Downloaded by guest on September 26, 2021 hita Janke Christian Waitz Markus Zeller Stefan laser electron He the Imaging www.pnas.org/cgi/doi/10.1073/pnas.1610688113 Grisenti E. Robert a D Reinhard and ald“unu aosae”() hs ttsaekon for known, are are wavefunction) where (2–4). states states physics their These nuclear (1). of from states” example, part halo being tunneling “quantum therefore called (and the threshold by that quantum- dominated near more existing and “classical” Systems less like. becomes infinity. the system by to the accompanied bind- expands that is fact system vanishing expansion quantum this of potentials, the limit short-range of For the size in wave- the (i.e., its energy), of ing threshold more the and at 1 more (Fig. sequently, region as tunneling the well, into as expands function state ground the effects (transition unbound become they as from other the are after states one excited depth, expelled potential the dra- small. decreasing change usually upon can matically: situation is this region potentials, short-range forbidden shallow For classically density probability this the in of electron optical fraction in distribution to the volts matter, nanoelectron physics bound to For and nuclear lattices. solids and in molecules volts in volts “tunnel- megaelectron energy all as on from occurs known and scales nature phenomenon in penetrate a omnipresent is can by Tunneling however, ing.” barrier particles, due potential Quantum particle the energy. classical into of a lack potential for a inaccessible binding regions to are The a points points. such turning these two beyond in in between forth shown situated and as back particles oscillate states, Classical bound 1. excited Fig. energetically cases) most (in A clusters calculations. recent most with agreement in is fur- and laser electron He free yielded a thermore is using imaging away explosion obtained far Coulomb proba- were matter results by the bound The exponentially. of universal: decreases constituents but truly zero the is never of that one of find world to one our bility shows two of particle features over tunneling few for a achieved the Imaging we magnitude. which of particle, orders tunneling decaying a exponentially of the cir- image density This directly region. to forbidden its us classical that allows this in cumstance probability found 80% be an will with nuclei two atoms, molecule helium gigantic two a of system: consisting quantum region, extreme tunneling an the here frac- into present small extends we chem- wavefunction a particle only to a typically of physics Although tion systems. nuclear state from solid scales and is istry energy behavior all This exponentially. across decay universal to known the is region,” density “tunneling this particle In accessi- mechanics. not classical energetically to are according which ble space in and regions reach nature 2016) par- to quantum 30, in allows ticles June It review phenomenon applications. for technological (received ubiquitous many 2016 for a crucial 3, November is approved and tunneling TX, Station, Quantum College University, A&M Texas Scully, O. Marlan by Edited e Austria; ntttf Institut eateto oeua hsc,FizHbrIsiu,115Bri,Gray and Germany; Berlin, 14195 Fritz-Haber-Institut, Physics, Molecular of Department ytm uha oeue rnce ihagon tt and state ground a with nuclei or molecules as bound such stable into systems condense to particles allow forces ttractive A | c to etce lkrnnSnhorn 20 abr,Germany; Hamburg, 22607 Elektronen-Synchrotron, Deutsches rKrpyi,Goethe-Universit Kernphysik, ur eimdimer helium ¨ B nFg ) ute eraeo h oeta depth potential the of decrease further A 1). Fig. in a,1 a edi Sann Hendrik , asmKunitski Maksim , a orner ¨ atnRichter Martin , a,d 2 | sbnigeeg f151.9 of energy binding ’s wavefunction iln Sch Wieland , a,1 a lxne Hartung Alexander , | a tunneling tFakut 03 rnfr,Germany; Frankfurt, 60438 Frankfurt, at ¨ rgrKastirke Gregor , a ollkopf 2 J , ¨ 11 r Voigtsberger org ¨ unu aosaeuigafree a using state halo quantum eand Be e ohrP.H Schmidt H. Ph. Lothar , ± e,which neV, 13.3 11 C ifr halo form Li and a iimWeller Miriam , .Con- D). a oisBauer Tobias , a no Kalinin Anton , d S emot etefrHayInRsac,621Drsat Germany; Darmstadt, 64291 Research, Ion Heavy for Centre Helmholtz GSI hsatcei NSDrc Submission. 