian National Science Fund. The Moscow team thanks Russian Foundation of Basic Research (RFBR) for a financial support according to the bilateral RussiaBulgaria research project N17-53-18006. The Warsaw team was partially supported by the National Science Center of Poland (Grant No. 2016/21/B/ST2/02190). Supplementary materials Supplementary materials associated with this article can be found, in the online version, at doi:10.1016/j.jqsrt.2019.106650. References [1] Quéméner G, Julienne PS. Ultracold molecules under control!. Chem Rev 2012;112(9):4949–5011. [2] Field R, Lefebvre-Brion H. The spectra and dynamics of diatomic molecules. Amsterdam: Elsevier; 2004. [3] Pazyuk E, Zaitsevskii A, Stolyarov A, Tamanis M, Ferber R. Laser synthesis of ultracold alkali metal dimers: optimization and control. Russ Chem Rev 2015;84(10):1001–20. [4] Borsalino D, Vexiau R, Aymar M, Luc-Koenig E, Dulieu O, Bouloufa-Maafa N. Prospects for the formation of ultracold polar ground state KCs molecules via an optical process. J Phys B At, Mol Opt Phys 2016;49(5):055301. [5] Gröbner M, Weinmann P, Meinert F, Lauber K, Kirilov E, Nägerl H-C. A new quantum gas apparatus for ultracold mixtures of K and Cs and KCs ground-state molecules. J Mod Opt 2016;63(18):1829–39. [6] Gröbner M, Weinmann P, Kirilov E, Nägerl H-C, Julienne PS, Le Sueur CR, et al. Observation of interspecies Feshbach resonances in an ultracold 39 K −133 Cs mixture and refinement of interaction potentials. Phys Rev A 2017;95:022715.
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The spin-orbit coupling of the 61 + and 43 states in KCs: Observation and deperturbation J. Szczepkowski a,∗, A. Grochola a, P. Kowalczyk b, W. Jastrzebski a, E.A. Pazyuk c, A.V. Stolyarov c, A. Pashov d a
Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, Warsaw 02–668, Poland Institute of Experimental Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, Warszawa 02–093, Poland Department of Chemistry, Lomonosov Moscow State University, Moscow, 119991, Leninskie gory 1/3, Russia d Faculty of Physics, Sofia University, 5 James Bourchier blvd., Sofia 1164, Bulgaria b c
a r t i c l e
i n f o
Article history: Received 2 August 2019 Revised 12 September 2019 Accepted 13 September 2019 Available online 14 September 2019 Keywords: Laser spectroscopy Alkali dimers Electronic states Potential energy curves Spin-orbit coupling Deperturbation
a b s t r a c t Energies of 1655 rovibrational levels of the mutually perturbed 61 + and 43 0+ states in KCs were measured in a series of experiments employing polarization labelling spectroscopy technique. The experimental term values of the 61 + ∼ 43 0+ complex in KCs were used in coupled-channels deperturbation analysis which took explicitly into account the spin-orbit coupling of the 61 + and 43 0+ states. The empirical deperturbation treatment was supported by ab initio electronic structure calculation of adiabatic potentials of the interacting states and of the corresponding spin-orbit coupling matrix element as a function of internuclear distance.
1. Introduction Diatomic alkali metal molecules became particularly interesting for physicists, since a new branch of so called ultracold physics has opened new possibilities of manipulating molecules, and investigating new phenomena in conditions of very low temperatures [1]. A transfer of ultracold molecules from the initial states, in which they are formed, to desired states, particularly to the absolute ground state (v = 0, J = 0 of the lowest electronic state), is usually a necessary step during such experiments. However this process is often connected with a change of the spin state, namely transfer from triplet to singlet electronic states of a chosen molecule. The task is demanding, since optical transitions between states of different symmetries are nominally forbidden. Using levels of mixed singlet-triplet character as intermediate states helps to overcome this problem. Such levels arise due to interaction of electronic states of different multiplicity. Therefore detailed information about energy structure of molecules which are involved in ultracold experiments is very important.
