Precision Measurements in Few-Electron Molecules: The Ionization Energy of Metastable 4 He 2 and the First Rotational Interval of 4 He 2 +
Luca Semeria, Paul Jansen, Gian-Marco Camenisch, Federico Mellini, Hansjürg Schmutz, Frédéric Merkt
PPrecision measurements in few-electron molecules:The ionization energy of metastable He and the first rotational interval of He + Luca Semeria, Paul Jansen, ∗ Gian-Marco Camenisch, Federico Mellini, Hansj¨urg Schmutz, and Fr´ed´eric Merkt
Laboratory of Physical Chemistry, ETH Zurich, CH-8093 Zurich, Switzerland (Dated: June 5, 2020)Molecular helium represents a benchmark system for testing ab initio calculations on few-electronmolecules. We report on the determination of the adiabatic ionization energy of the a Σ + u stateof He , corresponding to the energy interval between the a Σ + u ( v (cid:48)(cid:48) = 0, N (cid:48)(cid:48) = 1) state of He and the X + 2 Σ + u ( v + = 0, N + = 1) state of He , and of the lowest rotational interval of He .These measurements rely on the excitation of metastable He molecules to high Rydberg statesusing frequency-comb-calibrated continuous-wave UV radiation in a counter-propagating-laser-beamsetup. The observed Rydberg states were extrapolated to their series limit using multichannelquantum-defect theory. The ionization energy of He ( a Σ + u ) and the lowest rotational intervalof He ( X + 2 Σ + u ) are 34301.207002(23) ± . sys cm − and 70.937589(23) ± . sys cm − ,respectively. The comparison of the results of precision spectro-scopic measurements in few-electron atoms and moleculeswith the results of ab initio calculations challenges the-oreticians and experimentalists and forces them to con-stantly improve their respective methodologies. In recentyears, this symbiotic relation between theory and exper-iment has opened a route to test the Standard Model ofparticle physics and extensions thereof [1–4] and to de-termine the values of fundamental constants [5–9]. The2019 revision of the International System of Units (SI)fixed the Boltzmann ( k B ) and Avogadro ( N A ) constants[10], enabling a primary pressure standard that is directlytraceable via the SI to the density of a gas. The numberdensity of a gas can be determined via a measurementof the refractive index or the dielectric constant. Bothmethods rely on the precise value of the polarizability ofthe gas, which is known with sufficient accuracy only foratomic helium from recent ab initio calculations [11, 12].An accurate determination of the level energies of He might provide a way to test the polarizability calculationsof atomic helium via the long-range part of the molecularpotential V ( R ) = − α R + α W R + O ( R − ) for R → ∞ , (1)where α = 1 . a is the calculated static po-larizability of He (1 S ) [11], α is the fine-structure con-stant and α W R − is the relativistic correction from theBreit interaction [13]. Terms in R − , which can be cal-culated accurately, ought to be included in a fit based onEq. (1). Equation (1) becomes a good approximation ofthe internuclear potential for R > R LR ≈ a , with R LR being the Le Roy radius [14], implying that the highestvibrational levels of He are the most sensitive to thevalue of α . Model calculations indicate that an uncer-tainty of 1 . × − a might shift the absolute positionsof these levels by up to 0.2 MHz.Transition frequencies between low-lying rotationallevels of the vibronic ground state of He He + were de- termined with a precision of 18 MHz [15] but are notsensitive to α . The frequencies of transitions betweenhighly vibrationally excited bound levels of the X + 2 Σ + u and weakly bound A + 2 Σ + g levels of He have been mea-sured with a 1 MHz precision [16] but they are not sen-sitive to α either because the effects of the long-rangeinteractions largely cancel out in the energy differences.Experimental data on a broader range of levels of He are thus required.We present the results of a determination of the adia-batic ionization energy of metastable He in its a Σ + u state (He ∗ hereafter) and the first rotational intervalof He in its vibronic X + 2 Σ + u ( v + = 0) ground state(He hereafter) at a relative accuracy of 10 − and8 . × − , respectively. These results improve our pre-vious results on this system by more than an order ofmagnitude [17, 18] and approach the level of accuracy re-quired by the metrological application described above.They also serve as benchmark data for ab initio calcula-tions of three- and four-electron systems [19, 20].Our approach consists of measuring Rydberg series