1 Direct PNAS a is article This interest. of conflict wrote no R.D. declare and authors T.J., J.B.W., The Kunitski, reagents/analytic M. new S.Z., contributed and R.D. data; paper. analyzed and the S.Z., R.D. T.J., and scientist; and W.S., T.J., T.J., beamline A.C., J.B.W., S.Z., a T.B., tools; M.S.S., as A.K., L.Ph.H.S., support J.V., M.P., extensive M.B., Kunitski, T.B., Kitzler, provided M. A.H., M. M.B. H.S., Weller, research; Waitz, M. performed M. G.K., R.D. C.S., M.R., A.S., A.K., C.J., J.V., C.G., Kunitski, F.T., M. S.Z., research; designed ae ntesm eH neato oeta sdt calcu- to used potential are interaction properties He He–He the Those late same helium. the of on viscosity based and ther- the conductivity con- for values mal Boltzmann theoretical uses the today of Thermometry (9). terms stant is unit in Kelvin conflict temperature the of this thermodynamic redefinition of Resolving planned the (5). for importance calculations conflict of in also recent but most predictions the theoretical some with 52 with of agreement distance in internuclear is the wave of He matter value of mean using character halo Experiments the confirmed dissociation. diffraction to 1} leads with already associated force centrifugal the stable energy, zero-point bigger its of the Although 4 because all. whether unbound already at thus indeed molecule and is is stable a well dimer as potential exists the helium dimer of helium the depth the of 3 than energy about larger of zero-point distance the internuclear whether an at meV/2.96 meV mini- [0.947 1 a helium has about 1D) the of (Fig. mum potential physics: interaction atomic He–He the of and realm force, the in (He found dimer be can state un u htHe that out turns uhrcnrbtos .. .Kntk,JV,REG,LP...... n R.D. and T.J., M.S.S., L.Ph.H.S., R.E.G., J.V., Kunitski, M. S.Z., contributions: Author doerner@atom.uni-frankfurt.de. owo orsodnemyb drse.Eal elrao.n-rnfr.eor [email protected] Email: addressed. be may correspondence whom To He ao tsostepeitdepnnilsaeo tunneling a of shape exponential wavefunction. predicted we the He Here shows the particles. It of two halo. wavefunction only the of image made experimentally is conceptually system The systems physics. halo fascinating molecular simplest and these nuclear of in examples known region are tunneling Few the outside potential. in mostly the potential is of interaction density the particle of the range where the matter are of contrary the type on a halos which Quantum potential together. system interaction the the bound holds of the range of the density within molecules is the particles to of most nuclei objects, from solid scales, macroscopic length to all on matter bound In Significance b n ftems xrm xmlso uhaqatmhalo quantum a such of examples extreme most the of One a htnc nttt,Ven nvriyo ehooy 00Vienna, 1040 Technology, of University Vienna Institute, Photonics f aksS Sch S. Markus , eateto hsc,Uiest fNvd,Rn,N 89557 NV Reno, Nevada, of University Physics, of Department 2 a a atnPitzer Martin , ci Czasch Achim , a nlyfudeprmnal n19 n 94(,7.It 7). 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PHYSICS A BC inner turning point of the helium-dimer potential and the- oretical calculations agree well on the location of the turn- ing point. A comparison of our measured probability density distribution close to the inner turning point and some the- oretical predictions are shown in Fig. 2B. Here, two exem- plary theoretical curves (5, 15) are depicted along with a measurement conducted at our Ti:Sa laser because it pro- vides very high resolution and statistics for small internuclear distances. D The classically forbidden part of |Ψ|2 is shown in Fig. 2C on a logarithmic scale. For internuclear distances larger than 30 A˚ , the helium-dimer potential is two orders of magnitude smaller than the predicted ground-state binding energy and thus can safely be approximated to zero. Accordingly, the wavefunction is approximated in this region by the solution of the Schrodinger¨ equation below a step-like barrier, which is given by

q 2m − 2 Ebind R Ψ(R) ∝ e } . [2] Because the mass m and Planck’s constant } are fixed, the only variable defining the slope of the exponential decay is the bind- ing energy Ebind . Therefore, the binding energy can be extracted from the measurement by an exponential fit to the pair-distance distribution, as depicted in Fig. 2C. From the fit, we obtain a Fig. 1. (A) Shallow short-range potential holding a ground and an excited helium-dimer binding energy of 151.9 ± 13.3 neV (see Materials state. As the potential depth decreases (B), the excited state becomes and Methods for a discussion of errors and corrections to Eq. 2). unbound, leaving only the ground state. Further decrease (C) leads to the This value can be used for a fully experimental determination of particle probability density distribution leaking more into the classically for- the binding energy of the helium-trimer Efimov state, which was bidden region. In the extreme case of the helium dimer (note the loga- discovered recently using the same experimental setup (16). rithmic R scale) (D), this effect allows the wavefunction to extend to sizes of fullerenes, the diameter of DNA, and even small viruses (He2 potential and wavefunction taken from ref. 5), whereas the classical turning point is Discussion located at 13.6 A,˚ and the overall wavefunction extends to more than 200 A.