∗
Corresponding authors. E-mail addresses: [email protected] (J. Szczepkowski), [email protected] (A.V. Stolyarov), [email protected]fia.bg (A. Pashov). https://doi.org/10.1016/j.jqsrt.2019.106650 0022-4073/© 2019 Elsevier Ltd. All rights reserved.
© 2019 Elsevier Ltd. All rights reserved.
The spin-orbit coupling (SOC) is a phenomenon responsible for a vast majority of intramolecular perturbations observed in the excited states of alkali metal dimers. Both local and global SO perturbations [2] are especially pronounced in rovibronic spectra of alkali diatomics containing the heaviest Cs atom. The mixed singlet-triplet character of the excited states can be efficiently employed to realize the so-called ”intercombination” (spin-forbidden) transitions in the MCs (M=Li,Na,K,Rb,Cs) molecules. Indeed, in stepwise processes in these molecules, an optical transition from the ground singlet X 1 + or triplet a3 + states, to an intermediate excited state demonstrating the pronounced SOC effect, can be used to reach a manifold of molecular states of multipilicity other than of the initial state. The singlet-triplet complexes, arising in molecules from a local SOC between close-lying states of different multiplicity, are indeed often implemented in schemes for laser cooling and laser assembling of ultracold alkali diatomics in their absolute ground state [3]. One of the molecules for which the SOC is observed and may be useful in terms of manipulating of molecules in ultralow temperatures is KCs [4–6]. Spectroscopic experiments which were aimed at description of its structure have already been performed [7– 12]. One of the investigated states was 61 + , for which only limited information concerning the less perturbed energy levels was provided [13] and interactions with neighbouring states were not
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sufficiently explored because of lack of suitable theoretical model. In this paper a complex approach to solve this problem is described. It is supported by new calculations and new experimental data concerning the 61 + state, followed by a deperturbation procedure, which takes into account the direct spin-orbit interaction between the 61 + and 43 states. The empirically derived potential energy curve and SOC function describes the experimental data within their uncertainties. The analysis confirms high quality of the ab initio calculations and suggests that the accuracy of the theoretical SOC function is comparable or even better than that of the empirical one. 2. Experimental details We applied the V-type optical-optical double resonance polarization spectroscopy technique. The experimental setup was almost the same as in our previous report [13]. Two lasers were used simultaneously in our experiment. A single mode ring dye laser (Coherent 899-21, pumped by a DPSS Sprout-G laser and actively stabilized to HighFinesse WS-7 wavemeter) delivered the probe beam whereas a high power pulsed dye laser (Lumonics HD500 pumped by LightMachinery IPEX-848 XeCl excimer laser) served as a source of a stronger pump beam. The laser beams were superimposed in a heat-pipe oven placed between two crossed polarizers forming a polarizer-analyzer system for the probe beam. The oven contained KCs vapour typically at a temperature of 400 ◦ C with an addition of about 4 Torr of argon buffer gas. The frequency of the probe beam was fixed on well known (v , J ) ← (v , J ) transitions in the 41 + ← X1 + band system of KCs [14,15], thus labelling the involved (v , J ) levels in the ground state of the molecule. The frequency of the circularly polarized pump beam was scanned over the investigated 61 + ← X1 + transition (see Fig. 1). Whenever it was resonant with a transition originating from the labelled X1 + (v , J ) level, the originally linear polarization of the probe beam became changed. In such cases the probe light was partially passing through the crossed analyzer and detected by a photomultiplier coupled to the boxcar averager (Stanford SR250) and a personal computer. Absolute calibration of the pump laser frequency was achieved by simultaneous recording of optogalvanic spectra of argon and neon and linearity of laser scan was additionally controlled using transmission fringes of a Fabry-Pérot interferometer 0.5 cm long. The precision of the laser wavenumber determination was better than σ = 0.1 cm−1 . 3. Assignment of the spectra The first experimental characterization of the 61 + state [13] led to more than 10 0 0 term energies. They were placed in the region where the SOC to the neighbouring triplet states made the unambiguous assignment possible. The experimental data were described within a single channel (adiabatic) model with deviations between measured and observed energies reaching 10 cm−1 , an accuracy well outside the experimental precision. Moreover, the lowest assigned vibrational level of the 61 + state in [13] was v = 6. Although some transitions to lower vibrational levels were recorded, their analysis was impossible without taking into account interactions with the nearby triplet states (see Fig. 1). Therefore within the present study an additional set of measurements was carried out to supplement the existing data on the 61 + state. The efforts were focused on the energy range 20 0 0 0 − 20700 cm−1 , where the lowest vibrational levels (v < 6) were expected. In some cases two spectral lines were observed instead of a single line, one of them shifted to higher energy than expected and the other to lower energy. In Fig. 2 one can follow two vibrational progressions to the 61 + ∼ 43 complex from lev-
Fig. 1. The ab initio potential energy curves [16] for 1 + (solid black line), 3 + (dotted olive line), 1 (dashed-dotted red line) and 3 (dashed blue line) states in KCs molecule. The energy range investigated in the present work is marked by horizontal red dashed lines. The arrows present schematically transitions induced by the pump and probe lasers.