ofHe from the metastable a Σ + u state of He and extrapo-lating them to the series limits. The two lowest rotationallevels of He ∗ and the n p Rydberg series converging on thecorresponding rotational levels of He are schematicallydepicted in Fig. 1. Only odd rotational levels N (cid:48)(cid:48) and N + are allowed in He ∗ and He , respectively, because He + is a boson. We use double-primed symbols, un-primed symbols, and a superscript “+” to designate thequantum numbers of He ∗ , He Rydberg states, and He ,respectively. The level structure of He ∗ is adequately de-scribed using Hund’s angular-momentum coupling case(b), i.e., the total angular momentum excluding spin (cid:126)N (cid:48)(cid:48) couples to the total electron spin (cid:126)S (cid:48)(cid:48) to form the totalangular momentum (cid:126)J (cid:48)(cid:48) . The splittings between the re-sulting fine-structure components with J (cid:48)(cid:48) = N (cid:48)(cid:48) , N (cid:48)(cid:48) ± is also bestdescribed using Hund’s angular-momentum coupling case a r X i v : . [ phy s i c s . a t o m - ph ] J un (b). The rotational levels of He are split into two spin-rotation components with J + = N + ± and the split-tings for rotational states with N + ≤
27 were recentlymeasured, with an accuracy of 150 kHz for N + = 1 [23].Rydberg states of He with principal quantum num-bers n (cid:46)
200 are well described in Hund’s case (d) cou-pling. The rotation of the ionic core excluding spin (cid:126)N + couples to (cid:126)(cid:96) to give (cid:126)N , which couples to (cid:126)S to give thetotal angular momentum of the Rydberg state (cid:126)J . Thetriplet ( S = 1) n p Rydberg states of He are split in ninefine-structure components [23]. For n (cid:38) ∗ to n p Rydbergstates are governed by the selection rule ∆ J = J − J (cid:48)(cid:48) = N − N (cid:48)(cid:48) = ∆ N [23]. The observed Rydberg-transitionwavenumbers are extrapolated to their series limit usingmultichannel quantum-defect theory (MQDT) [24], yield-ing the adiabatic ionization energy of He ∗ and the firstrotational interval of He . The Rydberg transitions ob-served in our spectra connect the Hund’s case (b) levelsof He ∗ to the Hund’s case (d) Rydberg levels and canbe labeled as N (cid:48)(cid:48) J (cid:48)(cid:48) → n p N + N,J . Here we also report fine-structure-free values for the transition wavenumbers, inwhich case the subscript labels J (cid:48)(cid:48) and J are omitted. ν ~ ν ~~ ν ~ Figure 1. Energy-level diagram (not to scale) showing thelowest rovibrational states of He ∗ and n p Rydberg series con-verging on the corresponding rovibrational states of He .Relevant allowed transitions to n p Rydberg states are indi-cated by full arrows. Autoionizing levels are marked in blue(see text for details). The measurements were carried out using the exper-imental apparatus described in Ref. [23], specificallyadapted to (i) record first-order-Doppler-corrected single- photon transitions [25–27] and (ii) calibrate the transi-tion frequencies using a frequency comb [28].Pulsed supersonic beams of He ∗ molecules are pro-duced in a discharge of pure helium [29]. Beam velocitiesof about 1000 m/s and 500 m/s are obtained by coolingthe valve body to 77 K and 10 K, respectively [30]. A 3-mm skimmer selects the most intense part of the molec-ular beam, which intersects the laser radiation at nearright angles after about 1 m of free flight. Transitionsto Rydberg states with n (cid:38)
30 are detected by applyinga pulsed electric field of 750 V/cm across a cylindricallysymmetric stack of electrodes. The pulsed ionization fieldalso serves to accelerate the produced He ions towardan off-axis microchannel-plate (MCP) detector. A spec-trum is recorded by monitoring the ion yield as a functionof the laser frequency.The narrow-bandwidth ( < ∗ to n p Rydberg states is pro-duced by frequency doubling the output of a cw ring dyelaser tunable around 580 nm with a beta-barium borate(BBO) crystal mounted in a feedback-stabilized externalcavity. The Doppler-corrected transition frequencies aremeasured by retroreflecting the laser beam after it passesthe interaction zone, thereby generating two Doppler-shifted components in the spectrum. Their geometricmean corresponds to the first-order-Doppler-free transi-tion frequency in case of a perfect overlap between thetwo counter-propagating beams. Care was taken to mini-mize the waist of the laser beam, by means of a telescope,at the position corresponding to the surface of the back-reflecting mirror, which was used to overlap the forward-and