˚ The theoretical value for the binding energy was under dis- pute for many years (17–20). Predictions range from 44.8 neV (18) to 161.7 neV (21). Recently calculations became avail- able that include quantum electrodynamical effect and relativis- with previous experiments (8, 10) (see ref. 11 for a more detailed tic effects and go beyond the Born Oppenheimer approxima- discussion). tion. These supposedly most precise calculations predict a bind- At the same time, its quantum halo character makes He2 a ing energy of 139.2 ± 2.9 neV (5), which is in disagreement prime candidate for visualizing the predicted universal exponen- with the most recent experimental value of 94.8 + 25.9/-17.2 tial decrease of a tunneling wavefunction in an experiment by neV obtained in pioneering experiments by evaluating matter triggering a Coulomb explosion with a free electron laser (FEL). wave diffraction patterns and relying on a detailed theoretical Results modeling of the interaction of the dimer with the grating sur- face (8). The present value of 151.9 ± 13.3 neV is in good In the corresponding experiment presented here, helium clusters agreement with the prediction of Przybytek et al. (5) (139.2 ± were produced by expanding cooled helium gas through a 5µm 2.9 neV) and in clear disagreement with the predictions from nozzle. By matter wave diffraction, a pure helium-dimer beam some He–He interaction potentials, including the popular Tang– was separated from the lighter monomers and heavier clusters Toennies–Yiu potential (TTY) (19) and the mimic of the Liu– (6). In two experimental campaigns, both atoms of the dimer McLean potential (LM2M2) (22) yielding 114 and 113 neV, were then singly ionized using either single-photon ionization respectively. using photons provided by a FEL [<100 fs, 18.5 nm; Freie– The helium dimer is a remarkable example of a system exist- Elektronen Laser in Hamburg (FLASH)] or tunnel ionization ing predominantly in the quantum mechanical tunneling regime. using a strong ultrashort laser field (Ti:Sa laser; 780 nm; Dragon We were able to reveal the full shape of the wavefunction exper- KMLabs). In both cases, the ionization of the two atoms occurs imentally. The measured data confirm the universal exponential fast compared with the nuclear motion, thus triggering an instan- behavior of wavefunctions under a potential barrier on unprece- taneous Coulomb explosion of the repelling ionized particles. dented scales and yield a revised experimental value for the bind- The Coulomb explosion converts the potential energy of the two ing energy of the helium dimer, which has been under dispute for ions located at an internuclear distance R into a released kinetic more than 20 y. energy (KER) according to 1 Materials and Methods R = . [1] KER Dimer Preparation and Detection. A mixture of helium clusters was pro- duced by expanding gaseous helium through a 5 µm nozzle. The nozzle By recording a large number of Coulomb explosion events, via was cooled down to 8 K, and a driving pressure of 450 hPa absolute pres- cold target recoil ion momentum spectroscopy (COLTRIMS) sure was applied to maximize the dimer content in the molecular beam (16). To obtain a pure helium-dimer target beam, we made use of matter (12–14), a distribution of measured distances R (as shown wave diffraction. All clusters have the same velocity but can be sorted by in Fig. 2A) is obtained. This distribution represents a direct mass because their diffraction angle behind a transmission grating (100-nm measurement of the square of the helium-dimer wavefunction period) depends on their de Broglie wavelengths (λ = h/mv, with Planck’s 2 2 |Ψ| . The classically allowed part of |Ψ| provides a cross- constant h, mass m, and velocity v). That way, only dimers reach the laser check for our measurement because it falls off steeply at the focus, whereas the dominant fraction of atomic helium, as well as the