els labelled in the ground state: (v = 4, J = 110) and (v = 5, J = 54). An irregular vibrational spacing and appearance of extra spectral lines are visible. At the same time an intricate pattern of intensities of the lines was observed: P and R lines often had different intensity, for example in Fig. 2 for transitions from (v = 5, J = 54) the R lines are completely missing. Those symptoms are characteristic for a situation where rovibrational levels of the observed electronic state interact locally with levels of other electronic states. As it was reported in [13] and observed during present measurements, the largest shifts of positions of the rovibrational levels were found in regions where the energy levels of the 61 + state are close to levels of the 43 state, which enhances the effect of interaction between these two states. During the final analysis an iterative deperturbation approach was employed, a detailed procedure of which is described in the following section. At each consecutive step of deperturbation the two channels model was used to predict positions of spectral lines and then the spectra were reanalyzed to identify new lines or, if necessary, to reassign these for which the previous assignment was incorrect. The procedure was repeated until all lines were properly assigned and the final model was built. Eventually more than 600 new term energies were added to the data field forming a total of 1655 term energies (see Fig. 3(a)) based on 2551 spectral lines only of the main isotopologue 39 K133 Cs. In Fig. 3(b) one can see a comparison between the experimental energy levels and their positions cal-
J. Szczepkowski, A. Grochola and P. Kowalczyk et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 239 (2019) 106650
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Fig. 2. A portion of the experimental spectrum of KCs corresponding to transitions from two levels labelled in the ground state, (v = 4, J = 110) and (v = 5, J = 54), to the 61 + ∼ 43 complex. The lines above the spectrum indicate transitions to consecutive vibrational levels belonging to the P or R branch.
Fig. 3. (a) Reduced energies Ered = E − 0.015 × J (J + 1 ) [cm−1 ] of rovibrational levels taken into account in the deperturbation procedure. Red dots indicate levels observed in the present experiment, while black stars correspond to the levels assigned before [13]. (b) Comparison between the experimental energy levels and their positions calculated with potential curves obtained in the single channel model. Observed deviations exceed significantly the estimated experimental uncertainty of 0.1 cm−1 .
culated with the potential curves obtained in the single channel model. In the wavenumber range up to 21200 cm−1 the observed deviations show a structure which suggests a complete mixing of the original Hund’s case (a) states, since two separate groups of levels can be seen which can be attributed to two Hund’s case (c) electronic states. Beyond 21200 cm−1 the mixing becomes more local and typical patterns of avoided crossings appear.