backward-propagating beams over a path length ofabout 10 m. The systematic uncertainty associated withthe difficulty of achieving a perfect overlap of the twobeams (see Ref. [25] for details) effectively transformedinto a statistical uncertainty of 460 kHz when repeatingthe measurements multiple times after complete realign-ment of the laser beams.Absolute calibration of the fundamental laser fre-quency relies on the use of an optical frequency comb(Menlo Systems FC1500-250-ULN), referenced to a GPS-disciplined Rb oscillator. The Rb oscillator (StanfordResearch Systems FS725) has a stability of 2 × − over the time (1 s) required to record one data pointand the GPS receiver (Spectrum Instruments TM-4) hasa specified long-term stability of 10 − [28]. The beatsignal measured with the photodiode is filtered by thefrequency-comb electronics and is compared with a pas-sive frequency discriminator inside a home-built elec-tronic unit to generate an error signal, as describedin Ref. [31]. This signal is used to control the refer-ence cavity of the laser [31] and to ensure, by meansof a proportional-integral-derivative (PID) feedback-controlled loop, that the beat frequency is locked at60 MHz [32]. The frequency of the laser follows from f L = n c f r ± f CEO ± f beat , where f CEO and f r arethe carrier-envelope-offset and repetition frequency of thefrequency comb, respectively, and n c is the mode num-ber that generates the beat signal, as determined usinga wavemeter.A small dc electric potential is applied to the stack tocompensate the stray electric field along the symmetryaxis. Four pin electrodes are employed to compensatethe stray field in the plane perpendicular to the sym-metry axis, resulting in an overall field compensation ofbetter than 1 mV/cm [31], corresponding to a measuredfrequency shift of less than 200 kHz for Rydberg states of n ≈ Figure 2. Upper panel: Experimental data (black dots)and weighted-fit model (red curve) of transitions to Rydbergstates converging on the v + = 0, N + = 1 , .The assignment is indicated by the grey bars, on which thesmall circle denotes the centre of gravity of the transitionsof each triplet. Lower panel: Weighted residuals. The lightand dark-shaded areas correspond to one and two standarddeviations, respectively. Figure 2 shows a Rydberg spectrum in the region ofthe overlapping 1 → and 3 → , transi-tions. The spectrum was recorded at a valve tempera-ture of 77 K, resulting in a Doppler-limited linewidth of21 MHz (full width at half maximum). Each transitionappears as a pair of Doppler components generated bythe counter-propagating laser beams. The Doppler pairscan be grouped in triplets reflecting primarily the fine-structure splittings in the metastable state [23], and canbe determined separately using the ∆ J = ∆ N selectionrule (see Fig. 1 and Ref. [23]). The lineshape model usedto extract the fine-structure-free transition wavenumbersis the sum of three Gaussian pairs. The frequency spac-ing between the pair of peaks corresponds to twice thefirst-order Doppler shift and is represented by a single parameter in the fit. The width of all Gaussians is alsoaccounted for by a single parameter in the fit. The re-maining fit parameters are a constant background offset,the separations between the first-order Doppler-correctedtransition frequencies, and the individual peak intensi-ties. The Doppler-corrected transition frequencies of eachtriplet are converted into their fine-structure-free center-of-gravity (c.o.g.) positions using the 2 J (cid:48)(cid:48) + 1 statisticalweight of each fine-structure transition and are listed inTable I. The reported fit values were obtained in a two-step approach to account for the Poissonian statistics ofthe ion detection [33]. An initial unweighted fit was usedto estimate the Poissonian variance that, along with thevariance of the intrinsic noise, was used to determine thestatistical weights used in the fits. Table I. Observed transitions from the a Σ + u ( ν (cid:48)(cid:48) = 0 , N (cid:48)(cid:48) =1 ,
3) state of He to the n p N + N Rydberg states belonging toseries converging to the X + 2 Σ + u ( ν + = 0 , N + = 1 ,
3) statesof He and comparison with the results of MQDT calcula-tions. The symbol ∆ = (obs. − calc.) represents the differencebetween observed and calculated line positions. All values aregiven in cm − . → n p1 → n p1 n ˜ ν obs ( σ ˜ ν obs ) ∆ × ˜ ν obs ( σ ˜ ν obs ) ∆ ×
70 34 278 .
903 916(6) − .
662 500(8) 34 34 281 .
687 568(10) 2478 34 283 .
138 307(9) 1381 34 284 .
523 515(7) 783 34 285 .