14652 | www.pnas.org/cgi/doi/10.1073/pnas.1610688113 Zeller et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 t otehlu-ie rau,i h N the is breakup, helium-dimer the to ity hsproe epromdtoclbainmaueet ihisotropic For with it. measurements to (N calibration perpendicular channels two ones dissociation performed the we with purpose, direction this time-of-flight the the in calibration of nent the direction rel- time-of-flight yielded mean the measurement along a component This with spectrometer. momentum (23), 0.054%. the measurements of of reference deviation met kinetic ative of D and the calibration identified from measuring N of be Transitions position by the value obtained peaks. the of was absolute row and spectrum field the spectrometer release electric are energy the The calibration in detector. this the field needed. for electric is calibration parameters the energy precise crucial a Therefore, The KER. measured the from be can Calibration. KER, Detector the thus and ions, of the detector to of to ions focus vector from elec- reconstructed. momentum the distance initial known a the guides With and mm, charges, (12). laser) 39 ion using anodes masses, Ti:Sa impact line ion (at fields, of delay tric and position V/cm (MCPs) and 3.09 plates time-of-flight and microchannel measures FEL) which field in detector, (at electric acquired the homogeneous V/cm momenta A 4.41 ionic COLTRIMS. by The of measured explosion. each were repel Coulomb explosion ions this a (Ti:Sa charged ionization positively in two tunnel resulting The via other, KMLabs). or Dragon FLASH) nm; nm; 780 18.5 laser; (FEL; photoeffect via either of schematic conditions a expansion shows 3 gas Fig. chosen region. setup. ionization the the the at from away present deflected trimers, get helium of share in shown LM2M2 slope predictions the from theoretical conclude with to agreement account in is (C (B) region point forbidden turning inner the of 2. Fig. elre al. et Zeller B A h oiinclbainwsdn ycmaigtemmnu compo- momentum the comparing by done was calibration position The focus the in ionized singly get dimer the constituting atoms two The esrmn ftehlu-ie aeucin (B wavefunction. helium-dimer the of Measurement (A) .Ahlu-ie idn nryo 151.9 of energy binding helium-dimer A ). h idn nryo h eimdmri derived is dimer helium the of energy binding The 2 O/Ne 2 .Ms eeat u oeegtcproxim- energetic to due relevant, Most ). 3 n D and Πg 2 rau,wihpoie eynar- very provides which breakup, 2 hne t01 VKR iha with KER, eV 0.16 at channel O 1 C Σ otevleo h idn nry(see energy binding the of value the to u + nocniumcould continuum into ± and e sotie rmteepnnilsoe h lcrnrci a ob ae into taken be to has recoil electron The slope. exponential the from obtained is neV 13.3 C w ealdvesso h motn etrso hsqatmsse:teregion the system: quantum this of features important the show views detailed Two ) enrltv eito f62,weesadtoa hnesyedasmaller (N a 0.62% yield with channels deviation, additional whereas 6.2%, of deviation relative mean lt 3 m ltt rtn,3 m n rtn ofcs 9 m h focus The mm. 491 focus, to grating to 20 and about skimmer mm; was mm; 30 diameter 14 grating, skimmer, to between to slit Distances mm; nozzle 332 nanograting. follows: slit, as a were at elements diffracted beam beam the molecular a by ated 3. Fig. C ∗ 1)adPzbtke l 5,adteepnnildcyi h classic the in decay exponential the and (5), al. et Przybytek and (15) aeil n Methods and Materials vra ewe ae ou n uehlu-ie em cre- beam, helium-dimer pure a and focus laser between Overlap PNAS µm. | 2 t03 V n .5 (Ne 0.15% and eV) 0.36 at O eebr2,2016 20, December o details). for | o.113 vol. 2 t44eV). 4.4 at | o 51 no. | 14653

PHYSICS 0 the internuclear distance. The steep rise of the probability density at the inner turning point is not very sensitive to this effect and could consequently -20 be imaged by our experiment with an 800-nm laser pulse, which has supe- -40 rior statistics compared with the FEL experiment (Fig. 2). For the exponential HFIMD region of the probability density, we aim for a high-precision determination -60 of the slope. We therefore used photons from the FEL FLASH to ionize both TT HFDHE2 atoms of the dimer by single-photon absorption. Compared with an 800-nm -80 Expt. Luo et al. laser pulse, this ionization process has the additional advantage that the Expt. Grisenti et al. electron energy, and thus the recoil of the electrons onto the nuclei, is much -100 LM2M2* better controlled and has an upper threshold. LM2M2 TTY -120 HFD-B3-FCl1b Jeziorska ret. Electron-Recoil Correction. The initial ion energy during the Coulomb HFD-B3-FCl1a Przybytek -140 HFD-B2(HE) breakup has to be either zero or well-defined, because Eq. 1 assumes that HFD-B(HE) Gdanitz SAPT1 this the KER only results from the potential energy between the two point SAPT2 Jeziorska work -160 SAPTunret. charges and that there is no additional energy from other sources. The two most important sources of such additional energy are the zero-point kinetic PredictedbindingneV He-He energy / -180 1975 1985 1995 2005 2015 energy from the bound state before ionization and the energy transferred Year of publication during the ionization process by recoil of the escaping electron. The first source is negligible for He2, because the depth of the poten- Fig. 4. The predicted values for the helium-dimer binding energy using var- tial well is only 1 meV. We have also confirmed this finding