H = T + U (R ) + H ROT , where
T =− H ROT =
4. The coupled-channels deperturbation analysis The 61 + state is spin-orbit coupled directly only to the = component of the triplet 43 state. The spin-rotational interaction between the different components of the 43 state and its interactions with the two neighbouring 53 + and 63 + states were ignored when the initial Hamiltonian of the problem was set up. The diagonal elements H (corresponding to a single-channel model of the decoupled, i.e. isolated states) can be written as [2]: 0+
(1)
h ¯ 2 d2 ; 2μ dR2
h ¯2 (J (J + 1 ) − 2 + S(S + 1 ) − 2 − 2 ), 2 μR 2
(2)
and U (R ) is the potential energy of a given state. Since only the = 0+ component of the 3 state is considered, the relevant spin-orbit splitting function A(R) (arising from the diagonal part of the SO operator) is implicitly involved in the potential curve for the 43 0+ component: U3 + (R ) = U3 (R ) − A(R ). The 0
1
off-diagonal part of the spin-orbit term is represented by a single radial function:
HSO = ξ (R ).
(3)
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J. Szczepkowski, A. Grochola and P. Kowalczyk et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 239 (2019) 106650 Table 1 Scheme of the Hamiltonian matrix for the present problem.
61 + 4 3 0
61 +
4 3 0
T + H ROT + U (R ) + Hq,
ξ (R) T + H ROT + U (R ) + Hq,
ξ (R)
indirectly through their SOC with the 43 0+ state. We tried to decrease the systematic deviations by extending the model to three coupled channels, including the = 1 component of the 53 + state, but this did not seem to cause even qualitatively the needed J dependent shifts of the energy levels. Moreover, by adding a new electronic state we observed that the model became too overdetermined. Therefore, given the present data set, we decided to extend the model with J-dependent second order corrections reflecting rotational interactions with distant electronic states of the phenomenological form [18]:
Hq, =
Fig. 4. Comparison between the theoretical and fitted potentials (left axis). Both theoretical curves have been shifted up by 65 cm−1 in order to match closely the fitted PECs. The theoretical diagonal SO radial function is also shown (right axis).
The eigenvalues of the model Hamiltonian were calculated by the collocation (Fourier Grid Hamiltonian) method [17,18]. The grid ˚ With a mapping function, 150 grid points spans from 2.6 A˚ to 10 A. were sufficient to reach the accuracy better than 0.001 cm−1 . The initial fits of the experimental data with U (R ), U (R ) and ξ (R) being adjusted showed that the present model is adequate, leading to residuals which dimensionless standard deviation
σ¯ =
exp N 1 (Ei − Eicalc )2 N σi2 i=1
is about 1.5. The value slightly larger than 1 is mainly due to systematic deviations for larger J’s, reaching 0.5 cm−1 for J = 136. The only J-dependent operators which are missing in the model are the L-uncoupling operator which couples the 43 0 state with the 3 1+ states and the S-uncoupling operator which mixes the components of the 43 state since the spin-orbit splitting function A(R) is relatively small (as the present theoretical calculations show, see Fig. 4). These interactions influence the 61 + levels only
h2 ¯ 2 J ( J + 1 )q ( R ), 2 μR 2
(4)
and this corrections were added to the diagonal part of the Hamiltonian (1), thus introducing two more fitting functions to the model, one for the 1 + state, q (R), and one for the 3 state, q (R ). A simplified scheme of the Hamiltonian matrix is given in Table 1. With the second order corrections it was possible to reduce the dimensionless standard deviation to below 1. The effect of the second order corrections can be estimated as q = E/(J (J + 1 )), where E is the energy shift which they cause. This quantity is v and J dependent, but on average it varies between 5 × 10−5 and 2 × 10−4 cm−1 both for the 61 + and the 43 states. It is difficult to point which interactions are accounted for through these second order functions. They should be treated rather as effective phenomenological corrections, which include several intramolecular interactions in an implicit form. A total of 1635 energy levels of the coupled singlet-triplet system are reproduced by this model with an rms of 0.089 cm−1 , which is within the expected experimental uncertainty (see Fig. 5). About 20 experimental levels, which show systematic deviations above 0.5 cm−1 were excluded from the fit, but for such highly condensed structure of electronic states in a heavy molecule like KCs it is not unusual that other local perturbation may cause deviations from the present model. It may be noted that 80% of the residuals lie within ± 0.1 cm−1 , i.e. within one standard deviation. This is more than expected for a normal distribution ( ≈ 68%)
Fig. 5. Residuals from two fits of the model applying the same Hamiltonian from Table 1 but two different sets of model functions. PECs in case (b) have less parameters than in case (a) (see the text for details). The shapes of the fitted SOC are shown in Fig. 6.