298 807(20) 1085 34 285 .
994 074(15) 10 34 286 .
025 503(10) 2289 34 287 .
331 507(9) −
32 34 287 .
322 442(6) 3991 34 288 .
093 742(8) − .
769 663(9) − .
828 712(8) −
197 34 289 .
563 480(23) 103 → n p3 → n p3 n ˜ ν obs ( σ ˜ ν obs ) ∆ × ˜ ν obs ( σ ˜ ν obs ) ∆ ×
103 34 285 .
974 192(8) − .
024 156(9) −
11 34 288 .
048 567(12) 14117 34 288 .
305 868(8) −
23 34 288 .
327 945(10) 29121 34 288 .
827 917(12) −
32 34 288 .
846 249(10) 48
Fig. 3 illustrates the 1 → transition recordedat valve temperatures of 77 K (red) and 10 K (blue).The two spectra clearly demonstrate the reduction ofthe Doppler width and Doppler shift associated withthe reduction of the molecular-beam velocities by afactor of about two. The geometrical centers of thetwo doublets are 34284.518644(23) cm − (77 K) and34284.518657(8) cm − (10 K), which corresponds to a dif-ference of ∼
400 kHz, in agreement with the combined ex-perimental uncertainty of ∼
700 kHz of these values. Theslight asymmetry in the lines that is observed in the spec-trum recorded at 77 K is a consequence of the slightlyimperfect alignment of the laser and molecular beams.This asymmetry can be reduced by lowering the veloc-ity of the molecular beam and/or improving the laserwavefront, but it does not affect the Doppler-correctedline positions because the two lines of the Doppler pair
Figure 3. Comparison of the fitted 1 → , Rydbergspectra recorded at 77 K (red trace) and 10 K (blue trace),resulting in a FWHM of 21 MHz and 11 MHz, respectively.The spectra are normalized to their respective maximum in-tensity, which is reduced by a factor of four when coolingthe valve to 10 K. The insert shows the determined Doppler-corrected transition frequencies on a magnified scale. See textfor details. display an opposite asymmetry.The extrapolation to the series limit relies on MQDT asdeveloped and implemented by Ch. Jungen [24, 34, 35].The MQDT parameters of triplet n p and n f Rydbergstates determined by Sprecher et al. [36] were em-ployed without change. In order to ensure the robustextrapolation of ionization energies, we adopt the ap-proach of Beyer et al. [37]. The experimental dataset used for the extrapolation is chosen so as to coveran energy range at least as large as the energy inter-val to be extrapolated. The extrapolated ionization en-ergy of the N (cid:48)(cid:48) = 1 rotational level of He ∗ amounts to˜ ν = 34301 . − . The comparison betweenthe centers of gravity of the observed transitions andcalculated values resulting from the MQDT fit are alsoshown in Table I. The root mean square (RMS) of thediscrepancies amounts to 740 kHz and the residuals ap-pear to be normally distributed.The data also allow the determination of the first ro-tational interval of He , provided the rotational energycorresponding to the N (cid:48)(cid:48) = 1 → N (cid:48)(cid:48) = 3 transition inHe ∗ is known with sufficient accuracy. We determinedthis value to be ˜ ν (cid:48)(cid:48) = 75 . − by taking thedifference of the 1 → [34292.259569(17) cm − ]and 3 → [34216.446617(10) cm − ] transitionwavenumbers. We find that the interval betweenthe first two rotational states of He is ˜ ν +13 =70 . − .The estimated statistical and systematic uncertainties Table II. Systematic and statistical (1 σ , in kHz) contribu-tions to the uncertainty in the determination of the ionizationpotential ˜ ν of He ∗ (third column) and in the lowest rota-tional interval ˜ ν +13 of He (second column). All systematicuncertainties with the exception of the MQDT-extrapolationuncertainty cancel out in the determination of the rotationalinterval.Uncertainty (kHz) ˜ ν +13 ˜ ν Systematic ac-Stark shift (cid:28) (cid:28) < Statistical a