by calculat- ing the Coulomb explosion quantum mechanically. We found no difference ious theoretical calculations [HFDHE2 (Hartree–Fock dispersion for He2) (17), HFIMD (Hartree–Fock with intraatomic correlation correction and model in the KER between the classical calculation using Eq. 1 and the quan- dispersion) (18), TT (Tang–Toennies) (26), HFD-B(HE) (Hartree–Fock plus tum calculation, which automatically includes the initial state zero-point damped dispersion B for helium) (27), HFD-B2 (Hartree–Fock plus disper- motion (25). sion B2) (28), LM2M2 (22), LM2M2* (15), TTY (19), HFD-B3-FCI1a (Hartree- The energy transferred to the two nuclei during the ionization pro- Fock-dispersion B3 full configuration interaction) (29), HFD-B3-FCI1b, SAPT cess by the FEL is given by the recoil of the two electrons. The (symmetry-adapted perturbation theory) (21, 30), Gdanitz (20), Jeziorska et sum momentum distribution of two electrons with a kinetic energy of al. (31), and Przybytek et al. (5)] are displayed alongside experimental mea- Eγ - IP = 42.4 eV each was calculated and is reflected in the measured surements from Luo et al. (10), Grisenti et al. (8), and the present work. data. For two independent ionization events, the distributions of the sum momenta and the momentum difference of the electrons are equal. Although the sum momentum cancels out in the KER calculation, the rel- For the experiment at FLASH, despite excellent vacuum conditions ative momentum adds to it and increases the measured KER. Correct- −12 (8·10 hPa), an average of about 50 ions were collected for every FEL ing this effect reduces the slope of the exponential decaying function pulse. The majority of ions were charged hydrogen atoms or molecules by 12.1 neV. Taking this reduction into account, we obtain a binding- with short times-of-flight, which could be gated out by software dur- energy value of 151.9 neV ±1.7(stat) ±10.2(calib) ±1.4(corr) neV from ing data acquisition before writing to the hard drive. Nevertheless, the our experiment. The statistical error is the error of the fit caused by the MCPs endured constant stress, which led to a drop in detection effi- statistics of the data points, the calibration error is the uncertainty of ciency in the center of the detector. The detection efficiency was corrected the calibration of our COLTRIMS reaction microscope, as discussed above, to its normal level using a residual gas calibration measurement with a and the error labeled (corr) is the estimated error on the correction pro- Gaussian-shaped correction function containing a 5.5% uncertainty. This cedure compensating the detector efficiency and subtraction of random uncertainty leads to an error of ±1 neV on the binding energy. In addi- coincidences. tion, random coincidences from ionizations of two independent helium ions from the residual gas were subtracted. The error resulting from this Comparison with Theory. The exact value of the helium-dimer binding background subtraction is small in comparison with errors discussed above energy was the subject of dispute for decades. Fig. 4 displays the evolution (±0.4 neV). of theoretical predictions as small effects such as deviation from the Born– Oppenheimer Approximation, retardation and quantum, electrodynami- Binding-Energy Derivation. The solution to the Schrodinger¨ equation in the cal effects were included in calculations and more computational power region below a potential barrier is an exponential decay function (Eq. 2). became accessible. Our measurement is in good agreement with the most The helium-dimer binding energy has been extracted from the experi- recent calculations from Przybytek et al. (5) but cannot distinguish between mental data by applying an exponential fit to the reconstructed pair- that and several older calculations (20, 21). Our obtained results on the He2 distance distribution (50 a.u. to 300 a.u.). We excluded breakups recorded wavefunction and binding energy provide an experimental benchmark for ± . ◦ in the detector plane (with a tolerance of 33 5 ) because indistin- theoretical calculations. guishable background and potentially deadtime effects compromised the data here. ACKNOWLEDGMENTS. We are grateful for excellent support by the staff To image the exact shape of the probability density distribution by of FLASH during our beam time. We thank R. Gentry and M. Przybytek Coulomb explosion, imaging the ionization probability has to be indepen- for providing their theoretical results in numerical form. The experimen- dent of the internuclear distance. Two consecutive tunnel ionization steps tal work was supported by a Reinhart Koselleck project of the Deutsche can be influenced by enhanced ionization (24), an effect that depends on Forschungsgemeinschaft.