J. Szczepkowski, A. Grochola and P. Kowalczyk et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 239 (2019) 106650
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ECP-CPP-FCI approach is based on the full configuration interaction (FCI) treatment of two-valence-electron problem defined by the large-core two-component relativistic pseudopotentials (ECPs) and the core-valence correlation treatment through semi-empirical core-polarization potentials (CPPs) for both K and Cs atoms. The SO coupling matrix element between the 61 + and 43 states, ξ ab (R), and equidistant SO splitting of the triplet 43 state, Aab (R), were estimated as first-order SO interactions of scalar-relativistic states obtained during the ECP-CPP-FCI calculation. To take into account implicitly a systematic R-dependent error in the calculated energy [22] the potential energy curves (PECs) for the excited states were constructed by adding the vertical excitation energies UiXF CI = UiF CI − UXF CI calculated as functions of the internuclear distance to the highly accurate empirical ground X 1 + state potential emp UX from Ref. [7]: F CI Uiab (R ) = UiX (R ) + UXemp (R )
Fig. 6. Comparison between √ the theoretical SOC function ξ (R) (red line with circles, multiplied by a factor − 2) with two fitted curves: black, dashed line - from the first fit, blue, solid line - from the second fit. The theoretical curve was used as the initial guess for the second fit.
which points out that the experimental error has been assumed rather conservatively by us. All model functions are defined in a pointwise form and natural cubic spline functions were used for interpolation. This representation is very flexible and allows functions with various shapes to be modelled with relatively few parameters. However, the flexibility of the pointwise functions is also their disadvantage, as it may lead to unphysical oscillations. In the present version of the fitting code the minimization routine was based on the Singular Value Decomposition (SVD) technique [19], which orders the parameters (or their linear combinations) according to their influence on the positions of the calculated energy levels. During the fit, one can select and adjust only parameters which contribute significantly to the reduction of the standard deviation. The rest of the parameters are kept fixed to their initial values instead of giving them some arbitrary value. Such approach was successfully applied in single channel fits [20] and it was very useful in case of coupled channels [18] as well. However the increased number of fitted functions by the coupled channels treatment and strong correlation between some of the interactions make the shapes of the fitted functions somewhat ambiguous. While the potential energy curves may be kept smooth and close to our expectations (based on ab initio calculations), the shape of the other radial functions is usually unknown. Nearly identical sets of energy levels (within their uncertainty) may be calculated from Hamiltonians with very different off-diagonal spin-orbit functions. As an example in Fig. 5 two sets of residuals from the fits are shown. Within the experimental uncertainty (indicated with horizontal red lines) they look reasonably similar. However the SOC function (see Fig. 6) as well as the second order functions (Eq. 4) in both cases are different. The correlations between the model functions and the non-linear nature of the fit are the reason that the agreement between the experimental and the calculated energies cannot be the only measure for the goodness of the fit. In order to probe the physical reliability of the derived empirical PECs, and especially the radial SOC function ξ (R), we performed ab initio electronic structure calculations of the adiabatic potentials for the interacting 61 + and 43 states and the corresponding SO matrix elements in the framework of the scalar-relativistic ECP-CPP-FCI method described in detail in Ref. [21]. Briefly, the
(5)
It should be noted that only the energy of the = 1 component of the triplet 43 state can be directly obtained under the scalar-relativistic approximation. The ab initio potential required for the = 0+ component, which actually interacts via the SO coupling with the 61 + state, was determined from the difference: U3ab (R ) = U3ab − Aab (R ). 0+
1
The comparison between shapes of the theoretical and fitted potentials shows a good agreement (Fig. 4) in the region defined by the experimental data (marked with red horizontal lines). It is not uncommon that the theoretical calculations may suffer from systematic R-independent error, so both theoretical PECs were shifted by 65 cm−1 in order to achieve the observed agreement for the 61 + state. Outside this region the differences are not significant. At small internuclear distances the shapes of the fitted curves deviate from the theoretical ones because at some intermediate steps the repulsive branch has been smoothed manually. It is not unlikely that in this region the same fit quality may be achieved by potentials close to the shifted theoretical ones. The theoretical diagonal spin-orbit function Aab (R) for the 43 state is also shown in Fig. 4. In Fig. 6 we compare the theoretical spin-orbit coupling function with two fitted functions. √ To do this the theoretical function is multiplied by a factor√− 2 because the theoretical spin-orbit coupling is defined as − 2ξ ab (R ). The first experimental function (black dashes) was obtained when the fit started from the IPA potential