700 700 a Including the contribution of the residual first-order Dopplershift ( ∼
460 kHz) and the lineshape model ( ∼
400 kHz) (see text). affecting the recorded transition frequencies are summa-rized in Table II. The dominant contribution to the sys-tematic uncertainty comes from the MQDT extrapola-tion. In future this value might be reduced by improvingthe quantum defects of Ref. [36], which will require ac-curate measurements at principal quantum numbers n in the range 20–40. The largest remaining contributionsto the uncertainty are the uncertainty in the lineshapemodel, which corresponds to ∼ → transitionfrequency after realignment of the laser beams throughthe chamber (see Fig. 4). These measurements were also Figure 4. Statistical analysis performed on the 1 → transition ( f mean /c = 34289 . − ). The figureshows the difference ∆ f = f − f mean between the transitionfrequency recorded on different days with the average value f mean of all measurements. The different markers (triangles,squares and diamonds) denote different data sets after subse-quent realignment procedures. The grey-shaded areas includethe data points within one and two standard deviations, re-spectively (1 σ = 700 kHz). used to estimate the overall statistical reproducibility to700 kHz. The recoil shift and the second-order Dopplershift amount to 50 kHz and -5 kHz, respectively, and werecompensated when deriving the final transition frequen-cies. The remaining contributions to the error budgetwere too small to be quantified in our experiment andhave been estimated.We improved the uncertainty of the adiabatic ioniza-tion energy of the a Σ + u state of He (1.3 MHz) and thefirst rotational interval of He (1.9 MHz) by a factor of30 and 10, respectively, compared to our previous stud-ies (see Table III). This improvement was achieved by re-placing the pulsed UV laser used for photoexcitation by asingle-mode cw UV laser and by calibrating the laser fre-quency with a frequency comb rather than a commercialwavemeter. Our measurements provide benchmark datathat can be used to test ab initio calculations of He andHe ∗ and approach the precision required to probe the po-larizability of atomic helium via the two-body interactionpotential in He . Our results are relevant in the contextof the establishment of primary pressure standards andcomplement the current most accurate results that relyon dielectric-constant gas thermometry [38].We are grateful to Ch. Jungen for letting us use hisMQDT program and to Josef Agner for excellent techni-cal support. We thank B. Jeziorski (Warsaw), E. M´atyus(Budapest), N. H¨olsch (Zurich) and M. Beyer (Yale) foruseful discussions. This work was supported by the SwissNational Science Foundation (Grant No. 200020-172620)and by the European Research Council (Horizon 2020,Advanced Grant 743121). Table III. Comparison of the adiabatic ionization energy ofHe ∗ (˜ ν ) and of the lowest rotational interval of He ∗ (˜ ν (cid:48)(cid:48) ) andof He (˜ ν +13 ) obtained in this work and in previous studies.All values are given in cm − .Reference ˜ ν ˜ ν (cid:48)(cid:48) ˜ ν +13 Ref. [39] 34 301 . . . . . . . . . . . . a This work 34 301 .
207 00(4) 75 .
812 953(20) 70 .