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14654 | www.pnas.org/cgi/doi/10.1073/pnas.1610688113 Zeller et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 4 uJ ta.(03 togfil utpeinzto saruet lcrndnmc in dynamics electron to route a as ionization multiple field Strong (2013) al. et J, Wu energy kinetic Doppler-free 24. (1996) B Wannberg P, Baltzer D, Edvardsson M, Lundqvist potential helium the 23. for results initio on ab based of examination helium An (1991) for MJ curve Slaman RA, energy Aziz potential 22. accurate An (1997) RA Aziz AR, Janzen 21. Schr electronic the solving Accurately (2001) RJ potential Gdanitz Waals der van 20. He-He analytical Accurate (1995) CL Yiu JP, Toennies KT, Tang 19. K H, Kirst intermolec- R, accurate Feltgen An (1979) 18. GT McConville WL, Taylor JS, Carley VPS, Nain RA, Aziz 17. elre al. et Zeller a e al cluster. Waals der van a N of spectrum release curve. energy calculations. initio ab (r correlated revisited. explicitly using molecules and theory. perturbation on based Phys Chem J He helium. for potential ular 2 rudsaeptnilfo xeietb s farlal oeta model. potential reliable a of use by experiment from potential state ground o Phys Mol 76(5):2360–2378. hmPhys Chem J 99(11):923–930. he A al ,TrloF(92 nqedtriaino the of determination Unique (1982) F Torello H, Pauly KA, ohler ¨ 2 hmPhys Chem J 2+ hsRvLett Rev Phys . hmPhys Chem J 94(12):8047–8053. hsBAo o p Phys Opt Mol Atom B Phys J hsRvLett Rev Phys 107(3):914–919. 111(8):083003. 70(9):4330–4342. 12 74(9):1546–1549. − M-I I h eimdmr(He dimer helium The VI. )MR-CI. 29:1489–1499. dne qaino atoms of equation odinger ¨ 2 ) 1 eirk ,Cne ,PtosiK eirk ,SaeizK(07 arpotential Pair (2007) K Szalewicz B, Jeziorski K, Patkowski dimer W, Helium Cencek (1997) M, K Jeziorska Szalewicz 31. B, Jeziorski R, Bukowski HL, Williams T, standard A Korona helium: for calculations 30. initio Ab (1995) non-ideality MR Moldover the AR, for Janzen RA, relations Aziz ITS-90 29. the of analysis An (1990) state MJ ground Slaman the of RA, determination Aziz new A 28. (1987) CCK Wong FRW, McCourt RA, Aziz poten- Waals 27. der van the for model simple improved An (1984) JP Toennies KT, H Tang the of 26. imaging Spatial (2012) al. et LPhH, Schmidt 25. h unu limit. quantum the o eimfo ymtyaatdprubto hoycluain n rmsuper- from and data. calculations molecular theory perturbation symmetry-adapted from helium for sets. basis orbital Gaus- and large geminal using sian calculations theory perturbation symmetry-adapted from potential measurements. property transport for helium. of He for potential interatomic coefficients. dispersion the for 80(8):3726–3741. functions damping universal on based tial 3 eand He Metrologia 4 e eomne eain ae nanwitrtmcptnilfor potential interatomic new a on based relations Recommended He: hmPhys Chem J PNAS hsRvLett Rev Phys 27:211–219. | eebr2,2016 20, December 2 127(12):124303. . o Phys Mol 108(7):073202. hmPhys Chem J hsRvLett Rev Phys 61(6):1487–1511. 106(12):5109–5122. | 2 74(9):1586–1589. o.113 vol. + irtoa aefnto at function wave vibrational | o 51 no. hmPhys Chem J | 14655

PHYSICS