for the 61 + state and the ξ (R) function taken as a constant. The second function (solid blue) resulted when starting from the theoretical potentials and the theoretical SOC function (red). The fit quality is nearly the same (0.089 cm−1 and 0.087 cm−1 ). All three functions have similar values in the region of Rc ≈ 6.11 A˚ where the potentials of the two states cross, but one can see that at smaller and larger internuclear distances the difference between the experimental curves ξ (R) is comparable with the differences between each of them and the theoretical one. This observation demonstrates strong correlation between the fitted functions, which leads to some ambiguity in their shapes. With the present set of experimental data, mainly of 61 + character, it is impossible to fix unambiguously the shape of the off-diagonal spin-orbit coupling function in order to compare it with the theoretical calculations. On the other hand we did not manage to fit the experimental data with the ξ (R) function fixed to the theoretical prediction. This may indicate that the SO calculations need a second order correction reflecting the SO coupling with remote states, but also that within the simplified model Hamiltonian the fitted spinorbit function includes effectively additional interactions, omitted in the model. As a final result, we recommend the second fitted spin-orbit function (solid blue line in Fig. 6), because it is closer to the theoretical expectation. Additional reason to chose the second
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J. Szczepkowski, A. Grochola and P. Kowalczyk et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 239 (2019) 106650
set of model functions is that the potential energy curves are described in this case by less parameters (20 versus 30 for the 61 + state and 15 versus 22 for the 43 state). It should be also noted that the empirical SOC function can be arbitrary outside the range of internuclear distances covered by the present experimental data set (which approximately coincides with the classically allowed region of vibrational motion within the PECs). The resulting model functions can be found in the electronic supplementary material [23].
5. Conclusion We present an extension of the previous study of the 61 + state, coupled mainly to the = 0+ component of the 43 state [13]. A coupled channels treatment of the problem enabled the assignment of new experimental data and thus extended the number of experimental term energies by about 50%, up to 1655. Very often the recorded spectral lines were doubled due to strong perturbations. About 240 levels have leading or significant triplet contribution. A comparison between the single channel (Fig. 3(b)) and the coupled channels residuals (Fig. 5) convinces about the quality of the presented model. The experimental data come from a region with high density of electronic states, but nevertheless it turned out that almost all experimental observations can be reproduced within their uncertainty by a relatively simple model of two coupled channels, which includes two potential curves, one SOC function and two secondorder corrections. The structure of the experimental data and/or the nature of the problem did not allow us to fix unambiguously the shape of the SOC function ξ (R). We managed to fit the experimental data with two different functions (see Fig. 6), which differ also from the ab initio function. In spite of the apparent arbitrariness of this function it was not possible to fit the experimental data by fixing the SOC function to the theoretical one. For lower electronic states, SOC functions from similar electronic structure calculations turned out to be accurate enough to model the experimental data [9]. As already mentioned, the lower accuracy of the calculations for such high lying electronic states might be the reason. Another plausible explanation may be that the fitted SOC function is in fact an effective function (like every model function) which tries to account for all second order interactions neglected in the present two channels Hamiltonian. Since the value of the ξ (R) function is relatively small and the other electronic states are close, the fitted curve may differ significantly from the calculated one. A test for consistency of the presented model, along with its predictive power, would be a comparison of the experimental and calculated line intensities. It must be noted that the intensities of polarization labelling spectra should be handled with care, because they depend on many factors (variable intensity and absorption of the pulsed pump laser beam, intensity of the probe laser beam, frequency dependence of the detector response and so on) but at least minima in the intensity distribution of the experimental vibrational progressions should be reproduced by the model. This was done in Ref. [13] within a single channel model and therefore we tried to repeat this test using non-adiabatic vibrational wavefunctions calculated from the coupled channels model. Unfortunately, it was not possible to reproduce the minima of the experimental intensities with either of the fitted models. In fact their intensity predictions agree with each other, but not with the experiment. We would like to underline this discrepancy since so far, especially in single channels fits, the model parameters were adjusted only to match the experimental energies. Usually line intensities were calculated post factum and very rarely were they used as input data for the fit.