937 59(6) a Containing an estimated value of the relativistic and QEDcorrections. The nonadiabatic term value is 70.936 cm − [20]. ∗ [email protected][1] T. Steimle and W. Ubachs, J. Mol. Spectrosc. , 1(2014).[2] R. K. Altmann, L. S. Dreissen, E. J. Salumbides, W. Ubachs, and K. S. E. Eikema, Phys. Rev. Lett. ,043204 (2018).[3] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball,A. Derevianko, and C. W. Clark, Rev. Mod. Phys. ,025008 (2018).[4] S. G. Karshenboim and V. G. Ivanov, in Exploring theWorld with the Laser , edited by D. Meschede, T. Udem,and T. Esslinger (Springer International Publishing,2018).[5] J.-P. Karr, L. Hilico, J. C. J. Koelemeij, and V. I. Ko-robov, Phys. Rev. A , 050501(R) (2016).[6] J. Biesheuvel, J.-P. Karr, L. Hilico, K. S. E. Eikema,W. Ubachs, and J. C. J. Koelemeij, Nat. Commun. ,10385 (2016).[7] J. Biesheuvel, J.-P. Karr, L. Hilico, K. S. E. Eikema,W. Ubachs, and J. C. J. Koelemeij, Appl. Phys. B ,23 (2017).[8] S. Alighanbari, M. G. Hansen, V. I. Korobov, andS. Schiller, Nature Phys , 555 (2018).[9] K. Kato, T. D. G. Skinner, and E. A. Hessels, Phys.Rev. Lett. , 143002 (2018).[10] D. B. Newell and E. Tiesinga, The international systemof units (SI): 2019 edition , Tech. Rep. NIST SP 330-2019 (National Institute of Standards and Technology,Gaithersburg, MD, 2019).[11] M. Puchalski, K. Piszczatowski, J. Komasa, B. Jeziorski,and K. Szalewicz, Phys. Rev. A , 032515 (2016).[12] M. Puchalski, K. Szalewicz, M. Lesiuk, and B. Jeziorski,Phys. Rev. A , 022505 (2020).[13] W. J. Meath and J. O. Hirschfelder, J. Chem. Phys. ,3197 (1966).[14] R. J. Le Roy, Can. J. Phys. , 246 (1974).[15] N. Yu and W. H. Wing, Phys. Rev. Lett. , 2055 (1987).[16] A. Carrington, C. H. Pyne, and P. J. Knowles, J. Chem.Phys. , 5979 (1995).[17] P. Jansen, L. Semeria, L. Esteban Hofer, S. Scheidegger,J. A. Agner, H. Schmutz, and F. Merkt, Phys. Rev. Lett. , 133202 (2015).[18] L. Semeria, P. Jansen, and F. Merkt, J. Chem. Phys. , 204301 (2016).[19] W.-C. Tung, M. Pavanello, and L. Adamowicz, J. Chem.Phys. , 104309 (2012).[20] E. M´atyus, J. Chem. Phys. , 194112 (2018).[21] W. Lichten, M. V. McCusker, and T. L. Vierima, J.Chem. Phys. , 2200 (1974).[22] L. Semeria, P. Jansen, G. Clausen, J. A. Agner,H. Schmutz, and F. Merkt, Phys. Rev. A , 062518(2018).[23] P. Jansen, L. Semeria, and F. Merkt, Phys. Rev. Lett. , 043001 (2018).[24] Ch. Jungen, in Handbook of High-resolution Spec-troscopy , edited by M. Quack and F. Merkt (John Wiley& Sons, Ltd, Chichester, UK, 2011).[25] A. Beyer, L. Maisenbacher, A. Matveev, R. Pohl,K. Khabarova, Y. Chang, A. Grinin, T. Lamour, T. Shi,D. C. Yost, T. Udem, T. W. H¨ansch, and N. Ko-lachevsky, Opt. Express , 17470 (2016).[26] C.-F. Cheng, J. Hussels, M. Niu, H. L. Bethlem,K. S. E. Eikema, E. J. Salumbides, W. Ubachs, M. Beyer,N. H¨olsch, J. A. Agner, F. Merkt, L.-G. Tao, S.-M. Hu,and Ch. Jungen, Phys. Rev. Lett. , 013001 (2018).[27] N. H¨olsch, M. Beyer, E. J. Salumbides, K. S. E. Eikema,W. Ubachs, Ch. Jungen, and F. Merkt, Phys. Rev. Lett. , 103002 (2019). [28] J. Deiglmayr, H. Herburger, H. Saßmannshausen,P. Jansen, H. Schmutz, and F. Merkt, Phys. Rev. A , 013424 (2016).[29] M. Raunhardt, M. Sch¨afer, N. Vanhaecke, and F. Merkt,J. Chem. Phys. , 164310 (2008).[30] M. Motsch, P. Jansen, J. A. Agner, H. Schmutz, andF. Merkt, Phys. Rev. A , 043420 (2014).[31] M. Beyer, N. H¨olsch, J. A. Agner, J. Deiglmayr,H. Schmutz, and F. Merkt, Phys. Rev. A , 012501(2018).[32] G. Ritt, G. Cennini, C. Geckeler, and M. Weitz, Appl.Phys. B , 363 (2004).[33] H. Naus, I. H. M. van Stokkum, W. Hogervorst, andW. Ubachs, Appl. Opt. , 4416 (2001). [34] Ch. Jungen and O. Atabek, J. Chem. Phys. , 5584(1977).[35] Ch. Jungen and G. Raseev, Phys. Rev. A , 2407 (1998).[36] D. Sprecher, J. Liu, T. Kr¨ahenmann, M. Sch¨afer, andF. Merkt, J. Chem. Phys. , 064304 (2014).[37] M. Beyer, N. H¨olsch, J. Hussels, C.-F. Cheng, E. J.Salumbides, K. S. E. Eikema, W. Ubachs, Ch. Jungen,and F. Merkt, Phys. Rev. Lett. , 163002 (2019).[38] C. Gaiser and B. Fellmuth, Phys. Rev. Lett. , 123203(2018).[39] D. S. Ginter and M. L. Ginter, J. Mol. Spectrosc. , 152(1980).[40] C. Focsa, P. F. Bernath, and R. Colin, J. Mol. Spectrosc.191