At present we believe that the ambiguity of the fitted model functions (and also the associated discrepancy of line intensities) is due to the composition of the experimental data (only one component of the 43 state observed) and also to the nature of the non-linear problem which involves correlated model functions. It is worth mentioning that contrary to the single channel fit, where one model function is searched (the PEC), not only more model functions are fitted by coupled channels problems, but at the same time less quantum numbers can be assigned to the experimental data (usually only J and parity, whereas most quantum numbers associated with the electronic state and vibrational level are missing). Quite often matching experimental levels with the calculated ones based just on J value, parity and the proximity of the calculated energy level may lead to misassignments, especially at the initial stages of fitting when the calculated energies are still far from the experimental ones. Therefore in the future it would be necessary to find a fitting strategy which includes additional experimental information, for example line intensities, or which approximates the shapes of the model functions by results of more accurate ab initio calculations. Whenever possible (and reasonable), second order corrections should be introduced explicitly in the model. Otherwise the fit would try to compensate their effects by other model functions and this may lead to effective functional forms which have unexpected behaviour and/or cannot be compared directly with results of ab initio calculations. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Acknowledgments AP acknowledges partial support by the Grants DN18/12/11.12.2017 and DTNS/Russia/01/10 of the Bulgarian National Science Fund. The Moscow team thanks Russian Foundation of Basic Research (RFBR) for a financial support according to the bilateral RussiaBulgaria research project N17-53-18006. The Warsaw team was partially supported by the National Science Center of Poland (Grant No. 2016/21/B/ST2/02190). Supplementary materials Supplementary materials associated with this article can be found, in the online version, at doi:10.1016/j.jqsrt.2019.106650. References [1] Quéméner G, Julienne PS. Ultracold molecules under control!. Chem Rev 2012;112(9):4949–5011. [2] Field R, Lefebvre-Brion H. The spectra and dynamics of diatomic molecules. Amsterdam: Elsevier; 2004. [3] Pazyuk E, Zaitsevskii A, Stolyarov A, Tamanis M, Ferber R. Laser synthesis of ultracold alkali metal dimers: optimization and control. Russ Chem Rev 2015;84(10):1001–20. [4] Borsalino D, Vexiau R, Aymar M, Luc-Koenig E, Dulieu O, Bouloufa-Maafa N. Prospects for the formation of ultracold polar ground state KCs molecules via an optical process. J Phys B At, Mol Opt Phys 2016;49(5):055301. [5] Gröbner M, Weinmann P, Meinert F, Lauber K, Kirilov E, Nägerl H-C. A new quantum gas apparatus for ultracold mixtures of K and Cs and KCs ground-state molecules. J Mod Opt 2016;63(18):1829–39. [6] Gröbner M, Weinmann P, Kirilov E, Nägerl H-C, Julienne PS, Le Sueur CR, et al. Observation of interspecies Feshbach resonances in an ultracold 39 K −133 Cs mixture and refinement of interaction potentials. Phys Rev A 2017;95:022715.
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